Silicon Wafer Integration of Ion Electrospray Thrusters Noah Wittel

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Silicon Wafer Integration of Ion Electrospray Thrusters by Noah Wittel Siegel B.S., United States Military Academy (2018) Submitted to the Department of Aeronautics and Astronautics in partial fulﬁllment of the requirements for the degree of Master of Science in Aeronautics and Astronautics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY May 2020 © Massachusetts Institute of Technology 2020. All rights reserved.

Author...... Department of Aeronautics and Astronautics May 19, 2020

Certiﬁed by...... Paulo C. Lozano M. Alemán-Velasco Professor of Aeronautics and Astronautics Thesis Supervisor

Accepted by ...... Sertac Karaman Associate Professor of Aeronautics and Astronautics Chair, Graduate Program Committee 2 Silicon Wafer Integration of Ion Electrospray Thrusters by Noah Wittel Siegel

Submitted to the Department of Aeronautics and Astronautics on May 19, 2020, in partial fulﬁllment of the requirements for the degree of Master of Science in Aeronautics and Astronautics

Abstract Combining efficiency, simplicity, compactness, and high specific impulse, electrospray thrusters provide a unique solution to the problem of active control in the burgeoning field of miniature satellites. With the potential of distributed systems and low cost functionality currently being realized through development of increasingly smaller spacecraft, thruster research must adjust accordingly. The logical limit of this rapidly accelerating trend is a fully integrated silicon wafer satellite. Such a large surface area to volume ratio, however, both necessitates propulsion capability and renders other mechanisms of control unfeasible due to their respective form factors. While development of electrospray thrusters has exploded in the past two decades, current architectures are similarly incompatible with a silicon wafer substrate. This thesis examines the design and testing of a novel hybrid electrospray architecture which combines previous successes of both capillary and externally-wetted geometries. Our project achieved the first passively-fed, pure ionic emission with silicon emitters. More importantly the micro-manufacturing approach offers key advantages in flexibility and overall performance. Through adaption of a innovative approach to black silicon surface treatment, it is possible to tailor hydraulic impedance in order to maximize propellant flow rate and efficiency for a wide range of mission requirements. The manufactured design exhibits operation in the pure ionic mode with 1-ethyl- 3-methylimidazolium tetrafluoroborate and has an emitter density more than an order of magnitude larger than any previous electrospray architecture. Preliminary testing indicates that this will likely translate to a corresponding improvement in thrust density. Further, electrochemical degradation of emitter tips – a primary failure mechanism of electrospray thrusters – appears to occur at a relatively inconsequential rate.

Thesis Supervisor: Paulo C. Lozano Title: M. Alemán-Velasco Professor of Aeronautics and Astronautics

3 4 Acknowledgments

This work was supported by the MIT Lincoln Laboratory Technology Oﬃce, funded by the Oﬃce of the Under Secretary of Defense for Research and Engineering.

The ﬁrst person to whom I owe thanks is Dr. Paulo Lozano. An incredible advisor, mentor, and professor, his personal investment in both the research and individuals of his lab is continually apparent. Similarly, I appreciate everyone with whom I had the pleasure of working with in the Space Propulsion Lab over these past two years.

I am especially grateful for all of the expertise, guidance, and time given by those working on the WaferSat team at Lincoln Laboratory. I owe so much to Dr. Melissa Smith for all of the effort, encouragement, resources, and MEMS knowledge she dedicated to both my learning and the project at large. Without her, this thesis would have a completely different topic. I appreciate all the discussions and meetings with Dr. Dan Freeman; I always left with a much better understanding after fleshing out the details of any given topic. He has a knack for asking pointed questions which challenge the assumptions I tend to ignore. Jimmy McRae was invaluable to this research by manufacturing all of the final test articles. Without his patience to perfect the fabrication process, my data would be either unrecognizable or more likely nonexistent. Finally, I am truly thankful to both John Kuconis and my group leaders for offering this fellowship; it has been a transformative experience.

This acknowledgments section would be woefully incomplete without recognition of the many individuals at West Point who prepared me for graduate school. In particular, COL Bret Van Poppel and COL Michael Benson dedicated signiﬁcant time and energy advising my undergraduate research and guiding me towards the path I am on today.

On a different note, I am incredibly appreciative of the boys for helping enforce a generous work-life balance these past two years. Between bricking 3s in Rockwell (RIP to the Small Ballers’ playoff hopes), Sunapee ski trips, an occasional kegger, and 10¢ wings at the Red Hat, I’m confident in saying that my time in Boston has been vastly more enjoyable than I could have anticipated. A special thanks to Aaron

5 Schlenker for putting up with everything from my questionable music taste to random

LATEX questions. Jack, Logan, John, Nick, Gabe, Sam, Connor, Noah, and Wade: I hope we ﬁnd ourselves near each other again at some point – either in the Army or afterwards. Finally, I would like to take this opportunity to thank my family for their unwaver- ing support throughout this entire process and everything leading up to it. I certainly do not take it for granted. Additionally, the time they dedicated to proofreading this document is greatly appreciated.

DISTRIBUTION STATEMENT A. Approved for public release. Distribution is unlimited.

This material is based upon work supported by the Under Secretary of Defense for Research and Engineering under Air Force Contract No. FA8702-15-D-0001. Any opinions, ﬁndings, conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reﬂect the views of the Under Secretary of Defense for Research and Engineering.

6 Contents

1 Introduction 17 1.1 Propulsion ...... 18 1.1.1 Fundamentals of Space Propulsion ...... 18 1.1.2 Electric Propulsion ...... 23 1.1.3 Electrospray ...... 26 1.2 Satellite Miniaturization ...... 32 1.3 Thesis Motivation and Challenges ...... 36

2 Design and Fabrication 39 2.1 Electrospray Design Considerations ...... 39 2.1.1 Electrostatic Thruster Performance Metrics ...... 39 2.1.2 Electrospray Fundamentals ...... 41 2.1.3 Microﬂuidics ...... 52 2.1.4 Extractor Design ...... 56 2.2 Surface Treatments ...... 58 2.2.1 Theory ...... 58 2.2.2 Techniques ...... 61 2.3 Prototypes ...... 68 2.3.1 Design Iterations ...... 70 2.3.2 Final Geometry ...... 79 2.3.3 Fabrication Process ...... 85

7 3 Results and Discussion 87 3.1 Wetting ...... 87 3.2 Electrical Testing ...... 93 3.2.1 Second Generation Emitters ...... 93 3.2.2 Third Generation Emitters ...... 97 3.2.3 Failure Mechanisms ...... 103 3.3 Post-test Analysis ...... 106

4 Conclusion 111 4.1 Manufacturing Process ...... 111 4.2 Thruster Performance ...... 113 4.3 Future Research ...... 114

A Wafer Satellite Dynamics and Control 115 A.1 Structure and Assumptions ...... 117 A.2 Perturbations ...... 119 A.3 Implementation ...... 123 A.4 Initial Conditions ...... 127 A.5 Control Strategy ...... 129 A.6 State Estimation ...... 136 A.7 Performance ...... 139 A.8 Conclusions and Future Work ...... 146

8 List of Figures

1-1 Effect of specific impulse on required propellant...... 20 1-2 Matching specific impulse to ∆v requirements...... 22 1-3 Comparison of electrospray architectures...... 27 1-4 Polydispersive effects on thruster performance...... 30 1-5 Total Nanosatellites and CubeSats Launched as of January 2020 [61]. 33

2-1 Minimum polydispersive eﬃciency for two mono-disperse species. .. 48 2-2 Electrochemical wear of a silicon array...... 50 2-3 Eﬀect of impedance on emission characteristics...... 52 2-4 Conical emitter impedance...... 54 2-5 Extractor Geometry...... 56 2-6 Ideal contact angle measurement...... 58 2-7 Rough surface wetting models...... 61 2-8 Generic black silicon morphology...... 63 2-9 Anisotropic nature of plasma-based black silicon...... 64 2-10 Spatial non-uniformities in initial MACE process...... 65 2-11 Controlling impedance with MACE time splits...... 65 2-12 Thermal oxidation of silicon nanograss [1]...... 67 2-13 Dimensions needed for PIR emission...... 72 2-14 Initial capillary design...... 73 2-15 Cavitation related shorting in capillary style emitters...... 74 2-16 Nominal geometry of conical emitter tips...... 75 2-17 Estimated impedance of second generation emitters...... 77

9 2-18 Actual proﬁle of second generation emitters...... 78 2-19 Externally wetted architecture with plate electrode [1]...... 79 2-20 Final array architecture [1]...... 80 2-21 Sizing of capillaries for emission stability...... 81 2-22 Final architecture – dense D emitter array...... 82 2-23 Final architecture – dense G emitter array...... 83 2-24 Final architecture – tip radius of curvature...... 83 2-25 Simulated deﬂection of tight pitch G array...... 85

3-1 Spatially inconsistent wetting of second generation arrays...... 89 3-2 Transient wetting of second generation arrays...... 90 3-3 Passive wetting of ﬁnal array architecture [1]...... 91 3-4 Wenzel type wetting of oxidized black silicon emitters...... 91 3-5 Current-voltage curve of a 2nd generation type D tip...... 94 3-6 Transient current response of a 2nd generation dense type G array. .. 95 3-7 Overwet phase of a 2nd generation dense type G array...... 95 3-8 Consistent performance of 2nd generation type G emitters...... 96 3-9 Steady state current of a 2nd generation dense type G array...... 97 3-10 Electrical testing of a 3rd generation dense type G array...... 99 3-11 Electrical testing of a 3rd generation single type G emitter using a platinum distal electrode...... 101 3-12 Electrical testing of a 3rd generation single type G emitter using a carbon xerogel distal electrode...... 102

3-13 Carbonization of EMI-BF4 after electrical discharge...... 105 3-14 Extractor plate polishing...... 106 3-15 Electrochemical wear of second generation emitters...... 107 3-16 Highly localized electrochemical wear of a type D emitter...... 108 3-17 Third generation type G array after ∼3 hours of electrical testing. .. 109

A-1 Cross-section of normalized wafersat density...... 118 A-2 Relation between LVLH and body reference frames...... 124

10 A-3 Relation between ECI and LVLH reference frames...... 125 A-4 Satellite orientation and angular rate in the local frame...... 140 A-5 Total pointing error...... 140 A-6 Thruster activity...... 142 A-7 Satellite orientation and angular rate under loose pointing requirements.143 A-8 Thruster activity under loose pointing requirements...... 143 A-9 Total pointing error with disturbed model parameters...... 144 A-10 Thruster activity with disturbed model parameters...... 145 A-11 Collected thruster activity with disturbed model parameters...... 146

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12 List of Tables

1.1 Satellite size classiﬁcation...... 32

2.1 Classiﬁcation of surface wetting...... 59

2.2 Physical properties of EMI-BF4...... 68 2.3 Nominal dimensions of second generation emitters...... 76 2.4 Emitter arrays for ﬁnal geometry...... 82 2.5 Extractor dimensions (in microns)...... 84

A.2 Thruster module placement...... 130 A.3 Initial satellite attitude...... 139

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14 Nomenclature

Variables p linear momentum N s

m3 B magnetic ﬁeld T Q volumetric ﬂow rate s b beam width m q electric charge C

2 C capacitance F Qc collision cross section m m c exhaust velocity s r radius m V E electric ﬁeld m Rc radius of curvature m F thrust N T temperature K f frequency Hz t time s

G extraction electrode gap m tw saturation time s J G(E) free energy of solvation mol V voltage V m H extractor thickness m v velocity s h emitter height m Vw electrochemical window V I current A W work J

Isp specific impulse s x distance m A j current density m2 Y Young’s Modulus P a S kg K electrical conductivity m Z fluidic impedance s4 m kg L capillary length m m˙ mass flow rate s ˙ 1 Lb beam length m N particle flow rate s m mass kg P electric power W M kg m0 initial (wet) mass kg molecular weight mol mf final (dry) mass kg rr roughness ratio − mp propellant mass kg ∅ diameter m m P pressure P a ⃗qs surface flow rate s

15 − m β current fraction g gravitational acceleration s2 J J ∆G Gibbs free energy mol k Boltzmann constant K ∆P pressure drop P a

m ∆v velocity increment s Subscripts δ capacitor gap m atm atmospheric

δd deﬂection m a accelecrating component ϵ relative speciﬁc charge − c capillary

ϵr surface roughness m eﬀ eﬀective

ϵslip slip coeﬃcient − el electrostatic η eﬃciency − e electron

ηf non-dimensional ﬂow rate − h hydrostatic kg γ surface tension s2 in internal component 2 κps surface permeability m i ion λ mean free path m k kinetic (intertial) kg µ dynamic viscosity m s loss lost component kg ρ density m3 LV liquid-vapor interface kg σ surface traction m2s2 L Larmor

σv von Mises stress P a n normal component θ cone half angle rad poly polydispersive

θb beam half angle rad req minimum required

θc contact angle rad SL solid-liquid interface ε relative permittivity − start startup value

φs solid area fraction − SV solid-vapor interface tip at emitter tip Constants } Planck constant J s Superscripts N 1 A Avogadro’s number mol + positive polarity

θT Taylor angle rad − negative polarity F ∗ ε0 vacuum permittivity m critical value

16 Chapter 1

Introduction

In recent years, electrospray propulsion has experienced a gradual resurgence of interest concurrent to the trend of satellite miniaturization. Standardization of secondary payload sizing, driven largely by the emergence of CubeSats, provides a low-risk path towards nano-satellite development for both research and educational institutions. Due to potential for high specific impulse, fine control resolution, relative simplicity, and tendency towards improved performance with miniaturization, electrospray thrusters have emerged as a natural method of active propulsion for this platform. The logical endpoint of this miniaturization trend would be a fully micro-fabricated satellite. The percentage of active mass and volume could potentially be increased by an order of magnitude through direct integration of subsystems into silicon. However, control of a wafer satellite poses new issues. Current state-of-the-art electrospray technology relies primarily on physical properties of the emitter array material (especially porosity). For this reason, utilization of existing designs with a different material is not viable and thus a new architecture is necessary. It must be amenable to direct fabrication in silicon with micro electromechanical manufacturing systems (MEMS) techniques while still meeting the various requirements for stable and efficient emission. This thesis examines relevant constraints and design considerations, briefly discusses the manufacturing process, then explores testing and performance of these thrusters. Building off recent advances in the field, we developed a high-performance silicon-based thruster to facilitate production of a novel class of satellites.

17 1.1 Propulsion

This section is included to provide context and motivation for our focus on electrospray thrusters. A short exploration of the fundamental concepts of propulsion introduces key performance metrics that not only elucidate the thruster choice, but are later used in the design process and comparison of results to current technologies. A more detailed analysis of electrospray operation follows thereafter.

1.1.1 Fundamentals of Space Propulsion

Most basic concepts in space propulsion are drawn directly from the application of Newton’s laws. The vacuum of space provides an environment nearly free of external forcing apart from gravity. Non-conservative forces in particular are minimized, providing uncommon accuracy of extremely simple dynamic models. Given the absence of a surface to push oﬀ, propulsion in space relies on the principle of Newton’s third law. By ejecting some portion of its own mass (the propellant) at a high velocity, a rocket can accelerate in the opposite direction. The magnitude of this acceleration is proportional to both the mass expelled and the speed at which it leaves relative to our spacecraft. Formally, this quantity is known as thrust and is deﬁned by equation 1.1.

d F = (mv) (1.1) dt

− dm Given the case of constant propellant mass ﬂow (m˙ = dt ) and exhaust velocity, a generally accurate assumption for the type of thrusters which this research examines, this equation simpliﬁes to:

F =mc ˙ (1.2)

This intentionally neglects any thrust resulting from a pressure diﬀerential be-

18 tween ambient conditions and the nozzle exit. Inclusion of these terms is not relevant to the electric thrusters which this thesis examines, however more information and a derivation of the additional component may be found in [102]. With the assumption of no external forcing, we derive the classical rocket equation (1.3) through conservation of momentum. Note that we shift the analysis to an inertial rather than body-centered coordinate system.

dp dv dm = m + v +m ˙ (v − c) = 0 (1.3) dt dt dt

With simpliﬁcation, this becomes equation 1.4.

dv dm = − (1.4) c m We integrate directly to obtain, the famous Tsiolkovsky rocket equation (1.5)[106]. In doing do, the term ∆v is introduced as a convenient measure of the work done by our rocket. In free space (with no external forces including gravity) this term is directly equivalent to the change in velocity of the rocket. When other forces exist, it becomes the time integration of only the component of spacecraft acceleration which results from propulsion.

( ) ∆v m = ln 0 (1.5) c mf

It becomes apparent that for a given mission (and therefore a predetermined ∆v), the ratio of payload to propellant mass is ﬁxed by the exhaust velocity (c) of the rocket. For this reason, it is common to present the above equation in terms of the propellant mass fraction. Mathematically deﬁned as mp = m0 − mf , it is the portion of initial mass required as propellant to complete the mission. Reorganizing to isolate this parameter, we obtain equation 1.6.

[ ] −(∆v/c) mp = m0 1 − e (1.6)

19 From this equation, it becomes clear that the exhaust velocity is a critical parameter for the characterization of rocket performance. Traditionally, this quantity is normalized by gravitational acceleration to have units of seconds as seen in equation 1.7. This is the speciﬁc impulse (Isp) of the rocket and describes how eﬃciently the system utilizes its limited propellant mass. Formally, it is a measure of impulse per unit weight of propellant.

c I = (1.7) sp g

Figure 1-1 illustrates how slight improvements to speciﬁc impulse can have a drastic eﬀect on the ratio of propellant to useful mass.

100

0 0 1000 2000 3000 4000 5000

Figure 1-1: Eﬀect of speciﬁc impulse on required propellant.

Next, we deﬁne the eﬃciency of a rocket. This is simply the fraction of source power that is converted to useful work (in this case propulsive power) as shown in equation 1.8.

20 1 2 2 mc˙ η = P (1.8)

Given that the source power for any chemical propulsion system is the internal energy of the stored propellant, this is not a particularly useful metric. It does however find an application in electric propulsion. For certain types of electrostatic thrusters (electrospray included) the potential loss is relatively low in comparison to the overall extraction voltage. When this is the case, we can find an approximate efficiency by replacing the input power (P) with the product of applied voltage and extracted current according to Joule’s first law. In general, the specific impulse is a more important metric in space propulsion and nearly analogous to the efficiency parameter used to characterize many other systems which operate in the atmosphere.

Unlike efficiency, increasing specific impulse is not invariably beneficial. Looking at equation 1.8, we see that higher c requires increased power given constant m˙ and η. In fact, there is a squared dependence of input power on specific impulse (compared to the linear dependence of thrust on the same quantity). Given that power is a scarce resource on a satellite, it is necessary to optimize specific impulse based on mission requirements. Equation 1.9 illustrates the inherent trade-off between achievable thrust and system power that is governed by specific impulse.

F 2 η = (1.9) P c

To find an optimal specific impulse, we must first determine both maximum power available for thruster operation and the minimum thrust necessary to achieve mission goals. Applying equation 1.9 with some constant efficiency, this yields our specific impulse. Higher values will produce insufficient thrust at the given power while lower ones unnecessarily increase the propellant mass fraction, thus restricting ∆v. Equiv- alently, ∆v may be fixed and either the optimal power or thrust calculated. Tracing the system energy provides an alternative perspective on this trade space [102]. From

21 an inertial reference frame, the portion of total kinetic energy which accelerates the rocket (rather than its exhaust) is shown in equation 1.10. For a constant speciﬁc impulse, this value changes as a function of spacecraft velocity and captures inef- ﬁciencies which arise from residual motion of exhaust particles with respect to the inertial frame.

F v 2 (v/c) ηk = = (1.10) 1 − 2 v 2 F v + 2 m˙ (c v) 1 + ( /c)

The value of this metric is shown in figure 1-2 as a function of the velocity ratio. As expected, the peak occurs when specific impulse is equal to spacecraft velocity. Here, the exhaust particles have no excess speed and all kinetic energy is applied to propulsion. The naïve solution would be to adjust specific impulse to match the instantaneous velocity. However, this is almost never a possibility. We can solve for the specific impulse which minimizes the required energy over the total mission duration. Replacing the instantaneous vehicle mass with the analytical expression derived earlier, we arrive at equation 1.11.

100

0 0 1 2 3 4

Figure 1-2: Matching speciﬁc impulse to ∆v requirements.

22 [ ] 1 − −∆v/c 2 Ereq = 2 m0 1 e c (1.11)

This function is minimized by setting its derivative with respect to speciﬁc impulse equal to zero. The optimal exhaust velocity is found to be c = 0.6275 ∆v.

[ ] d ( ) ∆v (E ) = c e∆v/c − 1 − m e−∆v/c = 0 (1.12) dc req 2 0 However, this approach has several significant limitations. While efficiency is maximized, the lower specific impulse can incur a significant reduction to payload mass fraction which may not be acceptable. Additionally, it does not account for the additional structural mass associated with higher power levels (batteries, power supply components, et cetera). Note that while arbitrary boundary conditions may be used, the example above assumes acceleration from rest (the case used for launch vehicles, not electric thruster applications). A more thorough examination of optimal specific impulse can be found in [68]. Fortunately, many applications relevant to this project (particularly those which involve station keeping) require little variation in the magnitude of spacecraft velocity throughout the operational lifetime. Additionally, mission duration benefits immensely from high specific impulse when thrusters are used for attitude control (provided that the control authority is larger than disturbances for the available power level). For these reasons, we can reasonably make the claim that higher specific impulse is preferable for the drag compensation and control of micro- and nano-satellites.

1.1.2 Electric Propulsion

Most thrusters fall into two broad categories. The ﬁrst type, chemical propulsion systems, rely on the extracted internal energy of their propellant to provide the power needed to drive the engine. Electric propulsion systems, by contrast, depend on the propellant exclusively for its mass. The chosen particles are accelerated through one of various electric methods. The following subsections will examine both the

23 advantages of electric propulsion and its common classes. Note that the existence of more esoteric propulsion systems such as nuclear thermal rockets or solar sails is ignored in delineating these two primary categories.

Comparison to Chemical Propulsion

Both chemical and electric propulsion systems have inherent advantages. Chemical rockets typically produce higher thrust - often several orders of magnitude larger than electric systems. Since the internal energy of the propellant provides input power, maximum thrust is not limited by the amount of electrical power available. Internal energies per unit mass of chemical propellants are typically far larger than those of batteries. Further, chemical systems have higher thrust densities than any existing electric propulsion system. This makes them excellent candidates for escape from Earth’s surface. Lower thrust devices would be unsuccessful in this enterprise. Despite lower thrust levels, electric propulsion devices can accelerate propellant to higher exhaust velocities. While chemical propulsion is usually limited to specific impulses of about 450 seconds or lower, electric propulsion systems have been demonstrated with Isp an order of magnitude higher. Several systems currently in testing promise to stretch this limit significantly further [20, 64, 11]. Additionally, many types of electric propulsion can operate in a range of specific impulses without significant reduction in system efficiency. A tunable specific impulse improves performance by either matching the propulsion system to mission ∆v or introducing the possibility of variable Isp based on instantaneous spacecraft velocity. Finally, certain aspects of mission design and spacecraft dynamics differ for chemical and electrical systems. Despite higher specific impulses, electric propulsion requires additional or larger components to collect, store, and distribute power. These extra parts add to the total spacecraft mass (at a rate often nearly linear with respect to power required) that must be considered in mission design. Dynamics approxima- tions used for chemical propulsion – such as the use of impulsive maneuvers – do not apply for lower thrust systems. As a result, trajectory design and optimization for electric propulsion differs drastically from traditional approaches.

24 Variants

Electric propulsion systems can be subdivided into three primary categories – electrothermal, electrostatic, or electromagnetic – based on their method of accelerating the propellant particles. Again, a few esoteric designs fall outside these major categories. The purpose of this section is to provide a non-exhaustive overview of current electric propulsion systems in order provide background information and motivate the exclusive examination of electrospray thrusters in this research.

Electrothermal thrusters can be thought of as modified chemical rockets. Their basic operating principle is the use of either electric fields or direct electrical contact to increase the temperature of a chemical propellant or plasma within the combustion chamber. This added thermal energy is harnessed in some fashion to increase kinetic energy and therefore specific impulse. Resistojets and arcjects both fall into this category. Such systems can improve Isp by a few hundred seconds. This increase is typically limited by either material constraints of the combustion chamber and nozzle throat (due to the increased exhaust gas temperature) or available electric power, which inherently limits the affected mass flow rate. Given the relative simplicity of such systems and the number of missions with ∆v requirements in this range, electrochemical thrusters are fairly commonly used today.

Electrostatic thrusters accelerate propellant through the use of a static electric field. The two most common architectures are Hall and gridded ion thrusters, although electrospray and field emission electric propulsion (FEEP) also fall within this category. Typical propellants for Hall and ion thrusters are heavy inert gases such as xenon. The incoming particles are ionized by some means before being accelerated by an electric field. Magnetic confinement of electrons increases residence time within the ionization region and therefore improves system efficiency. The heavier ions are less affected and reach the acceleration region at a much higher rate. An external cathode is employed to neutralize the beam and prevent spacecraft charging. Specific impulse of these devices can range from 1,000 to 10,000 seconds with efficiencies of 60-70% being typical.

25 Electromagnetic thruster are a class of devices which rely on Lorentz force actuation to accelerate charged particles. This distinguishes such architectures from those in the previous paragraph which have an electric field generally parallel (or antiparal- lel) to the direction of thrust. These thrusters are an area of active research, showing significant potential for applications which require simultaneously high thrust and specific impulse. Pulsed plasma thrusters (PPT) and magnetoplasmadynamic (MPD) drives both fall into this category. Cathode degradation rates and extremely large power requirements of MPDs limit their applicability at the current time (although nuclear reactors are a potential solution). An operational system could have scalable specific impulses above 10,000 seconds and several newtons of thrust – a potentially revolutionary technology for interplanetary travel. PPT devices are comparatively simple and have specific impulses close to those of electrostatic thrusters, but suffer from very low efficiency.

1.1.3 Electrospray

As mentioned previously, electrospray thrusters are a type of electrostatic accelerator. Unlike Hall and gridded ion thrusters, the propellant is liquid rather than gaseous. This greatly simplifies the design as magnetic confinement of ionized particles is no longer necessary. More importantly, this causes such systems to be amenable to miniaturization since ionization losses tend to be the primary driver of efficiency reduction in smaller electrostatic systems. Intermolecular bonds and surface tension perform the same role with essentially no losses. Propellant choices range from doped solvents (colloid thrusters) to liquid metals (FEEP) to ionic liquids, with the latter having several significant advantages. Strong electric fields are used to extract and accelerate ions and/or charged droplets directly from the propellant. Certain geometries may be used to concentrate the electric field at specific points, thereby enhancing performance.

26 Architectures

Traditionally, electrospray research can be divided into three categories based on the emitter geometry. Each These families are illustrated in ﬁgure 1-3 and have distinct characteristics, challenges, and operational advantages.

(a) Capillary (b) Externally Wetted

Ionized plume

Liquid propellant

Substrate material

Extraction electrode

Figure 1-3: Comparison of electrospray architectures.

Many of the earliest studies into electrospray propulsion examined capillary architectures. Colloid thrusters almost exclusively adopted this architecture, with doped glycerol actively fed through hollow metal needles to produce droplet emission [21]. In 1972, an ESTEC study attempted to achieve passive propellant supply and was mostly successful [5]. As gridded ion engines and other alternatives for high Isp propulsion matured, colloid research decreased signiﬁcantly. A combination of factors (including the emergence of nanosatellites, advances in micro-fabrication, the need for precise and low-power propulsion) led to a resurgence of electrospray interest near the turn of the century. New research focused on ionic liquid propellants, but capillary designs remained common. Work at MIT [71] and Yale [90] employed increasingly thinner

27 capillary needles and lower feed rates in an attempt to achieve pure ionic emission. This goal was soon achieved with several ionic liquids (ILs) through use of active ﬂow control and often increased temperature [91, 35]. Concurrent work examined micro-fabricated silicon capillaries as an alternative to metal needles [80, 81]. This opened the door to compact multiplexing of emitters and attempts toward high aspect ratio capillaries to allow passive ionic emission – two major trends of continuing research [57, 41]. Overall, capillaries are very simple and controllable but emission stability degrades at lower ﬂow rates.

The second primary architecture is externally wetted. In this design, the propellant is applied directly to the emitter surface. Transport to the emission region is necessarily passive, which reduces system complexity. Because the propellant is exposed, flow disturbances are less frequent and rarely cause emission instability. The drawback, however, is a high dependence of performance on the emitter surface characteristics. This architecture was first proposed and tested in the 1980s with liquid metal ion sources [72, 104], but was successfully adapted by Lozano with ionic liquids and a sharpened tungsten wire in 2005 [70]. Tungsten needles continued to be a pop- ular emitter choice, with electrochemical sharpening and axial grooves used to obtain the proper meniscus size and flow control, respectively [73]. Using this setup, ionic emission was achieved for several ILs [17]. Concurrent to these efforts, Velásquez- García began using MEMS techniques to create externally wetted silicon emitters in various shapes and sizes – using black silicon for the roughness which allows porous surface transport [109]. The potential for multiplexing (and its implication for thrust density) motivated significant follow-on research both at MIT and elsewhere. No- tably, Gassend achieved nearly pure ionic emission with silicon emitters [38]. Recent work has focused on decreasing emitter spacing and improving emission uniformity. Despite its success, this architecture has a critical flaw; there is no method of replen- ishing propellant to the emitter tips. As such, these devices are not currently viable as thrusters. Additionally, the operation of such emitters is heavily dependent on the surface characteristics to provide the necessary impedance for steady emission.

Finally, porous emitters have been the most recent exploration in electrospray

28 design. Although patented in 2004 [51], this architecture was ﬁrst successfully implemented in 2007 at MIT. Electrochemical etching of porous tungsten sheets created

planar arrays which achieved passive ionic emission with EMI-BF4 [88]. The success of this study led to an exploration of various substrate materials and manufacturing methods which would allow similar performance with lower emitter spacing. A follow- on increased thrust density and determined that porous emitters could sustain similar current per emitter to other architectures [63]. This work was soon extended to two dimensions with conical emitters that also achieved purely ionic emission in both polarities without active flow control (using a porous propellant tank) [32]. After exploration of other porous metals [24, 15], ultra-fine pore borosilicate glass emerged as a promising substitute. While manufacturing of metal emitters required wet-etching (and its associated challenges with spatial uniformity, repeatability, and batch fabrication capability), new glass emitters could be processed by using laser ablation [23]. This substrate allowed higher thrust density with low (but non-zero) polydispersive loss [23, 27]. However, the non-uniform pore size distribution and small remaining droplet fraction created impetus for testing of carbon xerogel substrates [89]. This material is amenable to many fabrication techniques (including laser ablation), has high pore uniformity, and allows a range of hydraulic impedance [89, 100]. While other porous substrates are currently being explored, carbon xerogel seems to be the most promising option to date thus far. In general, porous architectures offer the performance of externally wetted tips but with a method of passive propellant delivery from an external reservoir. Emission, however, depends significantly on pore size and uniformity.

Operating Regimes

Electrospray operation can be divided into three regimes based on the type of particles emitted. Here we present a surface level introduction to the concepts necessary to understand the transition between regimes. The following chapter will include the derivations and equations used to quantify these trends. At high ﬂow rates and low impedances, electrospray tips emit primarily droplets. While the mass of these

29 particles can vary significantly based on operating conditions and physical properties of the chosen propellant, typical sizes are several hundreds of molecules for an ionic liquid. This larger charge to mass ratio benefits thrust, but is detrimental to specific impulse. Operation in this regime is known as the droplet mode and was the first to be explored by researchers.

As impedance increases, flow rate decreases and the meniscus formed necessarily becomes sharper. Accordingly, the electric field at the tip increases and ion evaporation (an activated process) begins to occur simultaneously to droplet emission. This is known the mixed regime of operation. While the increased specific charge ratio of ions (as compared to large droplets) would increase the Isp of the device, this is largely offset by polydispersive losses. This phenomena is illustrated by figure 1-4 – the derivation of which is provided in the following chapter. The left margin of the chart corresponds to droplet operation. Note that this figure assumes a constant ratio of light to heavy species, and therefore a monodisperse beam of droplets in the left extreme.

100 1

80 0.8

60 0.6

40 0.4

20 0.2

0 0 0 0.25 0.5 0.75 1

Figure 1-4: Polydispersive eﬀects on thruster performance.

30 When the fraction of emitted current attributed to ions exceeds about 90 percent, polydispersive efficiency begins to quickly approach unity. Simultaneously, the average charge to mass ratio decreases. Together, these two effects generate a rapid increase in specific impulse. Due to polydispersive losses, mixed mode thrusters typically aim to avoid operation below the inflection point for a given mass ratio. In the extreme (corresponding to the right margin of figure 1-4), no droplets are generated and all current can be attributed to ions evaporated from the meniscus. This is known as the pure ion regime (commonly PIR) and is an attractive operating point for several reasons. Specific impulse increases by nearly an order of magnitude. The exponential nature of the rocket equation means that this translates to an even larger decrease in the propellant necessary to achieve a given ∆v requirement. Fur- ther, efficiency returns to a value far above those of similar electric propulsion systems. Stability in PIR also tends to be better. The single drawback is that mass flow rates are lower for a given voltage and thus higher voltages and more power are required for similar thrust levels. This may translate to additional power supply mass relative to thrust (although the increase is certainly not linear for small satellites, primarily increasing at certain voltage thresholds due to component limitations). The inherent trade off between power and specific impulse must not be neglected is mission design. Chapter 2 discusses these operating regimes in more detail and includes a description of the mechanisms used to ensure the desired emission characteristics.

31 1.2 Satellite Miniaturization

The trend towards satellite miniaturization has experienced a rapid surge in the last decade – even necessitating new terminology for size. Table 1.1 shows a convention commonly seen in recent literature, although variations to these deﬁnitions certainly exist.

Classiﬁcation Mass (kg) Microsatellite 10 – 100 Nanosatellite 1 – 10 Picosatellite 0.1 – 1 Femtosatellite 0.01 – 0.1

Table 1.1: Satellite size classiﬁcation.

The ever-increasing availability of smaller and more capable commercial oﬀ-the- shelf (COTS) electronic components has allowed integration of capabilities previously reserved for larger satellites. GPS receivers, solar cells, cameras, and other common payload items have all decreased in both weight and cost concurrent to the improvement in microprocessor capability necessary for their operation. Not only does this enable launch cost reduction, but use of commercially-available components reduces development risk, time, and overall system cost. The magnitude of this trend is shown in ﬁgure 1-5 which shows an exponential growth in nanosatellite launches since the turn of the century. While 1U and 3U CubeSats fall squarely within the nanosatellite range, recent exploration into pico- and femtosatellites has produced interesting results. In 2014, a co-inventor of the CubeSat standard introduced the concept of a smaller platform, dubbed PocketQube, with each unit having half the width and height of its prede- cessor [107]. A handful of these picosatellites have been launched [84], with proposed active control capability using a combination of magnetorquers, reaction wheels, and micro pulsed plasma thruster (µPPT) pairs [6]. Other designs aimed for even lower mass and followed the "satellite-on-a-chip" approach. A group at the University of Surrey proposed a two-layer 4 cm2 printed circuit board satellite in 2007 [8]. Dubbed

32 Figure 1-5: Total Nanosatellites and CubeSats Launched as of January 2020 [61].

"ChipSat," it was soon followed by a larger variant: "PCBSat". Both designs lacked active propulsion capability (the larger model included a single-axis magnetorquer) and suﬃcient communication power to reach a ground station [7]. Several similar architectures have been pursed by other groups since then. Of particular note is the "Sprite" project at Cornell which included an orbital demonstration. The 3.5×3.5 cm chips were designed to last several days in LEO without propulsion while sending intermittent radio-frequency pulses to any available ground stations [74]. A successful preliminary test on an external pallet of the ISS was unfortunately followed with a deployment failure (resulting from a software master clock reset) by the 3U CubeSat designed to launch the chips. In 2014, a JPL study introduced a notional femtosatellite architecture for an analysis of distributed array guidance, navigation and control (GN&C) techniques [43]. Their proposed design included both electrospray and miniaturized hydrazine thrusters to achieve six degree-of-freedom control and silicon wafer integration of certain components for mass savings.

In both ﬁgure 1-5 and this brief survey of recent small satellite research, a few key trends emerge. First, two of the primary limiting factors towards miniaturization are

33 communications and propulsion. Smaller satellites have inherently lower power bud- gets due to both battery size and available collection area. Along with the difficulty of shrinking relevant components, this causes challenges for long range communication. Relaying collected data through a larger nearby satellite is a common solution to this issue. Femtosatellite propulsion is a more difficult issue. From figure 1-5, even the fraction of nanosatellites which include propulsion modules is limited to approximately five percent. To exacerbate this issue, the planar form factor of com- puter chips is not conducive to traditional methods of non-propulsive control (such as reaction wheels or magnetorquers). However, the other trend observed is that the best method of achieving further miniaturization is likely PCB or waferscale integration of any possible components and subsystems in order to maximize the fraction of active volume. Cold gas propulsion has been proposed [43] and even demonstrated [47] on a picosatellite platform, but suffers from very low specific impulse and a wide form factor. Temporarily ignoring electrospray thrusters, two very unique chip-based solutions have been explored as well. A 2007 Cornell study explores the feasibility of using Lorentz force actuation to achieve propellant-free propulsion for a low mass satellite [4]. A 1 cm2 silicon chip inside a Faraday cage with an attached fila- ment would be passively oriented using the local geomagnetic field. More recently, the "Breakthrough Starshot" project has examined the possibility of interstellar femtosatellite missions. Massive arrays of land-based lasers would accelerate a silicon chip to relativistic speeds through use of a small lightsail [44, 12]. While certainly a niche application, this demonstrates the extent of new opportunities presented by satellite miniaturization.

Apart from the exotic, small satellites oﬀer improved capability in current missions. Low cost, reproducible platforms could be employed for transient data collection such as the monitoring of ﬂuctuations in ionospheric plasma density [84, 50]. Alternatively, small satellites could be attached to larger ones and deployed to inspect any issues which arise on the host spacecraft [50]. This would provide an inexpen- sive method of mitigating risk, while collision probability and severity both decrease with inspector mass. In fact, a picosatellite implementation of such a system has

34 already been demonstrated [47]. Finally, sparse aperture imaging is perhaps the most promising application for constellations of small satellites [43]. A final advantage of small (and especially batch-produced) nanosatellites is their statistical survival rate [108]. Launching a large geosynchronous communication satellite, for example, entails significant risk. Not only can an isolated system failure lead to billions of dollars of loss, but single event risks such as launch failure or orbital debris pose a significant threat to monolithic satellites. Performing the same or similar operation using a distributed system can reduce risk by offering increased redundancy. If a single satellite in a swarm experiences some critical failure, the loss is limited and functionality possibly transferred to another member of the constellation. Individual satellites, by contrast, require internal redundancy and expensive protection of critical components along with the ability to maneuver away from potential threats [2]. (However, a large enough constellation with intersecting orbits inherently introduces some internal collision risk [87, 43]). Integration of disposal capability into the design of these new constellations is also critical to mitigation of future accessibility threats.

35 1.3 Thesis Motivation and Challenges

The previous section discussed the various advantages of small satellite platforms, with the logical extreme being a functioning waferscale satellite. Such a platform could provide incredible savings even relative to current micro-satellite platforms in two main areas. First, a significant increase in the proportion of active mass would lead to far lower launch costs – currently one of the most significant expenses for any type of satellite. With commercial launch costs currently at tens of thousands of dollars per kilogram sent to low earth orbit [52], going from a 1.33 kg CubeSat to a 0.33 kg WaferSat provides significant savings with little to no depreciation in capability. Additionally, this facilitates a shift towards batch fabrication, which could reduce production costs at scale by several orders of magnitude. Large constellations of wafer satellites could be quickly manufactured at a reasonable expense to complete missions traditionally reserved for large and expensive monolithic systems. One of the most significant challenges toward realization of this goal is the lack of propulsion options. Currently, no wafer integrated solution exists. While many CubeSats chose to forgo active control (relying on passive magnetic stabilization) this inherently limits capability and especially the potential for collective operation which makes WaferSat such an appealing platform. Devices such as magnetorquers, FEEP systems, reaction wheels, or cold gas thrusters – which have all been demonstrated on micro-satellite platforms – are not viable candidates for wafer integration due to either size or form factor. Electrospray, while showing potential for miniaturization, relies significantly on the physical properties of the substrate material. To date, no silicon-based electrospray thruster module with a propellant reservoir has demonstrated passive ion emission. This thesis addresses that gap. Several notable challenges are addressed in this project:

• Design and Fluidics: The only silicon-based electrospray designs have been either externally-wetted (and therefore non-functional due to a lack of propellant supply to the array surface) or capillary architectures. The latter have had aspect ratios far lower than necessary for passive ion emission. Porous designs

36 in silicon would be diﬃcult to implement and have not been explored to date.

• Performance Threshold: The limited surface area available on a WaferSat necessitates a higher thrust density than current state-of-the-art. Similarly, the low internal volume of this bus creates a need for high speciﬁc impulse to minimize propellant storage. Extraction voltage is limited to 1 kV to meet size and weight requirements.

• Astrodynamics and Attitude Control: The unique wafer form factor and low altitude orbit necessitate a detailed examination of drag and other disturb- ing torques to determine passive stability and the level of control authority necessary to maintain attitude requirements for the mission duration.

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38 Chapter 2

Design and Fabrication

Given the lack of precedent for functional and passively-fed ion electrospray thrusters on a silicon substrate, the design aspect of this thesis was quite involved. The novel architecture provided a multitude of fabrication challenges which necessitated several iterations of testing to correct and validate individual design features. I am incredibly grateful to all members of the WaferSat team at MIT Lincoln Laboratory for their time and expertise in the micro-manufacturing process.

2.1 Electrospray Design Considerations

This section is included to address the critical design considerations unique to electrospray thrusters. An examination of the fundamental physics of ion emission helps to establish a performance trade space which in turn allows an informed structure for decision making.

2.1.1 Electrostatic Thruster Performance Metrics

For electrostatic propulsion systems, we can directly derive most performance metrics with a conservation of energy approach. Assuming a negligible initial velocity of the exhaust particles, their kinetic energy relative to the spacecraft frame is related to the accelerating ﬁeld in the following manner:

39 1 2 − 2 mic = qi (Va Vloss) (2.1)

The loss voltage (Vloss) accounts for energy of the applied field which does not contribute to the particle’s acceleration. While this is typically on the order of single volts for electrospray systems, it is an important factor for gas-fed architectures such as a Hall Thruster or Kauffman Ion Engine. Such systems require ionization of the propellant prior to its acceleration. To first order, ionization efficiency is inversely proportional to the mean free path of high energy electrons with respect to neutral species [65]. Since the collision cross section between these two particle types (Qc) is nearly constant, the mean free path can only be lowered by increasing electron number density as seen in equation 2.2.

1 λ = (2.2) ne Qc

Since the mean free path must scale linearly with device size to maintain ionization efficiency, higher plasma densities are required for smaller thrusters. While this does increase thrust density, it leads to more charged particles reaching the thruster walls. This has several detrimental effects including: a decrease to thruster lifetime, significant heat flux to walls which necessitates active cooling and thus higher mass, and a significant decrease in ionization efficiency [65]. Similarly, strength of the magnetic field used for confinement is inversely proportional to area (a result of the smaller Larmor radius required).

me c¯ rL = (2.3) qe B

Both of these effects lead to the significant voltage losses reflected in equation 2.1 and diminishing returns on mass reduction with device miniaturization. By contrast, electrospray systems require no magnetic confinement or propellant ionization, mak-

40 ing them primary candidates for nano-satellite propulsion. With loss voltage assumed to be small and relatively constant, speciﬁc impulse can be expressed by equation 2.4.

√ q c = 2Va( m )i (2.4)

Neglecting the losses described above, electrostatic acceleration is a space-charge limited process. The steady state current density (j = Ni qi Va) is maximized when the charge of ions in the acceleration region completely shield the source from the applied electric ﬁeld. This condition is known as the Child-Langmuir space charge limit and was derived in [19] using a one-dimensional approximation. √ 4 ε 2 q V 3/2 j = 0 a (2.5) 9 m G2

For electrospray emitters, however, the deformable nature of the liquid surface introduces complexity. Since the propellant itself acts as an electrode, the meniscus can sharpen in the presence of space charge to increase the local electric ﬁeld and thus compensate for any shielding which occurs. This process is limited by the rate of charge relaxation to the liquid-plasma interface. Regardless, space charge accumulation can become a limiting mechanism for electrospray thrusters in the case of unipolar operation. A strong net potential outside the acceleration grid will eventually inhibit the further extraction of ions.

2.1.2 Electrospray Fundamentals

Electrospray emission relies on a balance of surface tension and electrostatic forces on a charged liquid meniscus. The electric field at the tip of an emitter is concentrated by the meniscus curvature. An approximation of this value for a flat plate extraction electrode can be derived by using prolate spheroidal coordinates and assuming an axisymmetric confocal hyperboloid meniscus. If the radius of curvature (Rc) is significantly smaller than the tip to electrode gap (G), we obtain equation 2.6. A full

41 derivation of this method may be found in [67].

2V /Rc E ≈ − (2.6) tip 4G ln( /Rc)

If the system is passively fed, no additional pressure acts on the liquid surface. Flow necessarily occurs when electrostatic traction exceeds surface tension. Expressed mathematically, this condition is equation 2.7.

1 2 2γ 2 ε0Etip > (2.7) Rc

Rearranging variables, we ﬁnd an equation for the approximate starting voltage. Note that this neglects space charge, applied pressure, and meniscus geometries other than a confocal hyperboloid. Thus, it serves as an order of magnitude estimate more than an exact expression.

√ ( ) γRc 4G Vstart ≈ ln (2.8) ε0 Rc

From the previous two equations, it becomes apparent that as the electric field increases is strength, the liquid propellant will gradually deform from a spherical cap to a cone. The mathematical limit of this process was first examined by G.I. Taylor in 1965 [103]. He theorized a conical liquid surface which equilibrates surface tension and the electrostatic field at every point except the very tip. The general equation for surface tension per unit area is shown below.

( ) 1 1 σ = γ + (2.9) Rc, 1 Rc, 2

For a cone, the slopewise curvature is necessarily zero. The component of orthogonal curvature (also equal to the total value) is shown below:

42 1 cot θ = T (2.10) Rc r

Here, θT is the cone half angle, and r the slopewise position. The above relation can be veriﬁed geometrically. Using the same condition as equation 2.7, we obtain an expression for the normal electrostatic ﬁeld.

√ 2γ cot θT En = (2.11) ε0r

Using Legendre functions as described fully in [103], it may be proven that a

single solution exists for the angle θT which allows an equipotential liquid surface.

This is θT = 49.29°, and a cone with such slope is commonly known as a Taylor cone. It is interesting to note the angle’s lack of dependence on fluid properties. While this idealized case is useful as an approximation, it neglects several important factors including space charge, finite transport speed, and a nonzero flow rate. Further, the solution results in a non-physical singularity at the cone apex where curvature is infinite. Still, many charged menisci seen in the colloid research somewhat resemble this closed-form solution and it serves as a basis for more complex models. The higher electric fields associated with ion evaporation result in a significant deviation from this profile. For detail on this structure, the reader should consult Gallud Cidoncha [33].

Droplet Emission

At some point close to the tip of a Taylor cone, the equipotential condition no longer applies as the ﬂow rate and charge relaxation begin to have similar timescales. This condition can be represented by the following expression and allows us to approximate the radius (r∗) at which the Taylor cone breaks down.

r∗3 εε ≈ 0 (2.12) Q K

43 A thin jet of propellant forms, allowing charge transfer which prevents the singularity condition and allows a dynamic equilibrium. Due to surface tension and Rayleigh instability, the liquid breaks up into smaller droplets through Coulombic explosion. This is commonly referred to as the cone-jet mode and results in droplet emission. The approximate size of these particles can be found using equation 2.13.

√ q 6 ε γ 0 = 3 (2.13) m max ρR /2

The current level of an electrospray emitter operating in the cone-jet mode can be approximated using equation 2.14. Its derivation is not presented here but can be found in [29]. The same paper veriﬁes this relation experimentally, with f(ε) ≈ 18 for most highly conductive liquids. √ γKQ I = f(ε) (2.14) ε

With several assumptions (notably that the majority of surface transport occurs due to convection and that charge is relaxed outside the emission region), the ﬂow rate can be estimated based on ﬂuid properties.

2 η γεε0 Q = f (2.15) ρK

The term ηf is a non-dimensional flow rate, defined in equation 2.16. It accounts for factors other than the propellant physical properties that effect the flow rate (i.e. field strength, emitter shape, and impedance). As the value of this parameter decreases below one, droplet emission tends to become unstable or even stop altogether. √ ρKQ ηf = (2.16) γεε0

44 As this limit is approached, specific charge of the emitted particles necessarily increases with a very positive effect on specific impulse. The endpoint of this trend would be mono-disperse emission of individual ions – a phenomenon which is examined in the following section.

Ion Emission

At high electric fields and low flow rates, field evaporation occurs at a rate similar to droplet emission. This is an activated process, and thus expression for current density has the following form.

∆G−G(E) kT − j = σ } e kT (2.17)

To calculate the reduction from the free energy of solvation needed to extract an ion from the liquid surface (G(E)), we calculate the minimum work needed to bring that ion to inﬁnity. The forces acting on the particle are the electric ﬁeld

Fel = qiE and an image charge from the opposite ion within the liquid Fim. This force is approximated with Coulomb’s Law as seen in equation 2.18.

− 2 qi Fim = 2 (2.18) 4πε0(2x)

The total work is simply the integral of the sum of these two forces from x+ to ∞ (assuming a negligible electric field strength at infinity). Taking the partial derivative with respect to position allows us to find the minimum extraction energy. A more detailed derivation may be found in [71]. √ 3 qi E G(E) ≡ Wmin = (2.19) 4πε0

From equation 2.17, we see that the current level attributed to ion evaporation

45 becomes significant when the exponent is on the order of unity – and thus when ∆G . G(E). With this knowledge, we can find an approximation for the critical electric field at which ion emission begins. This is equation 2.20.

4πε E∗ ≈ 0 ∆G2 (2.20) q3

Neglecting hydrostatic pressure, we have a new equilibrium condition for the liquid surface at the point where ion emission begins. The electric ﬁeld inside the liquid ≈ E∗ may be approximated by Ein ε [66].

2γ 1 ε E∗2 − 1 ε E2 = (2.21) 2 0 2 0 in R∗

This allows us to solve for the characteristic radius of the emission region. Since

charge transport within the liquid occurs at a rate of j = KEin and must be equal to the total current density from equation 2.17, an approximation for the ion current can be derived.

32πKγ2 ε I ≈ (2.22) 2 ∗3 − 2 ε0E (ε 1) Using the characteristic radius, we can further solve for a thrust density limit which is several orders of magnitude larger than that of existing electrostatic propulsion systems. This provides signiﬁcant motivation for densiﬁcation of emitter arrays as current state-of-the-art thrust density for electrospray is lower than either Hall thrusters or gridded ion engines.

( ) F ε − 1 = 1 ε E∗2 (2.23) A 2 0 ε

For ion-only emission, a diﬀerent (and far more exact) expression for ﬂow rate can be derived using the physical properties of the propellant. To use equation 2.24 however, we must know the desired current levels a-priori. Consistent with equation 2.22,

46 past studies have shown that a few hundred nanoamperes is a reasonable estimate for a single emitter tip operating in PIR [42]. To be conservative, a current level of 100 nA is targeted for this study. Note that use of the molecular weight (M) and

Avogadro’s Number (NA) in this equation necessarily assumes monomer emission.

I ·M Qi = (2.24) NA · 2 qe · ρ

For most propellants, especially the heavy and highly conductive ionic liquids, the flow rate calculated using equation 2.24 corresponds to a non-dimensional flow rate noticeably less than unity. The necessity of low flow rates to achieve ionic emission indicates the importance of hydraulic impedance to system performance. Section 2.1.3 will discuss this in detail within the context of each electrospray architecture.

Polydispersive Eﬀects

While the increased charge to mass ratio of ionic emission greatly improves specific impulse, this can be offset by polydispersive losses as mentioned in section 1.1.3. These arise from the discrepancy between relative energy used to accelerate particles of different masses as compared to the ratio of thrust they provide. Considering the case in which only two species are present, equation 2.25 describes the efficiency as a function of mass and current fractions.

( ) √ √ 2 ˙ ˙ √ 2 N1 m1q1 + N2 m2q2 − − 2 F ( )( ) [1 (1 ϵ)β2] ηpoly = = = (2.25) 2mIV ˙ ˙ ˙ ˙ ˙ 1 − (1 − ϵ)β2 N1m1 + N2M2 N1q1 + N2q2

Here β2 is the current fraction of the lighter species (typically the ions) and ϵ is the proportion of charge to mass ratios as shown in equation 2.26.

I (q/m) β = 2 and ϵ = 1 (2.26) 2 q I1 + I2 ( /m)2

47 Taking the partial derivative of polydispersive efficiency with respect to ion current fraction, we find the minimum efficiency as a function of η.

√ 4 ϵ η = √ (2.27) p, min (1 + ϵ)2 Assuming that all extracted species are singly charged (which is not always the case), this minimum eﬃciency can be plotted as a function of mass ratios as shown in ﬁgure 2-1. Note that this still assumes two mono-disperse species, one of which is an ionic monomer.

100

0 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 1 10 10 2 10 3

Figure 2-1: Minimum polydispersive eﬃciency for two mono-disperse species.

Equations 2.4 and 2.25 are combined to derive equation 2.28 – the reduction in speciﬁc impulse associated with discrepancy of charge to mass ratios in the beam. Figure 1-4 (presented earlier without derivation) illustrates this eﬀect and that of equation 2.25 simultaneously.

(√ √ ) √ ϵ − ( ϵ − ϵ)β q 2 c = 2V ( /m)2 (2.28) 1 − (1 − ϵ)β2 Overall, it is seldom advantageous to design for operation in the mixed regime.

48 The high efficiencies typically associated with electrospray are best achieved with purely droplet or PIR emission. The former may perform better for high thrust requirements, but the latter offers substantial improvement to specific impulse.

Electrochemical Wear

Electrochemical wear is another essential phenomena to consider in the design process for electrospray thrusters. As ions are extracted from the propellant and accelerated into space, charge necessarily accumulates elsewhere in the system. A double layer forms at the propellant-substrate interface acting similar to a capacitor. This has two potentially negative effects: spacecraft charging and electrochemical wear. The former has various methods of mitigation outside the scope of this thesis, but the latter is a significant concern to operational lifetime. When the voltage across this interface becomes larger than the electrochemical window, a reaction occurs which may alter the geometry and surface characteristics which control emission. Fortunately, there are methods to diminish the damage. Propellant selection (in tandem with the emitter material) determines the size of the electrochemical window. Ionic liquids have an advantage in this respect, with electrochemical windows commonly between 2-5 V . Doped solutions used for colloid thrusters are typically limited to ∼ 1 V where electrolysis of the solvent (such as water or glycerol) can occur [66]. This contributes to the primacy of ionic liquids in modern electrospray research (along with their zero vapor pressure, high conductivity, and wide range of temperature stability) The method and location of electrical contact also contribute significantly to the severity of electrochemical wear. Biasing the extractor relative to the propellant through use of a distal electrode within the tank allows emitters to remain at a float- ing potential. More importantly, this increases the interface area from exclusively the tip region to the entire electrode. Since capacitance is proportional to area (equation 2.29), this dramatically reduces the rate and spatial concentration of wear. Given that surface area is one of the most controllable design variables, research into high porosity materials for the distal electrode has generated much interest [100]. The

49 (a) Emitter before testing (b) Emitter after testing

Figure 2-2: Electrochemical wear of a silicon array. term δ is approximately equal to the chosen anion diameter [69].

εε A C = 0 (2.29) δ

Voltage alternation is a common strategy to suppress electrochemical reactions. Since the charge double layer takes some ﬁnite time to form, the polarity may be reversed before the voltage reaches a maximum. If the polarity is switched before the electrochemical window limit is exceeded at any point on the ﬂuid interface, decom- position is eliminated. We assume that charge transport within the propellant occurs more quickly than the double layer formation (an assumption validated by [69]), and

50 therefore use equation 2.30 to describe the rate of charge accumulation.

dV I = C (2.30) dt

Using the parallel plate capacitor approximation, we obtain the following expression for saturation time in each polarity.

V A t = εε w (2.31) w 0 I δ

For EMI-BF4, the electrochemical window limit is approximately 2.25V in either polarity [31]. We can use these charge buildup times to calculate an approximate minimum frequency of voltage alternation necessary to suppress electrochemical degradation (equation 2.32). If the window limit is asymmetric, the proportion of positive to negative emission must be adjusted. For most geometries, frequencies on the order of single Hertz are sufficient [69]. As thrust density increases and emitter spacing becomes smaller, the required frequency may approach the timescales for meniscus formation. At this point, further densification may be impossible without sacrificing thruster lifetime. For architectures not biased directly, the distal electrode surface area can be increased to eliminate the problem. Fortunately, this issue is not yet relevant to current thrusters due to manufacturing limitations.

+ − −1 fmin = (tw + tw) (2.32)

Finally, surface treatments can be used to mitigate the rate of degradation for directly-biased designs. Surface roughening increases the interface area, thereby increasing capacitance. Thin oxides may also be applied to the emitter surface, either to increase the dielectric constant or to act as a protective sacrificial layer. However, the effect of any surface treatment on system microfluidics must be considered.

51 2.1.3 Microﬂuidics

The previous subsection highlights the importance of propellant flow rate to electrospray performance. To minimize both size and system complexity, it is advantageous to use passive means of control whenever possible. For a given electric field strength, the flow rate can be effectively set by use of geometry and surface characteristics to obtain an acceptable combination of pressure and impedance. A simplified model,

Figure 2-3: Eﬀect of impedance on emission characteristics.

proposed by Hill [46], is illustrated in figure 2-3. The red line shows emission without any viscous losses. Adding a linear impedance decreases flow rate at a given field strength (resulting in the green line). The shaded purple region represents barrier- limited emission, where current and flow rate are dominated by electrostatic pressure and increase exponentially in accordance with equation 2.17. The orange region, by contrast, shows supply-limited emission. Here, the large impedance causes a viscous pressure drop comparable to driving electrostatic pressure. This causes the low flow rates necessary for field evaporation of ions to occur. If the surface impedance is consistent, this has the further advantage of increasing emission stability [22] and spatial uniformity [42]. The design considerations presented here do depend on emitter type.

52 Capillary Emitters

One significant allure of capillary emitters is the relative simplicity of their fluidics. Propellant in a capillary emitter fulfills all assumptions of Hagen-Poiseuille flow at steady state and thus pressure drop due to viscosity can be expressed by equation 2.33.

8µLQ ∆P = (2.33) πr4

From the ﬁrst derivative of the Washburn equation, we know the velocity of capillary rise. When multiplied by the pore’s cross-sectional area, this provides the expression for volumetric ﬂow rate shown in equation 2.34.

[P + P + P ](r4 + 4ϵ r3) Q = πr2 atm h c slip (2.34) 8r2µL At this point, we make several assumptions:

• Coeﬃcient of slip (ϵslip in equation 2.34) is zero.

• Atmospheric pressure (Patm) is zero due to vacuum operation and tank vents.

• Hydrostatic pressure (Ph) is negligible due to passive operation. • The channel walls are perfectly inelastic. • Fluid inertia is negligible.

Capillary pressure (Pc) becomes the only relevant force in this system, and can be described by the aforementioned Young-Laplace equation (2.35). At the scale of these structures, gravity can accurately be assumed to have a negligible eﬀect.

2γ cos θ P = c (2.35) c r

Since ﬂuidic impedance is simply the ratio of pressure drop to ﬂow rate (Z = ∆P ), Qi this value is known for constant circular cross-section.

8µL Z = (2.36) πr4 53 Impedance therefore can be adjusted by varying the aspect ratio of the capillary

L itself. (More speciﬁcally, the ratio r4 which relates directly to impedance.) Until recently, manufacturing constraints restricted the minimum radius such that passive PIR emission was not possible. As mentioned in section 1.1.3, this was one of several reasons that research has gradually moved away from capillary emitters.

Externally Wetted Emitters

For non-capillary emitter geometries, the analysis is slightly more involved. We make two key assumptions. First, the nominal emitter geometry is taken to be a cone with a spherical cap as shown in ﬁgure 2-4. Second, all propellant transport is assumed to occur via a thin porous layer on the surface. This model is therefore applicable to the class of externally wetted emitters.

z Rc 0

ℎ z

r z1

Figure 2-4: Conical emitter impedance.

We begin our analysis with the two dimensional form of Darcy’s Law (2.37).

54 κ ⃗q = − ps ∇⃗ P (2.37) s µ At the length scales of interest, capillary pressure is much greater than gravity. This allows simpliﬁcation to the following equation, where z is taken to be the slope-wise direction of the cone.

∂P µ = − · q (2.38) ∂z κps

For a symmetric electric field, the direction of propellant flow is always towards the emitter apex. The variable q is simply volumetric flow rate divided by surface area. As a function of z, this quantity can be expressed as A = 2πz sin θ. This is the perimeter of the cone at a given z slice as represented by the thick red line in figure 2- 4. If the emitter were porous, we would instead use the shaded area and derive a volumetric flow rate. The pressure drop across the emitter is given by equation 2.39.

Rc For a tip radius Rc, the limits of integration are z0 = tan(θ) (base of the emission h region) and z1 = cos(θ) (base of the cone).

∫ z1 µ · Q P dz = − (2.39) z0 κps Evaluating this integral yields an expression for total pressure drop over a single tip. ( ) µ · Q h · tan(θ) ∆P = · ln (2.40) 2π · κps · sin(θ) Rc · cos(θ)

Since impedance is simply a pressure drop divided by ﬂow rate, we get equation 2.41 for a single emitter. ( ) µ h · tan(θ) Z = · ln (2.41) 2π · κps · sin(θ) Rc · cos(θ)

Once again, ﬂow rate is determined by the external electric ﬁeld and can be estimated with the equations presented in section 2.1.2.

55 2.1.4 Extractor Design

For electrospray thrusters, the extraction electrode can limit miniaturization. As pitch distance between emitters becomes smaller, the extractor gap must be reduced, both to limit space charge effects (in accordance with equation 2.5) and to prevent current interception. The latter becomes a significant concern when maintaining a one-to-one ratio of tips to holes in the extractor plate. A method of achieving higher densities is proposed in [37]. Other arrangements of multiple emitters within one aperture are possible while maintaining relative symmetry of the electric field. The beam half-angle can be conservatively assumed less than 25° at steady state, based on the angular dispersion characteristics of other architectures which operate in the pure ionic regime [34, 70, 25]. In fact, 15° is a more realistic estimate assuming a single emission site per tip. The relevant dimensions for extractor design are illustrated in figure 2-5.

H r a b G

Figure 2-5: Extractor Geometry.

To avoid current interception and related shorting failures, the non-impingement condition is enforced. Mathematically, this can be expressed by equation 2.42.

r > tan(θ ) (2.42) G + H b

56 In general, a smaller extractor gap G is beneﬁcial. However, reducing this distance increases the magnitude of the electrostatic pressure experienced by the extractor grid. The distribution is approximated by assuming a spatially uniform ﬁeld and using the equation for a parallel plate capacitor.

ε V 2 P = 0 (2.43) el 2 G2

While it is possible for deflection of the extractor grid to be a mode of failure, a more probable case is uneven emission across the array which has negative implications on thruster lifetime. To ensure spatial uniformity of tip fields, we seek to limit deflection to a maximum value of one micron. Equation 2.44 provides a simple approximation for this bending based on the assumption of a simple beam with a length equal to the array width and a thickness of H. In reality, this tends to be a conservative estimate but is useful as a first approximation. Before manufacturing, finite element modeling is used to solve for the static equilibrium of the final geometries.

PL4 δ = b (2.44) d 32 YH3

Substituting our electric pressure approximation into this general deﬂection equation, we obtain equation 2.45. As proposed in [38], total electric pressure is scaled by the inverse of the closed area fraction.

a ε V 2 L4 δ = 0 b (2.45) d 64 (a − 2r) G2 YH3

In nearly every case, the constraint on deflection is far more stringent than those related to material failure. However, equation 2.46 can be used verify that the von Mises stress is lower than yield strength. Note that this calculation is subject to the same assumptions as equation 2.45 and therefore should be verified numerically using some finite element analysis.

PL2 a ε V 2 L2 σ = b = 0 b (2.46) v 2 H2 4 (a − 2r) G2 H2

57 2.2 Surface Treatments

As explored in section 2.1.3, the impedance (and therefore most other relevant aspects of performance) of this emitter variety is heavily dependent on the surface permeability. Given the requirement of silicon substrate material, it may seem as though we have little control over this aspect. However, various surface treatments may be employed to obtain required physical characteristics.

2.2.1 Theory

The wetting characteristics of a surface depend on a balance of interfacial tensions between the solid, liquid, and vapor phases to achieve some low energy state. For an ideal surface, this equilibrium and the associated contact angle are described by the Young-Laplace equation (2.47).

− γSV γSL cos(θc, eq) = (2.47) γLV

Figure 2-6 illustrates the balance of these interfacial tensions for a perfectly smooth surface. Note that this is an idealized state as external conditions or hysteresis may cause some discrepancy.

c SG

Figure 2-6: Ideal contact angle measurement.

This θc value is often used to characterize the wetting of a chosen liquid and solid. Hydrophilic surfaces will allow the liquid to spread, causing a low contact angle. On

58 a hydrophobic area, surface tension forces dominate which leads to a "beading" eﬀect. Table 2.1 lists the commonly deﬁned range for each wetting regime.

Category Contact Angle Complete Wetting ∼ 0° Super Hydrophilic 0° – 5° Hydrophilic 5° – 90° Hydrophobic 90° – 150° Super Hydrophobic 150° – 180°

Table 2.1: Classiﬁcation of surface wetting.

While the contact angle of most ionic liquids on a bare silicon wafer (with native oxide coating) is on the order of 34°, complete wetting is required for porous surface flow. Not only does this guarantee that the microscopic emitters are not submerged, but it permits passive flow control through the impedance of a semi-permeable layer rather than allowing free surface flow. This was an assumption of the derivation presented in section 2.1.3. By introducing micro-scale roughness, the bulk wetting characteristics of a surface can be tailored to achieve our desired result. The nature of roughness itself is a complex issue and can be characterized in a number of ways. Common metrics include the arithmetic mean deviation, ten-point height, maximum roughness depth, reduced peak height, mean peak spacing, skew, and several others.

For a simplified analysis, we can use a surface roughness factor rr which represents the ratio of actual to vertically-projected surface area. Under this definition, rr = 1 would describe an atomically smooth surface. In 1936, Wenzel used this factor to predict a new effective contact angle for rough surfaces θc, W based on the assumption of changing area for the solid-liquid and solid-vapor interfaces [111].

cos(θc, W ) > rr cos(θc, eq) (2.48)

Wenzel’s equation (2.48) predicts that increased roughness will enhance the in- trinsic wetting behavior of a surface. Additionally, it implies a condition – shown

59 below in equation 2.49 – necessary for complete wetting [76]. This implies that any hydrophilic surface can be suﬃciently roughened to achieve complete wetting.

−1 rr > cos(θc, eq) (2.49)

For most ionic liquids on silicon, this condition is satisﬁed when the roughness parameter exceeds ∼1.2. In reality, small pockets of air can become trapped under a droplet thereby violating the assumption of this relation. To account for this eﬀect, the Cassie-Baxter equation (2.50) is used [16].

cos(θc, CB) = ∓1 + φs [cos(θc, eq) 1] (2.50)

Here the sign is dependent on wetting regime. If the drop volume is less than that the surface texture, complete wetting will occur. If not, the sign changes and the droplet is suspended on a composite interface [76]. Both the Wenzel and Cassie- Baxter models are illustrated in figure 2-7. Each of these states has associated stability criteria which is not presented here. Either external shocks or internal Laplace pressure at the composite interface can cause a transition from the Cassie-Baxter to Wenzel state [79]. Experiments have shown these relations to hold except very near the extremes of super hydrophobicity and complete wetting. As the former does not relate to electrospray design, it is ignored here. Under the condition described in equation 2.51, a process known as hemiwicking may occur [10]. This phenomenon, illustrated in the lower pane of figure 2-7, is a liquid film movement in a thin interfacial layer due to surface roughness. This creates the necessary conditions for externally wetted electrospray architectures as surface flow disappears and the semi-permeable region acts as a porous medium governed by a modified form of Darcy’s law.

∗ ∗ 1 − φs θc, eq > θc where cos θc = (2.51) rr − φs More recent studies have determined that this phenomena is only valid above

60 (a) Wenzel (b) Cassie-Baxter

Figure 2-7: Rough surface wetting models. some critical spatial density of roughness peaks [53]. Observing the diﬀusion rate of a droplet on the relevant surface can indicate whether this condition is met.

2.2.2 Techniques

With the necessary microﬂuidic theory established, the remainder of this section focuses on speciﬁc surface treatments relevant to a silicon-based electrospray design.

Porous Silicon

Given the signiﬁcant progress with porous emitter architectures over the past decade, replicating this surface structure seems a promising approach. In fact, several past studies have formed porous regions in a silicon wafer through a variety of electrochemical etches. Unfortunately, limitations appear quickly. A 1987 French study

61 explored hydrofluoric acid etching of anodized, doped wafers. While the targeted porosity (≈ 56% in order to minimize strain and expansion during formation of the native oxide) was achieved with accuracy and uniformity, pore sizes were on the order of single nanometers [45]. At this scale the resultant fluidic impedance would be far above the range useful for electrospray emitters. Other studies managed to create mesoporous silicon [98, 97], but the capillary forces at this scale are still higher than acceptable for our targeted emitter size. A study by Feyh [30] found that thickness of the porous layer was quite limited and uniformity suffered with longer etch times. These issues, along with the incompatibility of many traditional MEMS techniques with bulk porous substrates, caused this path of development to be abandoned in prior studies [26]. Recent progress has made this a more viable approach; however alternatives were chosen to minimize project risk. An extensive description of the formation mechanisms of porous silicon can be found in Smith [97].

Black Silicon

With bulk porous fluid transport not available, the wafer surface must be sufficiently roughened to allow complete wetting – thereby leveraging the hemiwicking phenomenon discussed previously. Several previous studies have demonstrated the effectiveness of black silicon in this role [109, 36, 110, 37, 38]. Black silicon (sometimes referred to as silicon nanograss) is not a specific surface treatment, but rather a general structure of small thin spikes. Typical widths are sub-micron. Figure 2-8 shows an example cross section, and a demonstration of hemiwicking. The texture derives its name from its extremely dark appearance; the sub-diffraction limit structure spacing and high aspect ratios lead to minimal reflection of visible light [62]. Further, the high surface area makes black silicon an attractive option for mitigation of electrochemical wear in architectures with a distal electrode. Past studies have shown both super hydrophobic behavior (179 1°) and complete wetting of silicon nanograss depending on the surface chemistry [28]. Both modes exploit Cassie-Baxter wetting by use of thin conformal coatings to set the Young contact angle above or below 90° depending on the desired result.

62 (a) Cross section (b) Hemiwicking of EMI-BF4

Figure 2-8: Generic black silicon morphology.

Many fabrication methods have been explored. First identified in the 1970s, black silicon was often an undesired byproduct of plasma etching in the presence of (often metallic) dust nanoparticles [93]. Antireflective coatings were the primary application until its potential for controlled wettability was realized. Today, two primary categories exist: structured and random. The former is more difficult and involves using an extremely high resolution lithographic mask to etch predictable features. The latter is simpler and cheaper, but less controllable and uniform. Typical processes use some halogen gas and often aluminum or tungsten particles. A more detailed history and review may be found in [93]. For electrospray research, only random black silicon has been used. In the early 2000s, Velàsquez-Garcìa used a Cl2 based manufacturing approach [109]. While this treatment allowed emission for externally-wetted silicon arrays, it was found to be inconsistent and highly anisotropic [109, 38]. For this reason, many later attempts to create silicon emitters used a SF6O2 based process developed extensively in Jansen [49] then adapted by Gassend [38].

Our ﬁrst attempts at manufacturing black silicon were based on the SF6O2 recipe. However, this process was found to have several limitations. The texture varied signiﬁcantly causing spatially non-uniform impedance and therefore emitted current. The highly directional nature of the process also caused concerns given the vertical sections that occurred near the base of our second generation emitters (illustrated

63 in ﬁgure 2-9). While a conformal tungsten coating provided enough roughness for a meniscus to reach above each "sidewall" section, wetting and therefore impedance was far from ideal. For this reason, a metal assisted chemical etching (MACE) process was ultimately chosen.

Figure 2-9: Anisotropic nature of plasma-based black silicon.

Metal assisted chemical etching is a relatively recent process for making black silicon. It involves direct deposition of metal particles or ﬁlms followed by a wet etch-

ing process in HF and some oxidizing agent (commonly H2O2). An electrochemical step occurs as a local current resulting from the oxidation process ﬂows between the metal nanoparticle and silicon surface (or even adjacent holes forming in the silicon). The surface texture obtained is highly variable and depends on a great number of factors including the: metal chosen, deposition method, particle size, etch time, ambient temperature, and reagent concentrations. A detailed explanation of the process and resulting morphologies can be found in Chartier [18]. For this study, aqueous

silver nanoparticles and an H2O2 solution were used. While this new process allowed etching of vertical surfaces, other issues arose. The texture was far from spatially uniform when applied to a processed wafer. The likely source of this problem was an

64 (a) Side view (b) Emitter base region

Figure 2-10: Spatial non-uniformities in initial MACE process.

(a) 3 minute etch (b) 30 minute etch

Figure 2-11: Controlling impedance with MACE time splits.

65 inhomogeneity of nanoparticle deposition. With the oxidizing agent present, current flow occurs prior to surface metalization thereby causing a local electric field and disproportionate accretion in regions without a sharp corner. The process also pref- erentially etched along primary crystallographic planes – sometimes causing squaring of the emitters. Both of these effects are illustrated in figure 2-10. Application of the silver nanoparticles prior to the acid bath, along with a few other changes, alleviated these issues. Removing the oxidizing agent, deposition becomes an electroless process and occurs evenly regardless of surface geometry [18]. Having established an effective procedure, the etch time was varied to obtain the desired nanograss height and spacing for proper impedance. Figure 2-11 shows the effect of etch time on the tip region of an emitter.

Oxidation

The final surface treatment relevant to this design is oxidation. To date, very few electrospray thrusters have undergone extensive lifetime testing. Because of the high extraction voltages required for this propulsion method, electrochemical degradation is a concern. Along with array shorting (typically the result of either electrical discharge or beam interception), this process is the predominant life-limiting mechanism [14]. Prior work revealed that use of voltage alternation and distal electrode significantly reduce long-term wear [13]. Together, these mitigation strategies have allowed tests lasting 100s of hours without signs of deterioration [13, 27]. However, certain materials exhibit rapid electrochemical wear even with these measures in place [25]. Apart from being a function of the specific chemistry between propellant and substrate, manufacturing emitters with either high surface area or a non-conducting material would minimize the charge double layer depth and ensure ohmic voltage drop remains below the electrochemical window limit [13]. Thus a dielectric material is preferable. The current state-of-the-art electrospray emitters use a glass substrate (significantly less conductive than silicon) and have achieved ∼1000 hours of emission. Passivation of silicon is commonly achieved using conformal oxide coatings. These provide a higher resistance than the thin, SiO2 native oxide which forms naturally

66 upon exposure to the environment. Unfortunately, complete electrical insulation of the substrate would require an oxide thick enough to impede the super-hydrophilic wetting achieved. A thinner coating, however, would still slow (if not completely stop) electrochemical wear. Thus a balance must be found. Figure 2-12 shows this thermal oxide formed over silicon nanograss. Layer thickness is on the order of tens of nanometers. Note that oxidation is only relevant to architectures with a distal electrode. Externally wetted arrays typically must be biased directly, thus a dielectric coating would prohibit operation.

Figure 2-12: Thermal oxidation of silicon nanograss [1].

The author would once again like to acknowledge and thank the WaferSat team at MIT Lincoln Laboratory for implementing and reﬁning the aforementioned surface treatments. Developing procedures to achieve consistent results was far from a trivial accomplishment.

67 2.3 Prototypes

The thrusters in this project experienced several design iterations. This section details the intended results, proposed geometries, and actual products. The shortcomings of each iteration are discussed to motivate the chosen progression.

Propellant Choice

The ﬁrst design choice to be made is propellant selection. In this study, 1-ethyl-3-

methylimidazolium tetraﬂuoroborate (EMI-BF4) is used. Its various physical properties at standard conditions [112] are listed in table 2.2.

M (g/mol) ρ (kg/m3) µ (kg/m s) γ (kg/s2) K (S/m) ε (−) 197.97 1270 0.0038 0.050 1.36 10

Table 2.2: Physical properties of EMI-BF4.

EMI-BF4 has become a common propellant choice for electrospray thrusters in recent years. It has a relatively large and symmetric electrochemical window, reducing long-term wear and thereby improving lifetime. The high charge to mass ratio leads to correspondingly large specific impulse at relatively low voltages. It has zero vapor pressure and is stable over a wide temperature range. Its tendency to supercool down to approximately -70°C is beneficial to satellite survivability. Even if solidification occurs, it can be heated above 15°C to liquefy with minimal adverse effects [82]. Evaporation becomes possible at very high temperatures (above ∼200°C), however the rate is slow leading to minimal propellant losses. Dissociation occurs at an even higher temperature. With a range of thermal stability wider than that of most other satellite subsystems, these considerations are rarely necessary. Other thermal effects are relevant. Viscosity and conductivity both increase slightly at higher temperatures – which can result in a performance boost. How- ever, the increased current level seldom offers enough of an advantage to artificially heat arrays as the energy dissipated is typically larger than what would be gained in thrust. Due to the relatively high thermal conductivity of silicon in comparison

68 to typical substrates, heating only propellant on the array surface could potentially increase efficiency. A thorough analysis using the final geometry would be necessary for validation. At higher temperatures, electrochemical degradation would also occur at a slightly increased rate. Since impedance is inversely proportional to viscosity, running at these conditions has the potential to trigger mixed-mode operation rather than PIR (although the increasing conductivity would have the opposite effect). Surface tension is nearly constant with respect to temperature; thus the extent (although not the rate) of wetting is relatively unaffected. Ionic liquids also tend to increase in volume as they heat up. Any fuel tank must be designed with extra volume and venting to accommodate this expansion without flooding the array. Cold operation presents more significant concerns. Below about -5°C, current levels drop significantly [82]. This effect is dependent on architecture and has been only briefly explored in literature.

Finally, extensive prior work with EMI-BF4 motivated the decision to use it in this application. Although a signiﬁcant number of ionic liquids have been tested, only a small proportion have achieved PIR operation [85, 77]. Of that group, many ILs required either high temperature or active ﬂow control to move out of the mixed

emission regime. EMI-BF4 was the propellant to achieve PIR operation with an externally-wetted architecture [70] and did so in both polarities. Further, the propellant has demonstrated promising emission characteristics with silicon. In 2006, Velásquez-García et al. demonstrated use of black silicon to provide surface roughness and impedance with EMI-BF4 [110]. Shortly thereafter, Gassend et al. achieved PIR emission with the same liquid and a similar externally-wetted silicon array [37]. Later, Hill et al. attempted to tailor impedance levels by using a conformal carbon nanotube forest in place of black silicon. While emitter degradation occurred relatively quickly and transient droplet emission was observed, this project also achieved ionic operation [46].

EMI-BF4 is also amenable to unipolar operation. Typically, this is avoided as it leads to spacecraft charging and a speciﬁc impulse decrease. However, this allows use of a simpler (and much smaller) power supply – an important consideration for the

69 space-limited WaferSat bus. Since only a portion of the propellant mass is accelerated, eﬀective speciﬁc impulse is not longer equivalent to exhaust velocity and must be adjusted using equation 2.52.

√ ( ) ( ) ( ) 1 q m− m− I = 2 V = I (2.52) sp, eff g m− M M sp, nominal

+ Since the EMI cation weighs 111.2 amu and the BF4- anion only 86.8 amu, the effective specific impulse is reduced by just above half. (Note that this simplified analysis assumes exclusively monomer emission and equal charge transport by positive and negative ions.) The cations accumulate at the distal electrode and decompose. Operation in the negative polarity allows mitigation of spacecraft charging. For missions in LEO, ambient electrons are sufficiently plentiful and much more mobile than the heavier ions. As the satellite accumulates positive charge, they become attracted to and neutralize the platform.

2.3.1 Design Iterations

Capillary Architecture

The ﬁrst geometry examined was internally wetted. Most recent successes in electrospray development have used a porous geometry, with the characteristic pore size providing the proper impedance for ionic emission. Unfortunately, this avenue is not realistic for the chosen application. While MEMS processes exist to create pore-like structures near the surface of a silicon wafer, they are not easily isolated or spatially uniform. However, micro-manufacturing capabilities have improved since previous generations of silicon thrusters were developed, oﬀering new hope for capillary-based success. To date, no capillary design has achieved passive PIR emission at room temperature. Romero-Sanz et al. demonstrated PIR operation from a capillary emitter with

EMI-BF4 in 2003 (the ﬁrst to do so), but relied on active ﬂow control to increase hydrostatic pressure. The 30 cm long capillaries provided excessive impedance and

70 clogged easily. While the lowest flow rate (on the margin of stability) provided 232 nA of ionic current, all other conditions operated in the mixed regime [90]. Subsequent testing by the same group led to PIR emission of various other ionic liquids (mostly at increased temperatures). All however, relied on active flow control [91]. Soon thereafter, Garoz and a group at Yale successfully tested a litany of other ILs with mixed results (some purely ionic, others with significant droplet current fractions) [35, 17]. However, the team’s 50µm diameter needle was far too wide for passive flow control. The study did determine the benefit of high conductivity and surface tension ILs for PIR emission at room temperature.

In an attempt to increase impedance, researchers at EPFL in Switzerland manufactured silicon capillaries which were then filled with silica microspheres. While largely successful – droplet-free current was observed at high voltages[59] – there were several flaws. Emission was somewhat unstable and prone to clogging, fabrication inconsistent, and beam spreading significant [56, 58]. Additionally, the design still required hydraulic pressure to control flow[58, 92].

From previous research, it would seem that capillary size is the limiting factor to passive PIR emission. Using equation 2.36, we can derive an expression for the aspect ratio necessary to achieve a given ﬂow rate.

( ) 1 4µLQ 3 r = (2.53) πγ cos(θc)

This provides a family of geometries that would result in sufficient impedance to constrain the non-dimensional flow rate (ηf ) to slightly less than unity (corresponding to the flow speed of previous experiments employing active control). A plot of these aspect ratios is shown in figure 2-13.

The wafer used was 150 µm thick, constraining the length parameter. Two radii were chosen: 2 µm and 0.5 µm. The larger would be expected to operate in the mixed mode, while the later design had suﬃcient impedance for passive PIR operation. The high aspect ratio capillaries were achieved by depositing successive layers of conformal silicon after performing a through-silicon via (TSV) etch. A simpliﬁed illustration

71 1.5

0.5

0 0 100 200 300 400 500

Figure 2-13: Dimensions needed for PIR emission. and SEM image of this architecture are shown in ﬁgures 2-14a and 2-14b respectively. Note that an additional etch is required to re-open the reduced diameter pore after conformal deposition of the silicon. This unfortunately creates a basin for potential propellant accumulation, which would prevent proper anchoring of the Taylor cone.

Despite manufacturing success, experimental results were disappointing. The array emitted short spikes of current before quickly shorting. The basins formed at the capillary opening were hydrophilic and commonly flooded – precluding the formation of any Taylor cone anchored to the face. The bursts of current were likely the result of propellant cavitation within the pores. At such low flow rates and high impedances, the pressure drop across the capillary is significant. Ionic liquids have a tendency to readily absorb ambient gases such as carbon dioxide or water vapor. Under a strong pressure gradient, these can cavitate to create bubbles within the propellant. Not only does this interrupt flow and destabilize any Taylor cone present, but its expansion accelerates the fluid above. Given the small extractor gap, expelled propellant is likely to short the entire array. This failure mode becomes statistically more probable with high emitter densities. Figure 2-15 illustrates this progression. While unfortu-

72 Ionized Plume

Extractor Electrode

Va Oxide Layer

Conformal Silicon

Bulk Silicon

Propellant

(a) Nominal architecture.

(b) SEM cross section of an emitter [1].

Figure 2-14: Initial capillary design.

73 nate, this data conﬁrms trends seen in other papers regarding the stability of capillary electrospray architectures [25].

time

Figure 2-15: Cavitation related shorting in capillary style emitters.

Externally Wetted Architecture

With significant roadblocks towards integration of capillary or porous architectures into a silicon substrate, an externally-wetted variant was examined. Such arrays have shown significant promise in past studies, and several have been manufactured in silicon. In 2006, a group led by Manuel Martínez Sánchez at MIT fabricated an array of externally-wetted silicon emitters of various shapes and pitch densities which performed well [110]. Within a few years, a continuation of this project achieved PIR emission at steady state [37, 38]. Future iterations explored other methods of surface roughening (conformal carbon nanotube forests) and higher pitch densities [46]. While thruster longevity was a serious issue, the studies were quite successful overall. Nevertheless, the point remains that such an architecture does not constitute a viable thruster given the lack of propellant replenishment to the array surface. To remove this flaw, capillaries were introduced to the design. These channels could

74 passively transport propellant from an external tank to the array. Surface roughness between the capillary exit and emission region would provide the required impedance for stable ionic operation – thereby loosening the restrictions on pore size. Larger capillaries would be less susceptible to cavitation, and an interruption in the flow of one would no longer create a critical flaw. To mitigate risk, an externally-wetted array was first created and tested. With steady ionic emission first verified using this simple architecture, troubleshooting the likely issues of a complete system would be easier due to having fewer variables.

r h r

- b -

Figure 2-16: Nominal geometry of conical emitter tips.

Based on past studies, the conical emitter proﬁle shown ﬁgure 2-16 was adopted.

The value ϵr is a parameter which approximates surface roughness. Past work has shown a relation between the cube of this characteristic length and the surface permeability coefficient (κps) used in equation 2.41 [38]. As discussed in section 2.1, there are several critical design considerations. The radius of curvature influences meniscus shape, and therefore both startup voltage and slope of the current-voltage curve. To sustain ion emission at voltages below 1 kV , a radius on the order of single microns is ideal. A sharper tip may encourage electron emission while a blunter one would require larger electric fields. Pure ionic emission has not been demonstrated

75 17 with impedances below ∼ 1.5 · 10 kg/m4s [85]. For passive flow, emitter height and width must be sufficient to provide this value using surface roughness according to equation 2.41. Fluidic impedances above this threshold will reduce performance by limiting the maximum flow rate, and by extension both beam current and thrust (although spatial uniformity across the array may improve somewhat) [22, 46]. Because of the lack of experimental success with impedances far above this threshold, it has been postulated that a ceiling exists above which emission is choked altogether [55]. Using this method, we find that height and base width of 100 µm and tip curvature of 5 µm is an acceptable geometry given our expected surface roughness (ϵr ∼ 1 µm). We vary each of these three key parameters slightly to assess their relative effects on emitter performance. This results in seven variants – the characteristics of which are shown in table 2.3. Finally, the spatial density of emitters is an important variable for thruster performance. Pitch distances of 200 µm and 400 µm were chosen. Single emitters were included as a control group to test for any negative effects of densification.

Variant Rc (µm) h (µm) b (µm) θ (°) A 5 100 100 25 B 2.5 100 100 26 C 7.5 100 100 25 D 5 100 75 19 E 5 100 150 36 F 5 50 100 43 G 5 150 100 18

Table 2.3: Nominal dimensions of second generation emitters.

The corresponding impedances of these designs are shown as a function of surface roughness in figure 2-17. Note that this is a conservative estimate based on Poiseuille flow and neglecting any impedance provided by the base of the array. After manufacturing, significant discrepancies between the nominal and actual emitters were revealed. SEM images of the various tips (after tailoring the etching process) are shown in figure 2-18. First, the presence of steps is apparent. Most are

76 10 20

10 18

10 16

10 14

2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 0.1 1 10

Figure 2-17: Estimated impedance of second generation emitters.

on the order of 2 µm in size and result from a resolution limit of the lithographic mask. The first step is several microns tall (an effect of the decreasing etch rate) and affects the flow noticeably. At the beginning of test, extra propellant tends to accumulate at this inside corner, forming a meniscus which allows external rather than porous transport and reduces effective impedance. The sharp outside corner of each step can be difficult for fluid to cover. Additionally, these corners affect surface roughening as will be discussed in section 2.3.3. The tip radii of curvature also differ from nominal values. In fact, variants A, B, and C all exhibit nearly the same ∼10 µm curvature. The higher aspect ratio emitters (types D and G) have sharper tips while their shallower counterparts (E and F) have blunter ones. All emitters ended up slightly smaller than expected in both height and width. This was a significant issue for variant F, which ended up too short for testing altogether. However most high-level trends between the emitter types still hold, allowing down-selection for the final architectures based on the emission data collected (a summary of which will be presented in the following chapter).

As this was an intermediate design only intended to test emission stability, an

77 (a) Type A emitter (b) Type B emitter

(e) Type E emitter (f) Type G emitter

Figure 2-18: Actual proﬁle of second generation emitters.

78 integrated extractor was not created. Instead, a flat metal plate was used as illustrated by figure 2-19. Since the ion beam is completely intercepted by the extractor, RPA and ToF testing are not possible; only current voltage response was recorded. Additionally, only the negative mode was tested to allow suppression of secondary emission (although since the final variety will employ unipolar operation and use ambient electrons for neutralization, this is not an issue).

Figure 2-19: Externally wetted architecture with plate electrode [1].

2.3.2 Final Geometry

As mentioned previously, the final array architecture combines the passive flow control of an externally-wetted array with propellant supply via capillary transport. A representation of this hybrid geometry is shown in figure 2-20. A distal electrode offers substantial mitigation of electrochemical wear, even without voltage alternation.

79 Figure 2-20: Final array architecture [1].

Given the excellent emission characteristics of the second generation emitter, we seek to minimize the fluidic impact of adding capillaries. For stability and spatial uniformity of emission, the pressure drop over an individual tip should dominate over that of other sections in the propellant’s path [42]. Although this is less significant for our architecture than most porous designs, it would mitigate the effects of flow disruption in a single capillary. Since the ratio of emitters to capillaries is one, this condition can be expressed as Ztip ≫ Zpore. Note that neglects the region between pores and the adjacent emitter base; this distance is small in our design and thus the effect of this assumption is mild. To ensure conformity, capillary diameter is increased

80 to 30 microns (and the associated impedance is therefore more than two orders of magnitude lower than that of the emitter). Figure 2-21 shows the relationship between capillary size and its proportional eﬀect on total impedance. The red bands indicate uncertainty due to inexact knowledge of the surface permeability coeﬃcient.

10− 1

10− 2

10− 3

10− 4

10− 5

10− 6 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 1 10 100

Figure 2-21: Sizing of capillaries for emission stability.

The relative success of high aspect ratio emitters during testing of the externally wetted geometries led us to include only type D and G tips in this design iteration. A slight modification to the tip region ensured that the actual radius of curvature would be closer to the nominal value of 5 µm. To mitigate the large size of the lowest step seen in the previous design, a raised region was created on the array surface. Table 2.4 summarizes the chosen spatial densities included in this third generation design. The following three figures display this final architecture at increasing magnifica- tions. Figure 2-22 shows an entire type D array. To the author’s knowledge, this is the highest spatial density of functioning electrospray thrusters ever manufactured. Figure 2-23 gives a closer view of a high density type G array. At this magnification,

81 Emitter Pitch Distance Number of Spatial Density Variant (µm) Emitters (tips/cm2) D – 1 1 G – 1 1 G 450 810 493 G 127 729 6,200 G 127 2916 6,200 D 64 3025 24,414

Table 2.4: Emitter arrays for ﬁnal geometry. the capillaries and emitter proﬁle are visible. Figure 2-24 shows the tip radius of curvature of a type G emitter. Note that its sharpness can be adjusted within a range by small changes in the grayscale etching process. All pictured emitters have black silicon surface texture resulting from a MACE process.

Figure 2-22: Final architecture – dense D emitter array.

82 Figure 2-23: Final architecture – dense G emitter array.

Figure 2-24: Final architecture – tip radius of curvature.

83 Extractor Design

Design of the extractor electrode requires a few initial assumptions. Beam half angle is taken to be 20°, a conservative estimate based on prior research. To account for the possibility of oﬀ axis emission, an additional 5° is added. Initial optical alignment attempts indicate a tolerance on the order of single microns; thus a horizontal error of 5 µm is assumed. The current design seeks to minimize the extractor gap, with an estimated 5 µm vertical tolerance deemed acceptable. For a given aperture radius, we can use the previous assumptions to calculate a maximum extractor thickness before beam interception becomes likely. This expression is equation 2.54.

r + 5 hmax = −5 + ◦ (2.54) tan(θb + 5 ) The resulting thickness parameter was used to create a CAD model of the extractor. Validation of this geometry was performed with a COMSOL Multiphysics® static stress analysis. For deflection calculations, electrostatic pressure was assumed to be uniformly distributed evenly throughout the emission region (i.e. the array geometry is neglected). Despite this parallel plate assumption, the even distribution of emitters and application of an equivalent total field should provide sufficient accuracy in cases of limited deflection; a significant sag would cause noticeable increase to the pressure magnitude. For this reason, each of the three critical conditions were examined with allowable deflection limited to one micron. The type D array exceeded this threshold even at the maximum thickness. After support posts were moved closer to the emission region deflection was decreased to an acceptable value. Table 2.5 shows the results of this analysis.

Emitter Pitch Aperture Maximum Minimum Type Distance Diameter Thickness Deﬂection D 64 38 25.0 1.164 G 127 90 80.7 0.106 G 450 90 80.7 0.032

Table 2.5: Extractor dimensions (in microns).

84 For the more sparse type G arrays, initial deflection was well below the critical value. To minimize beam interception risk and ease alignment constraints, the array thickness was further reduced. Figure 2-25 shows simulated worst-case deflection as a function of extractor thickness for the 127 micron pitch type G array. The green region illustrates the range of acceptable performance. As a result of this analysis, a 50 micron extractor thickness was chosen for all type G arrays. Predicted deflection of the 127 and 450 µm cases is 0.181 and 0.737 µm respectively. The extractor electrodes themselves are etched silicon wafers with a thin oxide coating.

0 30 40 50 60 70 80

Figure 2-25: Simulated deﬂection of tight pitch G array.

2.3.3 Fabrication Process

Fabrication of all emitter variants was performed by members of the WaferSat group at MIT Lincoln Laboratory. Emitter tips were formed using grayscale lithography and a 200nm resolution mask. Details on the fabrication process can be found in Smith et. al [96]. The black silicon surface treatment was achieved with aqueous deposition of silver nano-particles and a wet acid etch. Several time-splits of this process were

85 created to ﬁnd an optimal surface roughness and corresponding ﬂuidic impedance (just above the threshold necessary for pure ionic emission). Between etching of the emitters and surface roughening, capillaries were created. To avoid causing damage to surrounding tips, these pores were etched from the bottom surface of the wafer. The capillaries taper from slightly above their nominal diameter at the wafer base to about 20 cm on the array surface. Both emitter shape and pore diameter are weak functions of spatial position.

86 Chapter 3

Results and Discussion

This chapter discusses the performance of our novel electrospray design. Due to current events and an unexpected shutdown of testing facilities, a complete characterization of thruster performance (to include retarding potential analysis and time of flight mass spectrometry) has been delayed and will not be included in this work. Microfluidics of all design iterations is presented here, along with electrical testing of second generation emitters. Inferences of the primary operating regime are made based on qualitative metrics. Full characterization of the final architecture will be included in a subsequent paper.

3.1 Wetting

After several iterations of black silicon process development (discussed with more detail in section 2.2.2), a relatively uniform surface texture capable of complete wetting was achieved. Further, the electrochemical etch step was brief enough that emitter proﬁle – especially tip curvature – remained unaﬀected. While the black silicon morphology was nearly ideal, two primary obstacles to consistent wetting emerged. First, applying the proper quantity of propellant proved crucial for the externally wetted design. Because the array itself is recessed, extra liquid accumulates in the emission region and violates the condition for complete wetting as expressed in equation 2.49. Excess propellant tends to form wide menisci anchored at the top corner

87 of the emitters’ large lower step. With enough volume, adjacent menisci join and provide a nearly impedance-free region on the array ﬂoor. Using the approximate surface area of the emission region and height of black silicon spikes, we estimate the proper volume of propellant. After some experimental validation, 70 nL per array is chosen.

For similar reasons, transient effects became a significant obstacle to consistent results. While anisotropic silicon nanograss growth is essential to wetting vertical surfaces of the emitter, it also provides texture to array sidewalls sufficient to allow spreading of propellant outside the emission area. Fortunately, this is a temporary problem. Future design iterations would include a hydrophilic oxide hard mask to contain propellant on the array. This solution, however, would requires a new lithographic mask and thus could not be implemented during that testing iteration. The primary effect of this flaw was the spatially inconsistent wetting shown in figure 3-1. Due to the excessively small propellant volumes used, distributed application proved

nearly impossible. The EMI-BF4 droplet was applied at the array center and gradually spread outwards. This led to pooling of the excess liquid in the center (allowing a low-impedance pathway via external surface flow) while tips near the edge of our array remained nearly dry. The top pane of figure 3-1 shows the spread of EMI-BF4 from top to bottom. Menisci at the base of emitters decrease impedance significantly. The lower left pane shows the tip of an emitter in the overwet region. It’s underwet counterpart can be seen in the adjacent panel. This significant difference in impedance causes spatially non-uniformity of flow rates. With enough excess propellant, a local transition to mixed-mode emission occurs.

Typically, this would be avoided by waiting for steady state. With the extent of propellant spread unlimited, however, an equilibrium condition is not achieved. The liquid continues hemiwicking outwards beyond the array edges. The issue is illustrated by ﬁgure 3-2. Both SEMs show the same emitter. The image on the left illustrates proper propellant coverage and was taken less than an hour after wetting. The right side shows propellant coverage twenty hours later. Between the two images,

EMI-BF4 continued to spread, leaving tips within the testing region under-wet. With

88 (a) 200 µm pitch thruster array ∼1 hr after propellant application

(b) Overwet emitter tip (c) Underwet emitter tip

Figure 3-1: Spatially inconsistent wetting of second generation arrays.

89 the remaining propellant closer to the sharp grooves at the base of the black silicon, capillary forces between spikes is larger and impedance is noticeably increased. Thus, discrepancies in wetting time must be accounted for when comparing of emission characteristics of diﬀerent arrays.

(a) Proper propellant coverage (b) Lack of propellant

Figure 3-2: Transient wetting of second generation arrays.

The inclusion of capillaries in the third generation architecture largely eliminated concerns about uniform wetting and excess propellant. The passive supply not only promotes spatially consistent impedance, but eliminates the possibility of over-wetting (assuming no external pressure gradient). Further, the large lower step seen on second generation emitters was eliminated. Without this sharp outer corner as an anchor, large (or connected) menisci and their associated low impedance regions did not form. Step size in general was reduced to the same order as surface roughness height; this ensured consistency of impedance and flow rate. The nearly ideal wetting of a third generation emitter array is shown in figure 3-3. The white specks in this image are excess aluminum nanoparticles from the MACE process. Slight adjustments to that procedure mostly eliminated these contaminants from future samples. Complete wetting with oxide coated samples proved slightly more difficult. The conformal layer reduced surface roughness as its thickness approached the nanograss spacing. Thinner oxides wet properly, but thicker layers appeared to cause a hydrophobic transition. This effect is seen in figure 3-4 where a drop of EMI-BF4 was

90 Figure 3-3: Passive wetting of ﬁnal array architecture [1].

Figure 3-4: Wenzel type wetting of oxidized black silicon emitters.

91 applied to the surface of a heavily oxidized type G emitter array. The exposed area of oxidized black silicon spikes became suﬃcient for Cassie-Baxter wetting (condition 2.51). However, the sharpness and spacing of emitters themselves causes a sig- niﬁcantly lower macro-scale solid area fraction (φs) than at the micro-scale – thereby allow Wenzel type wetting between emitters. Samples with thinner oxide coating did not exhibit this behavior. Due to inclusion of a distal electrode in the third generation architecture, minimal electrochemical wear was observed during initial tests of unpassivated silicon emitters. (This result will be discussed with further detail in the section 3.2.3.) As a result, exploration into modifying black silicon to achieve complete wetting with thicker oxide layers ceased.

92 3.2 Electrical Testing

As mentioned previously, electrical tests to date have used a solid extraction electrode. With accelerated ions unable to escape the emission region, retarding potential analysis and time of flight testing are not possible. This section presents initial current voltage response characteristics of second and third generation emitters. Note that the tip-to-extractor gap of the solid electrode was approximately 200 microns. For the integrated extractor that will be used in subsequent testing and satellite operation, this value drops to below 10 microns. This both introduces variability to our results and increases the required voltage by a factor of approximately 2.5 (from equation 2.8). Due to significant flaws discussed in section 2.3.1, the first generation capillary design never achieved steady state operation. In fact, the longest recorded emission prior to array shorting was approximately 4 seconds.

3.2.1 Second Generation Emitters

As mentioned in section 3.1, inconsistency was the most significant issue seen in electrical testing of second generation emitters. Of the six tip variations tested, the two high aspect ratio emitters (types D and G) produced the most consistent results. This can likely be attributed to a combination of two factors. First, these geometries had a tip radius of curvature close to the nominal 5 micron value. Other types had significantly blunter tips and thus required higher electric fields. Second, the steeper angle resulted in thinner steps and therefore more consistent flow rates. While steady emission was achieved with all emitter variants, results shown in this subsection (except figure 3-5) are all from type G tips to allow direct comparison to data in the following subsection. Note that the non-uniform wetting caused different numbers of emitters to fire for each test. Typically, a post-test SEM image or examination of the extraction electrode allows determination of this quantity. Figure 3-5 shows one of the cleaner current-voltage profiles collected from a single type D emitter. Consistent with many past experiments of electrospray operation in pure ionic mode (and the simplified model illustrated in figure 2-3) the curve is

93 0 time [min] −200

] 2

A −400 n [

n 1.5

e −600 r r u

c 1 −800

0.5 −1000

0 −1200 −1 −0.75 −0.5 −0.25 0 voltage [kV]

Figure 3-5: Current-voltage curve of a 2nd generation type D tip. composed of an initial exponential rise followed by a nearly linear increase after about -300 nA. Current levels of up to 1 µA were achieved. Most other IV curves had a more jagged appearance. Holding a specific voltage for extended periods results in drops from the rapid electrochemical wear to the tip region of this design. For the same reason, noise commonly increased throughout testing. A short, continuous voltage sweep typically alleviates this issue. An example of this current decay and noise increase can be observed in figure 3-6. After an initial spike, decreasing propellant supply causes a decay to some steady state level. Over the next several minutes, current fluctuations increase in both magnitude and frequency. This "overwet phase" prior to current stabilization was observed in some form during most experiments. Occasionally, it would manifest a period of simultaneously high current magnitude and fluctuation as shown in figure 3-7. Since corresponding extractors more commonly had propellant present after testing, we conclude that this phase likely consists of droplet emission. After establishing stricter controls on propellant quantity and wetting time, emission consistency drastically improved. Figure 3-8 shows two current-voltage curves

94 1000 voltage 0 [kV]

2.5

] −1000 A n

[ 2 t

n −2000 e

r 1.5 r u c −3000 1

−4000 0.5

0 −5000 0 2 4 6 8 time [min]

Figure 3-6: Transient current response of a 2nd generation dense type G array.

0 voltage [kV] −20k 2.5 ]

A −40k n 2 [ t n e r −60k 1.5 r u c 1 −80k 0.5 −100k 0 0 2 4 6 8 time [min]

Figure 3-7: Overwet phase of a 2nd generation dense type G array.

95 1000 time 0 [min]

] −1000 3 A n [ t

n −2000 e

r 2 r u c −3000 1 −4000

0 −5000 −3 −2 −1 0 voltage [kV]

(a) Sample A

1000 time 0 [min]

1.5 ] −1000 A n [ t

n −2000

e 1 r r u c −3000 0.5 −4000

0 −5000 −3 −2 −1 0 voltage [kV]

(b) Sample B

Figure 3-8: Consistent performance of 2nd generation type G emitters.

96 from nearly identical type G arrays (samples from the same wafer with adjacent MACE time splits). Post-test analysis indicated that nearly 20 tips ﬁred in both experiments. Not only do absolute current levels and slope closely match, but the turn-on voltages are nearly identical. Finally, second generation arrays achieved steady state operation at the targeted current levels. Figure 3-9 shows the tail end of data from a dense type G array. Later SEM images of this sample indicate that approximately 75 tips were ﬁring.

Figure 3-9: Steady state current of a 2nd generation dense type G array.

nA This corresponds almost exactly the 300 tip targeted in the design process. While tip degradation occurred quickly for these tips (discussed further in section 3.3) select tips sustained operation for approximately one hour. At the point consistent and acceptable emission was achieved, modiﬁcation of second generation arrays ceased and focus shifted to the design and testing of our third generation architecture.

3.2.2 Third Generation Emitters

To date, electrical experiments on third generation emitters has been limited to current-voltage response tests. With ﬁnal extractors still being produced at the time

97 of writing, these tests employed the same solid plate electrode as previous generation arrays. Initial results from this redesign were very promising. Steady and repeatable emission was achieved using all six array types described in table 2.4. Short term current fluctuations were noticeably lower than seen with second generation arrays, and startup voltages between samples more consistent. Extractor patterning indicated ionic emission – an effect that will be discussed in the following subsection. Compared to the externally-wetted architecture, third generation arrays showed very little signs of tip degradation in the current profile (a trend that was validated by post-test analysis). However, one significant flaw is glaringly apparent from initial tests; there is a rapid current decay in the first few minutes of thruster operation.

Figure 3-10 illustrates this phenomenon. The transient current response and IV curve of a dense type G array are shown in the top and bottom panels respectively (sub-ﬁgures 3-10a and 3-10b). While this is one of the less stable datasets collected, it show both the severity of initial decay and the reasonable current levels initially recorded. Further, this ﬁgure demonstrates a long-term stabilization at lower current levels.

Further investigation into this trend revealed two likely culprits – electrochemical degradation or system ﬂuidics. While third generation tips experienced relatively little visible wear, eﬀects on the distal electrode had not been previously examined.

When used with EMI-BF4, certain metals experience electrochemical reactions that can passivate portions of the surface (an eﬀect that might normally be mitigated through voltage alternation). This motivates use of materials with a high surface area or large electrochemical window [100]. While second generation emitters did not use a distal electrode, an aluminum wire directly contacted ionic liquid in third generation tests. To determine whether charge transport at this solid-liquid interface was the current limiting mechanism, electrical tests with other electrode materials were performed.

Figures 3-11 and 3-12 show transient responses and current-voltage curves for two tests of single type G emitters. The former used platinum wire for a distal electrode while the latter employed a carbon xerogel design developed previously in the MIT

98 (a) Transient current response

0 time [min] −5k 50

] 40

A −10k n [ t

n 30 e −15k r r u c 20 −20k

10 −25k

0 −3 −2 −1 0 voltage [kV]

(b) Current-voltage proﬁle.

Figure 3-10: Electrical testing of a 3rd generation dense type G array.

99 Space Propulsion Lab for porous architecture electrospray thrusters. Both materials are known to perform well with EMI-BF4, but the additional surface area of the latter increases longevity.

The platinum wire test resulted in behavior extremely similar to that of prior experiments. Initial current levels increased to the low end of the expected range (but still on the same order as second generation tests). However, a high rate of decay reduced the current to about 2 nA within 30 minutes. The carbon xerogel test had a higher starting voltage, but similar initial slope of the current-voltage response. Decay was significantly slower (appearing nearly linear rather than polynomial or logarithmic), but the peak value was also lower by a factor of 5. By the 30 minute mark, both tests had settled to nearly the same current levels. When normalized, ∼ nA this 2 tip was close to the value seen in testing of full arrays. The extremely limited effect of differing electrode material on normalized current, along with the similar decay rate of full arrays and single emitters led to the conclusion that electrochemical degradation was not the limiting mechanism.

With electrochemical effects rejected as an explanation for the observed current decay, we turned our attention to system fluidics. The introduction of capillaries added some impedance relative to second generation arrays. From figure 2-13, the magnitude of this change should be low enough to avoid perceptible changes. How- ever, it was suggested by Lozano [83] that a low open area fraction (the ratio of total pore area to solid material at the plane of the array base) could lead to limited flow rates from the propellant reservoir to the array surface. This situation would be fixed by creating a larger number of small pores to allow more open area at a similar total impedance. However, the results of single emitter testing later eliminated this possibility. While all third generation arrays were designed with an equal number of capillaries and tips, single emitters had four propellant supply channels. The relatively consistent post-decay current levels of all array types indicate that the open area fraction is not a flow limiting mechanism.

The current hypothesis is that inadequate roughening on the back side of the wafer prohibits propellant replenishment to the array surface. Current experimental

100 0 voltage −10 [kV] 2.5

] −20 A

n 2 [

n −30 e

r 1.5 r u c −40 1

−50 0.5

0 −60 0 10 20 time [min]

(a) Transient current response

0 time −10 [min]

] −20 20 A n [

n −30 15 e r r u c −40 10

5 −50

0 −60 −3 −2 −1 0 voltage [kV]

(b) Current-voltage proﬁle

Figure 3-11: Electrical testing of a 3rd generation single type G emitter using a platinum distal electrode.

101 0 voltage [kV]

2.5 ]

A −5

n 2 [

t n e

r 1.5 r u c −10 1

0.5

0 −15 0 20 40 60 time [min]

(a) Transient current response

0 time [min] 60

] 50

A −5 n [

40 t n e r

r 30 u c −10 20

0 −15 −3 −2 −1 0 voltage [kV]

(b) Current-voltage proﬁle

Figure 3-12: Electrical testing of a 3rd generation single type G emitter using a carbon xerogel distal electrode.

102 procedure involves attaching an ionic liquid-soaked piece of qualitative filter paper to the bottom surface of the array. This permeable material is contains the propellant and allows electrical connection without biasing the silicon substrate directly. After attachment, an extractor is fixed to the top surface and the module is put under vacuum for testing. If sufficiently roughened, the bottom surface of the wafer should absorb ionic liquid because of the porosity pressure gradient. This surface layer then permits flow through un-roughened capillaries. Without roughness, however, the capillary pressure provided by filter paper pores is stronger than that of the relatively large capillaries and flow does not occur. In this case, initial wetting of emitters and the array surface could be attributed to slight pressures applied to either the sample or filter paper during assembly of the test setup. Adjustments have been made to promote the same black silicon growth on the bottom surface of the wafer as the top, but electrical testing of this improvement has not been performed at the time of writing. A possible but unlikely alternative is that the black silicon surface treatment leads to impedance levels far higher than predicted by the conservative estimate of equation 2.41. In this situation, the high initial current is simply an over-wet phase and steady state emission is extremely supply-limited. The dissimilar magnitudes of current observed in second generation experiments as compared to recent tests (despite nearly identical MACE treatments) suggests that this is not the case. After additional roughening of the bottom wafer surface, expected behavior is a small initial

nA transient with steady-state current of at least 100 tip . Signiﬁcantly lower ﬂow rates would indicate that we should reduce surface roughness.

3.2.3 Failure Mechanisms

Consistent with the observations of Brikner [14], electric discharges and electrochem- istry were the two primary failure mechanisms for our thrusters. Events related to the former can be roughly split into two categories. The first is shorting due to fluid flow. The small component separation necessary for array miniaturization causes challenges for propellant containment. Additionally, current interception and the as-

103 sociated shorting risk become statistically more probable at higher emitter densities. In our testing of third generation arrays, shorting was the primary failure mechanism. Due to their negligible vapor pressure, ionic liquid propellants will not evaporate. This means that a shorting an array is almost certainly a permanent failure, even if current is halted before damage occurs. Electrical breakdown (through processes such as Townsend or corona discharge) appeared to occur far less frequently. With ambient pressure on the order of 10−6 Torr, electric fields were far below the breakdown threshold predicted by Paschen’s Law – even considering sharp local features. The absorption of ambient gases by ionic liquids may help explain this phenomenon. Most discharges occurred near the beginning of electrical testing. This would suggest that the propellant continued degassing as the extraction voltage was applied. A local pressure increase in the emission region would reduce the breakdown potential and therefore allow discharge. When this occurs, the strong electric field and high propellant conductivity cause a rapid increase in temperature and carbonization of the ionic liquid. Figure 3-13 shows the aftermath of this failure mechanism. Interestingly, the emitters underneath this layer often experience limited structural damage. Note that both types of electrical discharge can be triggered by the changing of emitter shape and fluidic properties associated with electrochemical degradation. This effect is discussed further in the following subsection.

104 (a) Emission region

(b) Individual emitter

Figure 3-13: Carbonization of EMI-BF4 after electrical discharge.

105 3.3 Post-test Analysis

Conclusive determination of an electrospray thruster’s operating regime requires time of flight spectrometry. An empirical mass distribution allows identification of ion species (monomers, dimers, and trimers) in addition to the approximate size and abundance of droplets. However, a post-test examination of the extraction electrode offers an indication of the likely steady state operating mode. It should be noted that this method is far less certain and inherently qualitative. Several previous studies have noted polishing of the collection electrode by thrusters operating in the ionic mode [37, 46, 38]. Without a path to escape, high energy ions

(traveling in excess of 20,000 m/s) collide the solid extractor plate and can remove material or cause secondary emission. Past research has employed EDX spectroscopy on similar solid electrospray extractors. The darker halo (seen more clearly with single emitters or bordering the emission region in tighter arrays) is often composed of fluorine and carbon while the lighter central regions are polishing of the underlying material [37, 46]. Similar patterns consistently appeared on our extractors, with little to no ionic propellant present. This, along with the low but steady current levels and shape of the current-voltage curves, would tend to indicate PIR operation. Figure 3- 14 shows polishing from single (fig. 3-14a) and multiplexed (fig. 3-14b) arrays.

(a) Single Emitter (b) Emitter Array

Figure 3-14: Extractor plate polishing.

The spacing of spots in ﬁgure 3-14b corresponds exactly to the pitch distance of

106 the array tested. This indicates steady, on-axis emission from all tips. Further, the beam spreading can be estimated from the polished area using equation 3.1.

( ) ∅ θˆ = tan−1 (3.1) b 2G

The rapid decrease in current density outside the tip region means that Coulomb interactions are very weak. As a result, the beam divergence angle can be approximated as a constant. Due to limitations of the experimental setup, an exact number for the extractor gap does not exist for the second generation architecture. Using an order of magnitude estimate, we find that the beam half angle during this test was approximately 12° – consistent with the hypothesis of steady state ionic emission. Electrochemical wear is also assessed post-testing. Due to the externally wetted architecture, second generation arrays were biased directly and thus most susceptible to electrochemical interactions. The speed of this effect varied with both current density and applied voltage, but in general tips lasted no more than about an hour. Variance, however, was significant. Figure 3-15 shows two second generation emitters which both sustained about 80 minutes of emission (in separate tests). Part of this discrepancy may result from inconsistent wetting causing local variations in emitted current density.

(a) Moderate wear of type G emitter (b) Severe wear of type A emitter

Figure 3-15: Electrochemical wear of second generation emitters.

107 Interestingly, this degradation sometimes occurred on very local scales – either to only certain tips in an array or select regions of an emitter. The latter is illustrated in ﬁgure 3-16. Once again, this is attributed to current density variations and local surface texture or curvature.

Figure 3-16: Highly localized electrochemical wear of a type D emitter.

As predicted, the introduction of a distal electrode in our third generation architecture signiﬁcantly reduced the rate of electrochemical wear. The high conductivity

of EMI-BF4 relative to silicon limited the charge double layer to a voltage below the electrochemical window, cutting current density across this interface to nearly zero. The magnitude of improvement can be seen in by comparing the emitters shown in ﬁgure 3-15 to those of ﬁgure 3-17. The latter tips experienced more than twice the operating time and show almost no signs of wear. Further, the black silicon texture appears unchanged.

108 Figure 3-17: Third generation type G array after ∼3 hours of electrical testing.

The success of this mitigation, along with ﬂuidic challenges associated with thicker oxide layers, motivated using only native oxide on subsequent emitters. Future results from longer tests may prompt reassessment of this choice. Note voltage alternation has not been employed as a mitigation strategy thus far. This reﬂects the planned unipolar operation for a WaferSat rather than any design-related limitations.

109 THIS PAGE INTENTIONALLY LEFT BLANK

110 Chapter 4

Conclusion

4.1 Manufacturing Process

The most significant contributions of this project are in design and manufacturing. Prior to this work, all electrospray architectures conformed to one of three overarching designs. The hybrid capillary and externally-wetted design proposed and manufactured in this project combine beneficial characteristics of both categories to create a more robust and higher performing final product. Further, this study includes the first demonstration of steady-state, passively-fed, purely ionic mode electrospray emission with a silicon substrate. While the several qualifiers to this statement may make it seem a niche or even trivial accomplishment, the combination of these particular four attributes is absolutely necessary for successful implementation of an active attitude control system on a wafer satellite. Apart from their necessity in wafer processing, the use of MEMS manufacturing techniques provides several tangential benefits. The extreme tolerances relative to any traditional machining technique allow excellent extractor alignment. This in turn permits smaller electrode gap distances, decreasing the voltage necessary for a critical electric field and therefore permitting the power supply miniaturization necessary for wafer integration. The nature of photolithographic grayscale etch technique permits further scaling of the emitters. Higher thrust densities would be relatively simple to adapt and manufacture if deemed necessary for mission success.

111 Development and validation of our black silicon morphology was a major accomplishment of this study. While past works explore the conditions necessary for production of several unique black silicon surface textures, very few do so for patterned structures (and therefore cannot be direct applied to matching the necessary fluidic conditions for electrospray operation). Past studies of the externally-wetted emitters employed a plasma etch approach which was relatively inconsistent, marginally repeatable, difficult to adjust, and highly isotropic. By contrast, our MACE method produced spatially uniform and repeatable surface roughness which facilitates consistent propellant flow. More significantly, the process allows significant control over texture. Variations in etch time, reagent concentration, and deposition of catalyst nanoparticles control height, thickness, and spacing of the silicon spikes. With the ability to tailor fluidic impedance, it becomes possible to achieve optimized performance for any specific mission. Finally, the nature of production methods chosen are amenable to scaled manufacturing. With the exception of the MACE process, all fabrication steps involve dry etching. (Transition to a contained method is planned for the surface roughening.) This permits full integration of the thruster array into a work-flow which produces an entire satellite on one single wafer; dicing becomes unnecessary. Additionally, the stacked wafer structure adopted to accommodate other subsystems is amenable to full integration of a thruster module. The extraction electrode and eventual fuel tank can be produced from the wafers above and below that of the emitter array and bonded into place. Thus, the design can be considered completely wafer integrated – or at least possesses the necessary characteristics for that process.

112 4.2 Thruster Performance

At this time, drawing strong conclusions regarding thruster performance is not advisable. The lack of an transparent extraction electrode inherently limits the compre- hensiveness of results. Without direct access to the ion beam, neither the mass nor energy distribution of its component particles can be measured. Knowledge of both functions is essential to quantification of system performance. Neglecting beam fragmentation, the mass distribution allows calculation of an empirical specific impulse (and thus also the magnitude of polydispersive losses). Under similar assumptions, the energy distribution provides system efficiency. Together, these two metrics allow calculation of the total thrust provided by the device. A comparison of the different array densities would then allow both a determination of maximum thrust density and any losses associated with densification. Still, it is possible to extrapolate a likely range of performance through comparison to past research. Based on results of the second generation architecture, emitted current levels should range from 100 to 300 nA after fixing the minor fluidic issues seen in third generation arrays. This range is consistent with many recent studies of porous designs which operate at or near the pure ionic mode. With similar flow rates and extraction voltages, the thrust per emitter of our design can be reasonably assumed to be similar those of previous studies. Accordingly, our tip density improvement of over one order of magnitude, should correspond to a nearly proportional increase to thrust density. Regardless, upcoming extractor integration and further testing of our design will very soon either verify or refute this prediction.

113 4.3 Future Research

While initial results demonstrate favorable emission characteristics and potentially record-breaking thrust density, these thrusters still require significant development and verification. As mentioned in the previous section, time of flight mass spectrometry and retarding potential analysis are the two most important tests for characterization of electrospray performance. While results are not included in this thesis, execution of these two experiments are nearly underway. If successful, a subsequent publication will include verified numbers for key performance metrics. While baseline performance can be established with the two aforementioned test, several others are necessary for full system characterization. Thermal vacuum testing will quantify an acceptable temperature range for operation (and the associated thrust and efficiency changes). Lifetime and duty cycle testing will be required to determine endurance. The angular distribution of the beam must also be characterized. Direct measurement of performance in the form of a thrust balance test or MagLev demonstration would be beneficial. Development of a propellant tank architecture has not been included in this work and will require thorough design and experimentation. Apart from the microfluidics challenges, electrical isolation, pressure relief, and electrochemical wear must be accounted for to allow system integration. Similarly, development of a wafer-integrated high-voltage power supply is necessary for operation. Finally, methods of improvement to the current design should be explored. Fur- ther experimentation with the metal assisted chemical etch process could allow more optimal surface roughness by changing the height and spacing of black silicon spikes. The combination of these parameters should be such that impedance is just above the threshold for transition from barrier-limited to supply-limited emission. Changing capillary design (such as several thinner pores feeding each tip) could have positive effects on emission stability. With the MEMS process developed, further shrinking the emitters to improve thrust density should be possible – even if not necessary for mission success at the current time.

114 Appendix A

Wafer Satellite Dynamics and Control

Nomenclature

ρ Density kg/m3 I Inertia tensor kg · m2 m Wafer mass kg F Force N

m3 2 µE Standard gravitational parameter of Earth /s r Distance m 0 − Pn Legendre polynomial m − Pn Legendre function

Jn Zonal harmonic coefficient − m − Cn Tesseral harmonic cosine coefficient m − Sn Tesseral harmonic sine coefficient

CD Coeﬃcient of drag − A Wafer area m2 v Orbital velocity m/s τ Torque N · m

115 R Hemispherical reﬂectivity − α Hemispherical absorptivity −

I Solar irradiance W/m2 c Speed of light m/s

θs Angle between sun and wafer normal rad 2 mD Magnetic dipole moment A · m

BEarth Earth magnetic ﬂux density T η Eﬃciency −

N˙ Ion extraction rate 1/s mi Ion mass kg

m ve Ion exit velocity (speciﬁc impulse) /s i Current A qi Ion charge C V Voltage V R Rotation matrix − t Time s

2 H Angular momentum kg · m /s

ω Angular velocity rad/s q Quaternion − S State vector various

θerr Pointing error to local vertical rad

Superscripts and Subscripts

B Body coordinates LV LH Local vertical local horizontal coordinates ECI Earth centered inertial coordinates ER Earth rotating coordinates [k] kth time iteration

CM Center of mass

CP Center of pressure (geometric centroid)

116 A.1 Structure and Assumptions

The current model is a much simplified astrodynamics simulation of a wafer satellite with the goal of better understanding dynamic stability and basic propulsion requirements. Given the current fluidity of proposed mission details, specifics of the orbit itself is of relatively low importance. This permits safely limiting certain flexibilities common to orbit simulations in favor of examining transient behaviors. Specifically, this model emphasizes accurately estimating torques caused by various orbital perturbations. Because of both the extremely low thrust levels available and high surface area to volume ratio of a wafersat, such perturbations may have a far more significant effects than commonly seen with traditional satellite geometries.

The key assumptions and limitations of this model are as follows. First, the initial orbit is taken to be perfectly circular at a altitude of 400 km. Gravitational perturbations from both the Sun and Moon are assumed to be negligible given the low altitude. Various assumptions related to speciﬁc perturbations will be discussed in the following section. Finally, the Earth is taken to be an inertial frame.

The satellite itself is assumed to be a rigid disk of nonuniform planar density. Its z dimension is normal to the top surface and x dimension is parallel to the vector from center of mass to centroid. As a result, the products of inertia are zero and body axes are principal. To oﬀset the center of mass the density function shown in equation A.1 is used. The constant 2330 corresponds to the density of bare silicon in kg m3 . Figure A-1 shows the spatial density distribution in the z = 0 plane. The values have been normalized by the bulk density of silicon

√ ρ = 2330(1.5 − 5 (x + 0.05)2 + y2) (A.1)

Using this equation, we can derive a reasonable inertia tensor for this model using the typical equations as shown below:

117 Figure A-1: Cross-section of normalized wafersat density.

∫ ∫ ∫ 2 2 Ixx = ρ(x, y, z)(y + z ) dx dy dz ∫V ∈ ∫wafer ∫ 2 2 Iyy = ρ(x, y, z)(x + z ) dx dy dz (A.2) ∫V ∈ ∫wafer ∫ 2 2 Izz = ρ(x, y, z)(x + y ) dx dy dz

V ∈wafer

A sample of this inertia tensor is represented below. Note that far higher numerical precision was used in the simulation than is shown here.     Ixx Ixy Ixz 0.761 0 0      I =   =   · 10−3 CM Iyx Iyy Iyz  0 0.786 0 

Izx Izy Izz 0 0 1.547

Finally this density function allows calculation of both the total mass and the center of mass position relative to the centroid. Relevant equations along with the calculated values for the density function speciﬁed in equation A.1 are reproduced

118 below for convenience. ∫ ∫ ∫ m = ρ(x, y, z) dx dy dz = 0.3239 kg (A.3)

V ∈wafer

Note that the density function is symmetric about both the y and z axes; the center of mass therefore lies directly on those two axes and only calculation of the x component is necessary. ∫ ∫ ∫ 1 xB = x · ρ(x, y, z) dx dy dz = −0.0069 m (A.4) COM m V ∈wafer

The optical properties of the wafer must be known to calculate solar radiation pressure. The chosen quantities are listed below and assumed to be average hemispherical values as explained in the following section.

Optical Properties Bottom Absorptivity 0.70 Bottom Reﬂectivity 0.30 Top Absorptivity 0.74 Top Reﬂectivity 0.04

A.2 Perturbations

Apart from gravity, several forces of smaller magnitudes create perturbations in the satellite orbit. This section will examine the sources of those forces, estimate their relative importance, and ﬁnally provide an equation for integration within the overall dynamic model. Before moving on to these smaller magnitude forces, it is useful to discuss the implementation of gravity in the code. Because the primary purpose of this model is not orbit propagation but to inform thruster and overall system design by estimating control requirements, precision of torques is far more important than forces. As such,

119 gravity is modeled in the most simple case - the force between two point masses.

µ m F = E r (A.5) gravity |r|3

Because this assumption is not precisely accurate, our gravitational force as shown in equation A.5 is slightly off. To improve accuracy slightly, the largest two causes of error in this simplified model are added as separate disturbances. (Both however have relatively little effect in the short time period of the simulation.) Gravity gradient accounts for the non-spherical geometry of the satellite. For any given orientation, differential mass elements of the satellite that are closer to the earth will experience slightly more gravitational acceleration than those farther away. This effect causes a slight torque and a tendency for the most dense portion of the satellite to orient toward the Earth. Although the derivation is omitted [40], an expression for this torque is shown in A.6.

3µ τ = E zˆB × I zˆB (A.6) gravity |r|3 CM

The second gravitational perturbation in our model corrects for non-spherical Earth eﬀects (and spatial variations in density). The true gravitational potential of the Earth (or any other celestial body) can be accurately expressed as an inﬁnite sum of spherical harmonic functions as shown in equation A.7 [40, 9].

µ ∑Nz J P 0(sin θ) ∑Nt ∑n P m(sin θ)(Cm cos mφ + Sm sin mφ) u = − E + n n + n n n (A.7) r rn+1 rn+1 n=2 n=2m=1

Empirically we know that the first zonal term, which physically corresponds to first order oblateness about the polar axis, is dominant. For Earth, the coefficient of · 10 km5 this term (J2 = 1.75553 10 s2 ) is three orders of magnitude larger than that of any other zonal or tesseral harmonic [86]. Given the discrepancy in magnitude, only this term of the expansion is used as a perturbation. Neglecting other terms provides a greatly simplified gravitational potential function as shown in equation A.8.

120 µ J P 0(sin θ) µ J u = − E + 2 2 = − E + 2 (3z2 − r2) (A.8) r r3 r 2r5

To find the force associated with this perturbation in the Earth-centered inertial frame, we can differentiate only the portion associated with the J2 constant. The result is shown in equation A.9 and is the final expression of this gravitational perturbation. ( ) µE F J = −∇ u + 2 [ (r ) ( )] x + y 3 z 9 (A.9) = J 6z2 − (x2 + y2) + 3z2 − (x2 + y2) 2 r7 2 r7 2

The first external disturbance in this model is drag. As the proposed mission involves operation within the rarefied atmosphere of low earth orbit, drag cannot be ignored as is often the case in deep space missions where high vacuum conditions are present. The drag force itself can be split into two components, pressure drag and skin friction drag. The latter of these two components results from shear stress on the spacecraft surface and is inversely proportional to the square of velocity. Given the high orbital speed of the wafersat, it is reasonable to ignore this component. Pressure drag by contrast is proportional to velocity squared and therefore must be considered in this analysis. Equation A.10 is presented below to describe this component quantitatively. An analytical equation for the drag coefficient can be found in [54], however past research has shown that drag coefficients of most any satellite shape in LEO are approximately 2.0, regardless of aerodynamic profile [78].

− 1 · F drag = 2 CDρ(A v)v (A.10)

This drag force acts on the spacecraft’s center of pressure (synonymous with the geometric center for a ﬂat disk). Therefore, a moment arm between these points creates a restoring torque on the satellite as shown in equation A.11

τ drag = rCM→CP × F drag (A.11)

121 A second source of perturbation is solar radiation pressure. This force occurs as an effect of the momentum exchange between incoming electromagnetic waves and any object which absorbs, reflects, or refracts them. The magnitude of this phenomena depends on the optical characteristics and orientation of the incident surface [39]. As shown in equation A.12, it has components from absorption and reflection which each act in different directions. Intensity (I) fluctuates with events such as solar flares but

W can be approximated as a constant 1361 m2 at a distance of 1 AU.

IA IA F = 2R cos(θ ) + α cos2(θ ) zˆB (A.12) solar c S c S

Note the simplified nature of this equation. It assumes a constant value of re- flectance (R) and absorptivity (α) rather than having functions of wavelength and incident angle. Additionally, the formula assumes a flat, opaque, diffuse surface of homogeneous optical properties [39]. Similar to drag, this force acts on the centroid of the wafer and therefore creates an additional torque as seen in equation A.13. Given the different line of action, however, this is not a restoring torque.

τ solar = rCM→CP × F solar (A.13)

Next, we consider the magnetic disturbances acting on the spacecraft. The magnetic torque which tends to align the spacecraft dipole with Earth’s magnetic ﬁeld is described by equation A.14.

τ magnetic = mD × BEarth (A.14)

Given that the current design does not include any permanent magnetic elements for the use of passive stabilization, only transient dipoles should occur in wafersat. Without better knowledge of the system architecture (and electronics specifically) this moment cannot be predicted at this time. With the availability of active control through the use of thrusters, this specific disturbance is not a significant concern. For similar reasons, damping effects resulting from magnetic hysteresis will be ignored.

122 Finally, the low thrust levels oﬀered by electrospray relative to gravitational acceleration allows us to consider the thrusters as an additional perturbing force (not an impulsive maneuver). Thrust can be estimated according to equation A.15 [48].

− ˙ qi 1 F thrust = η N mi ve = η i ve ( ) (A.15) mi

We expect an eﬃciency of η ≈ 0.8. For EMI-BF4 propellant ﬁring exclusively in the ∼ · 3 C negative mode, our charge to mass ratio is 500 10 kg (depending on the monomer fraction). We expect a current levels between 100 and 300 nA per tip with arrays of 1000 tips. Equation A.16 is used to calculate exit velocity.

√ ( ) qi ve = 2V (A.16) mi

m For a voltage of 1 kV , this velocity is 47100 s . Note that exit velocity and speciﬁc

impulse are not equivalent as the former accounts only for mass of the BF4 anions. From this analysis, our expected thrust level per array is approximately 3.4 µN.

A.3 Implementation

The ﬁrst step in implementation is the establishment of the various reference frames. For this model, our primary reference frame is Earth-centered inertial (hereafter ref-

erenced with the subscript ECI ). From this reference frame we can relate to the three other primary frames currently incorporated in the model. First is the rotating earth

frame (ER) used to reference the position of Lincoln Laboratory for any pointing requirements dictated by the payload (although these are currently assumed constant with respect to time, causing this frame to be unused thus far). This is the simplest

transformation as it involves a single time-dependent rotation about the zˆECI axis as seen in equation A.17. ( ) 2π t xˆ = R ∗ xˆ (A.17) ER z 86400 ECI

The location of the sun is required for thermal considerations and perturbations

123 such as solar radiation pressure. However, the significant distance allows replacement of this reference frame with a direction vector for computational simplicity. Further, the assumption of a stationary sun is made. This neglects the changes in angle of solar incidence throughout the year but simplifies the dynamics noticeably. Given that the goal of this study is an understanding of transient satellite stability and not long term orbital propagation, this assumption is acceptable. Thus the Sun-Earth unit vector is both constant and arbitrary given an inclination of between 23.5o with respect to the equatorial plane. In this model, the Sun is located in the negative xÊCI direction. The local vertical, local horizontal frame (LVLH) is essential to this simulation. It serves as an intermediary frame for convenient attitude reference between ECI and the satellite itself. The origin of this coordinate system coincides with the body origin. The xˆLV LH axis is parallel to satellite velocity. The zˆLV LH vector points in the gravity direction (towards the origin of the ECI frame). This allows us to conveniently define the satellite pointing error as the angle between zˆLV LH and zˆB, as illustrated in figure A-2. Relation between LVLH and ECI is similarly shown by figure A-3.

̂zLV LH

̂zB

Pointing Error ̂xB

̂xLV LH ◔◔

̂yB

̂yLV LH

Figure A-2: Relation between LVLH and body reference frames.

124 Sun

zˆECI

xˆLV LH

~r zˆLV LH yˆLV LH

yˆECI

xˆECI

Figure A-3: Relation between ECI and LVLH reference frames.

Finally, the satellite’s own body-centered coordinate system (xˆB) must be tracked. Its equations of motion are simply those of a rigid body. To calculate translation, we use Newton’s second law (equation A.18). High speciﬁc impulse leads to the assumption of constant mass.

∑ d(m v) dv F = ≈ m (A.18) dt dt

Euler’s equation of rotation is used to track rotation of the spacecraft relative to the inertial. Having deﬁned an axisymmetric body and bi-symmetric density function, we can use the simpliﬁed equation for rotation relative to the body principal axes (equation A.19).

d d HB = I ωB = τ − ωB × (I · ωB) (A.19) dt CM dt CM

The algorithm tracks a thirteen element state vector in addition to time. The

125 state vector is composed of x, y, and z position and velocity of the satellite in the ECI frame, a quaternion to track the relative rotation from ECI to B, and the angular rate of the satellite relative to inertial (S = [x, v, q, ω]). The governing expressions are manipulated into a set of ﬁrst order equations (A.20) which can then be propagated explicitly through the use of a 4th order Runge-Kutta solver.

  x˙ [k+1] = v[k]   ∑  1 d v˙[k+1] = F [k] S[k] = m (A.20) dt  q˙[k+1] = 1 q[k] ⊗ ⟨0, ω[k]⟩T  2  ∑  −1 − −1 × · ω˙ [k+1] = ICM τ [k] ICM [ω[k] (ICM ω[k])]

Note quaternion implementation in the attitude portion of the state vector. This is done for robustness (avoiding the gimbal lock limitation associated with Euler angles) and computational eﬃciency. A right-handed convention is used; the quaternion’s scalar component is the ﬁrst element as shown below.

ˆ ˆ ˆ q = ⟨q1, q˜⟩ = ⟨q1, q2, q3, q4⟩ = q1 + q2 i + q3 j + q4 k

Using this deﬁnition, the quaternion multiplication operation seen in equation A.20 is deﬁned according to equation A.21.

  − − − q1 r1 q2 r2 q3 r3 q4 r4    −  q1 r2 + q2 r1 + q3 r4 q4 r3 q ⊗ r =   (A.21)  −  q1 r3 + q3 r1 + q4 r2 q2 r4

q1 r4 + q4 r1 + q2 r3 − q3 r2

This method allows eﬃcient rotations of vector quantities between reference frames, which occurs frequently due to the necessity of applying forces and torques in an inertial frame. Using right-handed quaternions, rotation of a vector v is performed with equation A.22.

126 ⟨ , v′⟩ = qˆ ⊗ ⟨0, v⟩ ⊗ qˆ−1 (A.22)

This requires that the quaternion has unit length (hence the notation qˆ). For this reason, we re-normalize our quaternion after each time step, ensuring that numerical dissipation does not degrade the accuracy of rotations. Under this constraint, the inverse q−1 is equal to the conjugate q∗ – both of which are deﬁned by negating the vector portion as seen below in equation A.23.

−1 ∗ qˆ = qˆ = ⟨q1, −q˜⟩ (A.23)

When necessary, roll, pitch, and yaw angles can be extracted from a quaternion according to equations A.24 where atan2 is the two argument arctangent function.

− 2 − 2 ϕ = atan2(2 q1 q2 + 2 q3 q4, 1 2 q2 q3)

−1 θ = sin (2 q1 q3 − 2 q4 q2) (A.24) − 2 − 2 ψ = atan2(2 q1 q4 + 2 q2 q3, 1 2 q3 q4)

A.4 Initial Conditions

The code was written to allow specification of initial attitude and angular rates of the body frame relative to LVLH. The orbit of interest to this project is a circular LEO trajectory. A nominal initial altitude of 400km is used in all simulation results. Motion in the equatorial plane is examined, although an extension to any arbitrary inclination is achieved through rotation of the initial velocity vector. As magnetic effects are currently ignored, the only differences in disturbances attributed to inclination would be from J2 effects (precession of the longitude of the ascending node) and the angle of incidence for solar radiation pressure. With a circular orbit and no inclination, the argument of periaspis and longitude of the ascending node are

undeﬁned. Our chosen initial position is along the positive xECI axis at a distance equal to the orbital radius. Initial velocity is deﬁned by equation A.25 to ensure a zero eccentricity orbit.

127 √ µE v0 = (⟨0, 0, 1⟩ × xˆ0) · (A.25) ∥xˆ0∥

Since the set [ϕ, θ, ψ] is not a true vector, we cannot rotate these initial Euler angles as speciﬁed (relative to the LV LH) into the ECI frame using a coordinate

transformation. To ﬁnd the initial quaternion q0 relating inertial coordinates to the body frame, we instead use equation A.26.

       1 + tr(R)  T   N xÊCI N xˆB  −      1 R(3, 2) R(2, 3)     qˆ = √   where R = N yÊCI  N yˆB  (A.26) 0 2 1 + tr(R)  −      R(1, 3) R(3, 1) N zÊCI N zˆB R(2, 1) − R(1, 2)

The superscript N above the unit vectors in equation A.26 denotes that they referenced with respect to inertial. Since the ECI frame is itself inertial, the ﬁrst

matrix simpliﬁes to the identity matrix I3. The unit vectors of the body frame can be calculated by applying successive rotations to the LVLH unit vectors, which are

N LV LH N LV LH known by deﬁnition ( xˆ = vˆ0 and yˆ = −xˆ0).     N B N LV LH  xˆ   xˆ      N B  N LV LH  R = I3  yˆ  = Rx(ϕ0) Ry(θ0) Rz(ψ0)  yˆ  (A.27) N zˆB N zˆLV LH

Finally, the initial angular rate of the body frame relative to inertial derives from

the addition of the angular rate vectors ωB/LV LH (speciﬁed arbitrarily when initializing

the simulation) and ωLV LH/ECI . It is important to note that the former must be speciﬁed in the ECI frame prior to addition. As angular velocity is a pseudovector, this can be done with a quaternion rotation. The later angular rate can be calculated with equation A.28.

(−x0) × v0 ωLV LH/ECI = 2 (A.28) ∥x0∥

128 Accordingly, our ﬁnal initial condition (body angular rate in the inertial frame) is given by equation A.29.

− × ⊗ ⟨ ⟩ ⊗ −1 ( x0) v0 ω0 = qˆ0 0, ωB/LV LH qˆ0 + 2 (A.29) ∥x0∥

A.5 Control Strategy

The first decision in creating a control algorithm for a wafer satellite is the positioning of thruster modules. While many configurations allow six degree of freedom control, some are more optimal. The most obvious choice for a first thruster module is the farthest point from the center of mass. This allows maximum torque for a given thrust level. A complementary array in the opposite orientation provides the ability for +z maneuvers in addition to maximizing damping potential along the primary axis. Another in-plane module should be positioned in the negative x axis to compensate for drag in the primary velocity direction. The remaining two in-plane thrusters can be positioned near the positive x extremity to provide yaw damping in either direction. This model uses 10° offset from the axis to allow at least 1 cm spacing between adjacent arrays. To find their optimal orientations, we can use equation A.30

ˆ τˆyaw = ∥rthruster + rCM→CP ∥ × F thruster (A.30)

Finally, an out of plane thruster is required for control about the roll axis. This model adds a second roll torque thruster to simplify control. Positions of the various thruster modules are summarized in table A.2. While this conﬁguration provides excellent stabilization, it may not be optimal for missions with a high ∆V requirement. While the drag compensation thruster is present, maneuvers such as an inclination change would be diﬃcult with the current setup and thruster orientations aligned with the vector rCM which do not produce torque should be considered. Next, the control algorithm will require some objective function. For simplicity (and due to current ambiguity in mission requirements) this model seeks to minimize

129 Position [cm] Orientation Type 9 0 0.1 ⟨ 0, 0, 1 ⟩ OP 9 0 -0.1 ⟨ 0, 0, -1 ⟩ OP 0 9 0.1 ⟨ 0, 0, 1 ⟩ OP 0 -9 0.1 ⟨ 0, 0, 1 ⟩ OP -10 0 0 ⟨ -1, 0, 0 ⟩ IP 9.85 1.74 0 ⟨ -0.163, 0.987, 0 ⟩ IP 9.85 -1.74 0 ⟨ -0.163, -0.987, 0 ⟩ IP

Table A.2: Thruster module placement.

the roll, pitch, and yaw angles between the body and LVLH reference frames. The function which describes this total angular error is shown in equation A.31.

[ ] θ = 2 · cos−1 (qˆ ⊗ qˆ−1 ) •⟨1, 0, 0, 0⟩ (A.31) err B⇐ECI LV LH⇐ECI

This error can also be split into three orthogonal components to isolate the necessary response about each primary body axis. This is accomplished by applying equation A.24 to the pointing error quaternion. qˆ . From these equations, B⇐LV LH we can also come up with an alternative to equation A.31 which uses an arctangent function rather than an arc-cosine.

θerr = 2 · atan2(∥ q˜∥, |q1|) (A.32)

Unsurprisingly, our control algorithm choice is primarily dependent on the dynamics of the system at hand. We see from equation A.20 that a nonlinearity in the attitude dynamics arises from the cross product of the body angular rate vector. With this in mind, we would typically choose some method of nonlinear or robust control algorithm. In fact, a model reference adaptive control (MRAC) algorithm is well suited to this application due to its adjustment to uncertainty in either the system dynamics or satellite properties (notably small errors in the predicted inertia tensor). However, this approach is not computationally feasible due to limitations of the microprocessor currently planned for implementation. Simplicity and eﬃciency

130 are typical characteristics of linear control strategies. For reasons discussed above, such an algorithm would have poor performance at any state with signiﬁcant angular rate of the body frame with respect to inertial coordinates.

Our initial solution to this issue was a dual implementation of proportional derivative (PD) control alongside a linear quadratic regulator (LQR). The PD controller would be employed primarily for de-tumbling of the satellite. Whenever angular rate is larger than one rotation per orbit by an order of magnitude or more, nonlinearity of the dynamics becomes nontrivial. While not optimal with respect to either maneuver time or fuel consumption, a PD controller would be robust to these nonlinearities and will eﬀectively damp the motion to a point near equilibrium. From here, the LQR controller takes over. The system can be linearized about the desired operation point with minimal error introduced. Also, LQR is an optimal control scheme allowing us to weight propellant usage against attitude and position requirements. Most importantly, both PD and LQR require very little computational power or memory relative to other control algorithms.

Note that this approach is viable only due to our objective function being equivalent to the equilibrium point of the overall system dynamics. Given some attitude

requirement either at a large angular offset from the zLV LH vector or relative to the inertial frame, linearization would introduce significant error and an LQR controller would be ineffective. The PD control scheme could be used, but gains would be constant and therefore the response not optimal. Additionally, this would increase propellant consumption as control effort is longer weighted against error. A larger microprocessor would be need to be incorporated to allow implementation of some nonlinear or adaptive controller.

A PD control scheme can be described with only the system of equations shown in A.33. Here the proportional and derivative gains (Kp and Kd respectively) are

simply positive constants. For the application at hand, Kp = ⟨1, 1, 1⟩ and Kd = ⟨60, 60, 60⟩ seemed to provide adequate de-tumbling performance.

131 ∑

F control = F ∠i   ⟨0, 0, 1⟩ · ∥F ∥thrust (Kd,ϕ · e˙ϕ + Kp,ϕ · eϕ) < 0 F ϕ =  ⟨0, 0, −1⟩ · ∥F ∥thrust (Kd,ϕ · e˙ϕ + Kp,ϕ · eϕ) > 0   (A.33) ⟨0, 0, 1⟩ · ∥F ∥thrust (Kd,θ · e˙θ + Kp,θ · eθ) < 0 F θ =  ⟨0, 0, −1⟩ · ∥F ∥thrust (Kd,θ · e˙θ + Kp,θ · eθ) > 0   ⟨−0.163, −0.987, 0⟩ · ∥F ∥thrust (Kd,ψ · e˙ψ + Kp,ψ · eψ) < 0 F ψ =  ⟨−0.163, 0.987, 0⟩ · ∥F ∥thrust (Kd,ψ · e˙ψ + Kp,ψ · eψ) > 0

The Linear Quadratic Regulator (hereafter LQR) requires slightly more compu- tation. First, we must linearize the dynamic system about its operation point. We choose to do this in the LVLH reference frame as it renders position vector constant and allows us to assume a nearly constant velocity equal to the initial orbital speed. Given that control loop execution occurs on a time scale at least two orders of magnitude faster than the relative movement of the ECI and LVLH frames, the error introduced by application of the laws of motion in a non-inertial reference frame is minimal. As it is a nearly constant value, we can add its nominal value after linearization to remove most of the introduced error. At the current time, we are relatively unconcerned with the position tracking of this satellite. Therefore, we do not attempt to regulate either position or velocity. This greatly simpliﬁes the control dynamics. We take our state vector to be the the Euler angles and body angular rates (Sc = [ϕ, θ, ψ, ωx, ωy, ωz]). Equation A.34 describes the evolution of our control variables. The assumption:

d LV LH/ECI ≪ d B/LV LH dt Sc dt Sc allows us to consider the LVLH frame inertial and thus apply the three angular rate equations. Note that products of inertia are taken to be zero.

132    ˙ ϕ = ωx + ωy sin ϕ tan θ + ωz cos ϕ tan θ   θ˙ = ω cos ϕ − ω sin ϕ  y z   ˙ d ψ = ωy sin ϕ sec θ + ωy cos ϕ sec θ Sc ( ) (A.34) dt  − ω˙ = ¯I 1 τ − (¯I − ¯I )ω ω  x xx x zz yy y z  ( )  ¯−1 − ¯ − ¯ ω˙ y = Iyy τy (Ixx Izz)ωxωz   ( )  ¯−1 − ¯ − ¯ ω˙ z = Izz τz (Iyy Ixx)ωxωy

The only missing component is calculation of the applied moment τ. Since drag is the largest disturbance by at least an order of magnitude, it is the only torque considered in this analysis. Solar pressure is the next largest, but ignored due to both relative magnitude and time variance of the reference frames.

( ) × − 1 · τ = rCM→CP 2 CDρ(A v)v (A.35)

Calculation of the drag torque involves rotation of the vector (rCM→CP ) from body to LVLH frame. Deﬁning α as the angle between the body z axis and the velocity direction, we can simplify this equation. We exploit the fact that the velocity direction is constant by deﬁnition in LVLH.

( ) − 1 2 · LV LH × ⟨ ⟩ τ = 2 CDρAv cos α rCM→CP 1, 0, 0 (A.36)

To ﬁnd the vector component of equation A.36, we perform a quaternion rotation. To linearize this operation with respect to Euler angles, we assume that the rotation is small. This is an acceptable assumption given that this controller is only active when the body and LVLH frames nearly coincide. The approximate quaternion is deﬁned by equation A.37.

⟨√ ⟩ ≈ − ϕ2+θ2+ψ2 ϕ θ ψ qˆB→LV LH 1 4 , 2 , 2 , 2 (A.37)

Since both the satellite center of mass and center of pressure lie along the x axis,

133 ⟨ ⟩ the vector rCM→CP in the body frame has the form a, 0, 0 . Rotating this vector and further assuming that α ≈ θ for small angles, we obtain a ﬁnal approximation for the drag torque in the LVLH frame.

τ ≈ − 1 C ρAv2 cos θ drag ⟨2 D ⟩ √ √ (A.38) · − ϕ2+θ2+ψ2 ϕ ψ a − ϕ2+θ2+ψ2 − ϕ θ a 0, θ a 1 4 + 2 , ψ a 1 4 2

We now have all the necessary information to linearize the system. We choose the

equilibrium point Sc, eq = [0, 0, 0, 0, 0, 0] and (unsurprisingly) most terms cancel out. The system requires one ﬁnal adjustment. As discussed earlier, the rotation speed of the local frame itself can be subtracted from the body angular rate to partially compensate for the error associated with applying Newton’s Laws in a non-inertial frame. Using the nominal value, this becomes an additive constant and we are left with equation A.39. Note that this angular rate must be speciﬁed in the local, not inertial frame.

  0 0 0 1 0 0     0 0 0 0 1 0   [ ] 0 0 0 0 0 1 T d ≈   · − Sc   Sc (ωLV LH ) 0 0 0 dt   /ECI 0 0 0 0 0 0 (A.39)   0 k 0 0 0 0  ¯Iyy  0 0 0 0 0 0

− 1 2∥ ∥ where k = 2 CDρAv r CM→CP

With the approximate system now linearized, an LQR controller can be derived. Fundamentally, this control strategy seeks to minimize a cost function which consists of both weighted state and control costs as shown in equation A.40 [95].

∫ ∞ T T JLQR = S Q S + u R u dt ∀ t ≥ 0 (A.40) 0

134 Provided that the state and input cost matrices (Q and R respectively) are positive

semidefinite, then we can find a linear controller gain Klqr of the form u = −Klqr · S. Most commonly, diagonal cost matrices are chosen to eliminate unnecessary cross terms between state or actuator cost [95]. Positive terms on the diagonals ensure that the matrices themselves are positive definite. We use Bryson’s Rule to develop an appropriate weighting for control effort (and by proxy, propellant consumption relative to attitude tracking fidelity). The optimal controller gain can then be solved using the Algebraic Riccati Equation (A.41) where A is the state transition matrix from equation A.34 and P is some symmetric positive semi-definite matrix [95].

0 = AT P + PA + Q − PBR−1BT P (A.41) −1 T Klqr = R B P

Our control response vector u is simply the set of thruster roll, pitch, and yaw torques respectively. Unlike many control responses however, our thrusters operate at a single level. Total response is moderated by pulsing to produce the proper time- averaged value. As a result, the true control response in any axis is either zero or the nominal torque of the relevant thruster. This is shown by equation A.42 for any given time step.

  τ thruster, i ui > τ thruster, i ui = (A.42)  0 ui < τ thruster, i

After the process of deriving an LQR controller, we find that it is remarkably similar to the PD controller naively proposed at the beginning of this section. Thus for both robustness and computational efficiency, we choose to employ only the PD scheme with a few modifications. First, we subtract the nominal angular rate of the

N N N LVLH frame (ωLV LH ) to ωB to ﬁnd ωB/LV LH . This pseudovector is then rotated into the LVLH frame and used in place of the derivative error (e˙) from equation A.33. Next, some tolerance is chosen below which no thrusters are ﬁred. This performs the same function as a control cost for a LQR implementation. Drawing from Bryson’s

135 rule, the chosen tolerance should have the following proportionality.

⟨ϕ, θ, ψ⟩ ¯I · τ ε ∝ · thruster (A.43) tol ⟨ϕ, θ, ψ⟩ 1 2∥ ∥ max 2 CDρAv r CM Upon further examination and testing, it appears that the control authority from the thrusters is large enough in comparison to the drag torque near equilibrium that we can simply specify the tolerance as equal to the maximum acceptable angle about the relevant axis. Finally, we ensure that the tolerance for roll and yaw angles is kept tight. The motivation for this is the increased system nonlinearity from these angles as compared to pitch. This serves to improve propellant eﬃciency by reducing the rate of divergence from equilibrium after de-tumbling.

A.6 State Estimation

The previous section presumes accurate knowledge of system states at each time step. In practice, this information must be estimated from sensor readouts. The accuracy of this state estimate will have significant effects on controller performance. Errors either propagated from sensor noise or introduced in the estimation loop itself inherently limit the satellite’s tracking accuracy. In low earth orbit, GPS is available for accurate position sensing. Apart from that, on-board instrumentation would include a coarse sun sensor and inertial measurement unit (to include a gyro, magnetometer, and 3-axis accelerometer). Star trackers are not an option due to size limitation. From accelerometer and GPS data, inertial position and velocity are well-known throughout the orbit. Attitude estimation is more complicated. MEMS gyros typically exhibit noise and zero-rate-offset levels far higher than the angular velocities expected from our satellite near its operating point [105]. The integration of this error leads to significant discrepancy between predicted and actual attitude (commonly referred to a gyro drift). An estimator must correct this by fusing data from other sensors. The magnetometer may be used to provide an rough attitude estimate. We can find the optimal relation between measurements of the same vector (v′ and v) in two different reference frames according to the method described in [60]. Optimality

136 here is defined by a quadratic loss function and w is a set of weights determined by magnetometer details. In this case the v′ is a measurement of the magnetic field vector. Two possibilities for v exist. We can either use a previous measurement (in which case the calculated quaternion relates the current frame to that of an earlier time step), or we can compare to some predetermined geomagnetic field model based on our known position from GPS. For a variety of reasons not discussed here, we choose to do the former. Without derivation, equation A.45 is presented to calculate p (the set of Rodrigues angles)[94]. The required parameters are shown in equation A.44 and the optimal gain g was calculated by Shuster and Oh [94] to be approximately equal to the sum of weights w.

∑ B = w(v′vT )

S = B + BT (A.44) Z = [B23 − B32,B31 − B13,B12 − B21] σ = tr(B)

−1 p = [(g + σ)I3 − S] Z (A.45)

The same paper presents an algorithm (known as the QUEST solution) for determining the relevant quaternion from these angles. When converted for consistency with the right-hand convention explained previously, this becomes equation A.46.

  1 1 qˆ = √   (A.46) 1 + pT p p

When not in eclipse, the coarse sun sensor may also be used to ﬁnd an estimated

quaternion between the satellite −zˆB direction and the sun. With knowledge of the current relation between true inertial and ECI frames, an estimated update quaternion may be calculated by multiplication of the conjugate. For outlier rejection and to improve accuracy, we can average the this with the magnetometer estimate. The mathematical method for doing this relies on a nearly identical process as the deriva-

137 tion of equation A.46, but is presented more concisely for the case of two quaternions by a NASA paper [75]. The final form is shown in equation A.47. This provides a unique solution except in the antipodal case, where the scalar is taken to be positive by arbitrary convention. √ ≡ − 2 T 2 z (w1 w2) + 4w1w2(qˆ1 qˆ2) (A.47) [(w − w + z)qˆ + 2w (qˆT qˆ )qˆ ] qˆ = 1 2 1 2 1 2 2 avg ∥ − T ∥ (w1 w2 + z)qˆ1 + 2w2(qˆ1 qˆ2)qˆ2 Unfortunately, both of these methods are typically only accurate to within a few degrees due to environmental fluctuations and sensor limitations. Additionally, the magnetometer method does not provide an inertial reference and thus the satellite experiences the same error accumulation problem as with gyro when in eclipse. The solution to this problem is incorporation of system dynamics knowledge into the estimation loop. This can be done with an extended Kalman filter (EKF) which improves estimate accuracy by examining the overlap of probability distributions from various sensors with that of our forward-propagated dynamics. For linear systems, estimation is commonly performed using a Kalman filter. This estimator achieves optimality through minimization of the state error covariance matrix defined below in equation A.48.

[ ] [ ] ˜ ˜ T T E S[k] S[k] = E (Sest[k] − S[k]) (Sest[k] − S[k]) (A.48)

An extended Kalman filter follows nearly the same process, but involves a re- linearization of the state transition matrix at each time step [99]. This additional complexity allows significantly reduced uncertainty and more accurate results. In order to decrease the computational cost, a simplified version of the dynamics are used. Similar to the previous section, only drag and thruster torques are considered. Thus equation A.38 can still be used. Pre-calculation and storage of the system Jacobian can also improve efficiency Due to problems arising during the Simulink implementation of this method, the estimation problem will be addressed in a future iteration of the project. The current

138 code focuses on ﬁdelity of dynamic model and performance of the control algorithm for attitude tracking. Currently, the control loop itself is provided with state information that is either accurate or has additional band-limited white noise.

A.7 Performance

By forward propagation of the full nonlinear system dynamics (described by equation A.20) and calculation of control response (from equation A.42), we can quantify performance of the proposed attitude tracking system. Although a wide range of initial attitudes were explored, the set shown below was used to generate all results in this section.

B B B ϕ0 θ0 ψ0 ωx, 0 ωy, 0 ωz, 0

0.1 rad 0.2 rad 0.1 rad 0.0 rad/s 0.0 rad/s 0.0 rad/s

Table A.3: Initial satellite attitude.

The satellite was permitted to tumble for one orbital period prior to the ADCS system being engaged. This also provides insight into the passive stability of our design. The chosen time step for both system dynamics and the control algorithm was one second. The tolerance for angular error was set to one degree in each axis. Time scales in this section are normalized by the nominal orbital period, which corresponds to just above 90 minutes at the chosen altitude of 400 kilometers. Figure A-4 shows the transient ﬂuctuations in each of the Euler angles and values of their associated angular rates. Note that all six quantities are referenced relative to the LVLH frame.

The total pointing error, equivalent to the dot product between zˆLV LH and zˆB, is shown in figure A-5. The rapid convergence of pointing error to zero is clearly apparent at the end of the first orbital period. Regardless of disturbances, the control algorithm appears to have no trouble keeping error within the specified tolerance. It is notable that drag acts as a nearly restorative force at this initial condition and others within a reasonable range from the equilibrium point. While not shown here, any significant angular rate

139 Figure A-4: Satellite orientation and angular rate in the local frame.

Figure A-5: Total pointing error.

140 or initial angle above approximately 50 degrees quickly leads to a transition from oscillation to tumbling. Even smaller angles tend to become gradually less stable and will begin to flip after tens or hundreds of orbital periods. Regardless of rotation rate, the controller manages to re-establish equilibrium quickly. Another interesting aspect of figure A-4 is the nonzero steady-state pitch rate. The amplitude of this offset is equivalent to the rotation rate of the local frame relative to inertial. This provides a point of comparison for the magnitudes of the various body angular rates. The effect of this relatively small rotation becomes apparent when examining frequency of thruster use in each axis. In the right column of figure A-4, we see a step response in time steps when thruster are active. While relatively infrequent in the roll and yaw axes, these corrective measures are required often in the pitch axis (which is aligned with the rotation of the LVLH frame due to the equatorial orbit used for this simulation).

Propellant consumption is a key metric for characterization of controller performance. An assessment of mission lifetime provided significant motivation for this project. We can estimate the mass of propellant used over a typical orbital period from the total firing time and nominal specific impulse. In this case, polydispersive effects are ignored and specific impulse assumed equal to the exit velocity multiplied by the mass ratio of the anion to full molecule. This yields the expression shown in equation A.49.

· M Fthruster tfiring BF4 −7 mpropellant = · ≈ (10 · tfiring) g (A.49) v M EMIBF4 e EMIBF4

From ﬁgure A-6 we see that the thrusters collectively ﬁred for about 30 seconds per orbital period after the de-tumbling concluded. Using equation A.49, this corresponds to 3 µg of propellant consumption. Operation for 100 days at the same rate would require only about 5 mg. This can be reduced substantially by allowing a larger pointing error. Figure A-7 shows the satellite orientation and angular rates when tracking tolerance for pitch is increased to 15 degrees. Roll and yaw tolerances are

141 80

20 thruster usage [sec]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 time [orbital periods]

Figure A-6: Thruster activity. kept small to limit nonlinearities. Due to the slow rotation of the local frame relative to inertial and drag acting as a nearly restorative force, the wafer oscillates about the equilibrium with minimal correction required. From ﬁgure A-8, we see that the rate propellant consumption after de-tumbling is more than an order of magnitude slower than before. Note that this simulation lasts ﬁve orbital periods rather than two. Due to the assumption of perfect system knowledge, however, this estimate of propellant usage is quite optimistic.

In addition to orbital perturbations, performance is adversely affected by errors in the assumed model parameters. An analysis on the sensitivity of tracking fidelity to various disturbances is crucial to determining controller robustness. In this work, three primary sources of error were introduced. After individual examination, the various disturbances were applied simultaneously to assess cumulative effects. The first error is an offset of the center of mass by up to 1 cm in each body axis. A uniform distribution is used for selection of the offset magnitude. Next, the inertia tensor is changed. An error of up to one percent of the tensor’s largest element is added to each component. These numbers are also chosen from a uniform distribution. Symmetry of the tensor is enforced. The introduction of non-zero products of inertia indicates that the thrusters are no longer in alignment with the body principle axes. Finally,

142 Figure A-7: Satellite orientation and angular rate under loose pointing requirements.

Figure A-8: Thruster activity under loose pointing requirements.

143 small errors in the thruster direction are introduced to simulate off axis emission. These came from a standard normal distribution (with units of degrees) re-sampled at each time step. Of the three, a center of mass offset was responsible for introduction of the most nonlinearity. Not only did it affect the line of action for the various torques (which caused drag to be non-restorative), but it also impeded control efficiency by adding cross term components to the corrective moment provided by the thrusters. The altered inertia tensor caused a very small decrease in overall stability. However, it increased the frequency of control activity as the thrusters no longer caused rotation about a single axis. Finally, off-axis emission had the least negative performance implications. It is possible that the transient nature of the disturbance causes effects to average out over several time steps.

30 angle [deg] 20

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 time [orbital periods]

Figure A-9: Total pointing error with disturbed model parameters.

Figure A-9 shows the total pointing error when random disturbances are introduced as described above. The tolerance in each direction was doubled to two degrees.

144 Despite having the same initial conditions as those of figure A-4, we see significantly more nonlinear behavior exhibited during the first orbital period when the control system is inactive. The algorithm still manages to maintain proper satellite orientation, but motion near equilibrium is significantly less stable.

There is a signiﬁcant increase in propellant consumption associated with the introduction of this model uncertainty. Figure A-10 shows that cumulative ﬁring time doubled with disturbances introduced (despite a looser tracking tolerance). This effect will eventually be compounded with inclusion of estimation error. Depending on mission requirements, this may motivate a more complicated controller (as will be discussed in the following section).

100 80 60 40

thruster usage [sec] 20 0

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 time [orbital periods]

Figure A-10: Thruster activity with disturbed model parameters.

Since the applied disturbances are random, we obviously must examine more than one case. Figure A-11 shows thruster activity for a collection of 25 simulations, each with identical initial conditions but diﬀerent magnitudes of disturbances. While there is a large discrepancy in de-tumbling time, the rate of thruster usage after stabilization is relatively consistent between trials. Thus, we can be fairly comfortable taking the results from ﬁgure A-10 as representative with regard to propellant consumption.

145 Figure A-11: Collected thruster activity with disturbed model parameters.

A.8 Conclusions and Future Work

Due to binary nature of thruster response and discrepancy in relative rotation rates of the various reference frames, the LQR controller for this system is nearly identical to a proportion-derivative scheme. The implementation of this controller is computationally efficient and robust to both disturbances in model parameters and significant nonlinearities in the system dynamics. For these reasons, a modified PD controller is a sufficient option for this satellite. Given the high control resolution demonstrated, attitude estimation will likely be the limiting factor in tracking accuracy. Increas- ing the pointing error tolerance results in significant propellant savings due to drag acting as a restoring force after de-tumbling. This could extend the lifetime of missions without strict pointing requirements. If significant error in the inertia tensor or center of mass position are observed in the final system, implementation of a more complicated control strategy may be advisable to optimize propellant usage. There are several avenues to improve this work. The first few are relatively easy to implement, but increase the simulation’s robustness. The necessary code has already been included for most of these, but results are omitted here due to either insufficient parameter knowledge or lack of a significant effect on results. The first would be inclusion of magnetic effects. If the WaferSat’s dipole moment were known, a Lorentz

146 force and associated torque terms could be included in the summation of equation A.20. Effects of the control loop time step should also be examined. Currently, this loop is implemented during every forward propagation of the system states. Since thruster response has a constant magnitude, a longer time step may increase the maximum pointing accuracy. However, such an examination would be more useful after implementation of an estimator; if the pointing accuracy of the control system is higher than that of the estimator, this consideration becomes irrelevant. In a similar vein, it may be useful to examine a controller which allows multiple power levels for each thruster. This paper considered only one orbit. Changing the initial Keplerian elements may affect pointing accuracy to a small degree. Finally, examination of different objective functions would provide valuable insight into control system limits. Transient pointing requirements may be useful for thermal mitigation or other reasons. The increased drag at a non-equilibrium target orientation would degrade pointing precision and increase propellant consumption. Other improvements would involve significant extensions to the current work. Per- haps most necessary is the incorporation of an estimation loop. Given the excellent performance of the control scheme when fed perfect state information, it is extremely probable that estimation becomes the limiting factor with respect to pointing accuracy. Most other useful metrics such as the rate of propellant consumption will be strongly influenced by estimator performance. Although the current model allows introduction of disturbances in the satellite parameters, only small perturbations were examined. A significant change, especially to the inertia tensor, could degrade tracking precision or increase propellant usage. In this case, examining alternate controllers may be advisable. Given the difficulty in exact determination of a wafer’s inertia tensor, a model reference adaptive controller (MRAC) may offer increased performance at the expense of computational efficiency. Finally, an extension of the control algorithm to linear position would be useful for scenarios which involve multiple bodies. For example, this would be necessary to control relative position in satellite formation flight. This expansion of control requirements would involve weighting relative state costs and may motivate a re-examination of thruster module placement.

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