Computational simulation of coupled nonequilibrium discharge andcompressible flow phenomena in a microplasma thruster

Thomas Deconinck,a� Shankar Mahadevan, and Laxminarayan L. Rajab�

Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin,Austin, Texas 78712, USA

�Received 16 March 2009; accepted 12 August 2009; published online 23 September 2009�

The microplasma thruster �MPT� concept is a simple extension of a cold gas micronozzle propulsiondevice, where a direct-current microdischarge is used to preheat the gas stream to improve thespecific impulse of the device. Here we study a prototypical MPT device using a detailed,self-consistently coupled plasma and flow computational model. The model describes themicrodischarge power deposition, plasma dynamics, gas-phase chemical kinetics, coupling of theplasma phenomena with high-speed flow, and overall propulsion system performance. Compared toa cold gas micronozzle, a significant increase in specific impulse is obtained from the powerdeposition in the diverging section of the MPT nozzle. For a discharge voltage of 750 V, a powerinput of 650 mW, and an argon mass flow rate of 5 SCCM �SCCM denotes cubic centimeter perminute at STP�, the specific impulse of the device is increased by a factor of �1.5 to about 74 s. Themicrodischarge remains mostly confined inside the micronozzle and operates in an abnormal glowdischarge regime. Gas heating, primarily due to ion Joule heating, is found to have a stronginfluence on the overall discharge behavior. The study provides a validation of the MPT concept asa simple and effective approach to improve the performance of micronozzle cold gas propulsiondevices. © 2009 American Institute of Physics. �doi:10.1063/1.3224863�

I. INTRODUCTION

Small satellites �mass less than 100 kg� have recentlygained interest for various commercial, military, and sciencespace missions.1,2 Several advantages, including a reducedlaunch cost, are realized with small satellites. Small satellitesare power limited, typically delivering less than 1 W/kg ofspacecraft mass. Propulsion requirements for these small sat-ellites can vary significantly given the diversity of small sat-ellite missions, but are generally characterized by their verylow thrust values and low impulse bits for attitude control�approximately tens of �N /s�. Scaling traditional propulsionsystems down in power and size to suit the needs of smallsatellites is a major engineering challenge.

Cold gas thrusters are low-performance devices limitedby low values of specific impulse ��50 s�.3 The advantagesof these devices compared to other propulsion systems in-clude their low system complexity, low cost, and absence ofsatellite contamination problems when using inert propel-lants �e.g., N2�. Micronozzle cold gas thrusters are suitablefor attitude control applications of small satellites. They canalso be used as a primary propulsion system if the �v re-quirements are less than �100 m /s. Since nozzle throat di-ameters in the micrometer range are required for micronozzlepropulsion, the production of these microthrusters relies onmicroelectromechanical fabrication technologies.4–6

The extremely small dimensions of microdischargescombined with intense and controllable gas heating can beexploited in microthruster technologies. Like conventional

cold gas thrusters, electrothermal microdischarge propulsionsystems can have extremely low mass and volume footprints.They are electrically simple, and, unlike other electric pro-pulsion technologies, they do not require auxiliary systemssuch as neutralizers, heaters, or magnets. The thrust fromthese devices is widely tunable by varying power levels. Asignificant improvement in the specific impulse compared toconventional micronozzle cold gas thrusters can potentiallybe achieved with these systems.

The microplasma thruster7 �MPT� concept consists of adirect-current microdischarge in a geometry comprising aconstant area flow section followed by a diverging exitnozzle. Geometric dimensions are on the order of hundredsof microns. In the MPT concept, a stable microdischarge isgenerated at low propellant flow rates �few SCCM�. The pro-pellant gas in the microdischarge is eventually expelled fromthe diverging nozzle. Thrust is produced by the expandinggas flow, which is heated by the microdischarge. The MPThas potentially important advantages over competing con-cepts such as the microresistojet thruster,8 which also relieson preheating an expanding gas stream. The important dif-ference between a resistojet and the MPT is the nature ofpower deposition into the gas. In a resistojet, the heatingelement is essentially the hottest part of the system and heataddition to the gas occurs via conduction/convection to thegas stream. There is clearly an upper limit on the heatingelement temperature, which in turn limits the amount of heatdelivered to the gas. In the MPT, the gas is heated directly bythe plasma. Importantly, if the plasma thermal power depo-sition occurs far away from the surfaces, much higher ther-mal energies can be delivered to the gas.

a�Present address: Numeca International, Brussels, Belgium.b�Author to whom correspondence should be addressed. Electronic mail:

lraja@mail.utexas.edu.

JOURNAL OF APPLIED PHYSICS 106, 063305 �2009�

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Successful design of the microthruster requires a de-tailed understanding of microdischarge phenomena as wellas plasma-flow interactions. Experimental investigations ofthe microdischarge are hindered by the extremely small ge-ometries, where well-established plasma diagnostic tech-niques are not applicable. High-fidelity computational mod-els can therefore have a big impact in developing afundamental understanding of the MPT.

In previous studies, Kothnur and Raja9 studied direct-current microdischarges in a flowing gas stream for applica-tions in electrothermal microthrusters. For currents around 1mA, the microdischarge is found to operate in an abnormalglow mode with positive differential resistivity. An increasein input electrical power results in an almost linear increasein the gas temperatures; this property of microdischarges is akey feature that can be exploited in the MPT concept. Ara-koni et al.10 studied a nominal microdischarge configurationwith a flowing gas that resembles a thruster, although theflow modeling approach provides an inadequate descriptionof the gas dynamics expansion through the nozzle and sub-sequent expansion into vacuum. Gas temperatures exceeding1000 K are reported for power densities of tens of kW /cm3

at upstream pressure of tens of Torr. They found that thenozzle length and the location of the discharge in the nozzlehave an important influence on the incremental thrust �abovethat of the cold flow�.

Section II describes the mathematical model and numeri-cal approach used in this study. The model comprises aplasma module coupled to a flow module. The plasma mod-ule provides a self-consistent, multispecies, multitemperaturedescription of the microdischarge phenomena while the flowmodule provides a description of the low Reynolds numbercompressible flow through the MPT. Section III describessimulation results and a discussion of the MPT phenomena.Finally, we conclude in Sec. IV.

II. METHODOLOGY

Coupled plasma and gas flow phenomena are encoun-tered in a number of applications requiring the developmentof an integrated plasma model coupled to a gas dynamic flowmodel.11 In the MPT, the microdischarge plasma is in ahighly nonequilibrium state with disparate electron and gastemperatures and nonequilibrium finite-rate chemical effects.The gas flow is characterized by low Reynolds number vis-cous dominated effects combined with high Mach numbercompressibility effects owing to the expansion of the rela-tively high pressure ��100 Torr� micronozzle gas streaminto vacuum. Both the plasma and flow phenomena arestrongly coupled: The plasma causes gas heating, whichmodifies the gas density, and hence the flow field; the gasflow velocities, in turn, affects the distributions species andtemperature of plasma in the discharge. The integratedplasma-flow model used in this study is described below.

A. Plasma module

1. Governing equations

The densities of individual species in the discharge aredetermined by the species continuity equation

�nk

�t+ �� · �nkV� � + �� · �� k = Gk, k = 1, . . . ,Kg �k � kb� ,

�1�

where nk is the number density of species k, V� is the mass-averaged gas flow velocity, �� k is the drift-diffusion flux of

species k, Gk is the gas-phase species generation rate throughplasma chemical reactions, and Kg is the total number of gasspecies. A dominant neutral species is identified as the“background” species, and is designated as kb; its densitybeing determined to satisfy the ideal gas law

P = nkbkBTg + nekBTe + �

k�ke,k�kb

nkkBTg, �2�

where P is the local pressure, kB is Boltzmann’s constant,and subscripts e and g denote electrons and the backgroundgas, respectively.

The electron temperature, Te, is explicitly computed inorder to determine the reaction rates due to electron impactand the electron transport properties. For this purpose, thefollowing electron energy equation is solved:

�

�t�3

2kBneTe� + �� · �5

2kBneTeV�� + �� · �5

2kBTe�� e�

− �� · ��e�� Te� = e��� e + neV� � · �� � −3

2kBne

2me

mkb

�Te

− Tg��e − e�j=1

Ig

�Ejerj , �3�

where �e is the thermal conductivity of electrons, me and mkbare the molecular masses of electron and dominant back-ground gas, respectively, �e is the electron momentum-transfer collision frequency with the background gas, �Ej

e isthe energy lost per electron �in eV� in an inelastic collisionevent represented by a gas-phase reaction j, rj is the rate ofprogress of a reaction j, and Ig is the total number of gas-phase reactions. In our model, we assume a common tem-perature for all heavy species �ions and neutrals�, this tem-perature being solved for in the flow module.

The self-consistent electrostatic potential is determinedusing Poisson’s equation

�2� = −e

�0�

k

Zknk, �4�

where � is the potential, e is the unit electric charge, �0 is thepermittivity of free space, and Zk is the charge number ofspecies k �e.g., �1 for electrons�.

The ion flux is evaluated as niV� +�� i=niu� i, where u� i is theion fluid velocity, which is determined using the first momentof the ion species Boltzmann equation, i.e., the ion speciesmomentum equation

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�niu� i

�t+ �� · �niu� iu� i� = −

Zieni

mi�� � −

kBTi

mi�� ni − ni�u� i − V� ��i.

�5�

The first term on the left-hand side of Eq. �5� is the unsteadyterm and the second term is the ion inertia term, which isnon-negligible as the pressure approaches vacuum conditionsnear the MPT exit. The first term on the right-hand side�RHS� of Eq. �5� is the electrostatic force on the ion fluid, thesecond is the gradient diffusion of ion fluid momentum, andthe last term is the loss of ion fluid momentum owing tocollisions �friction� with other species. The last term assumesthat the ion density is small compared to the dominant back-ground species density, whose fluid velocity corresponds tothe mean-mass velocity of the gas. In Eq. �5�, Ti is the iontemperature, which is assumed equal the local heavy species�gas� temperature Tg and �i is the ion momentum-transfercollision frequency. The mass-averaged flow velocity and thegas temperature are solved for using the flow module dis-cussed in Sec. II B.

2. Transport formulation

For species other than the ions �neutrals and electron�,the species number flux is evaluated using the drift-diffusionapproximation, given as

�� k = − �knk�� � − Dk�� nk, �6�

where �k=Zke /mk�k is the charged species mobility and Dk

=kBTk /mk�k is the species diffusion coefficient. The electronthermal conductivity appearing the electron energy equationis given as �e= �5 /2�neDe. The operating �propellant� gasconsidered for this study is argon. The electron transportproperties are determined a priori by solving a zero-dimensional electron Boltzmann equation �“BOLSIG+”12�with appropriate electron energy-dependent cross sectionsfor a range of reduced electric fields �E /N�. The electronproperties and the electron mean energies are determined asa function of E /N by the Boltzmann solver. These results arethen used to tabulate the properties as a function of the elec-tron mean energy �temperature� for use in the plasma dis-charge simulations. The ion transport properties are derivedfrom experimental mobility data13 based on which the colli-sion frequency �i can be determined. Hard-sphere cross sec-tions k based on the Lennard-Jones interaction potentials14

are used to determine the neutral transport properties using�k=nkb

gkk, where gk is the relative thermal speed of speciesk.

3. Boundary conditions

For the electron continuity equation, the electron flux atthe solid walls is specified using a kinetic Maxwellian fluxconditions for consumption of electrons at the walls with areturn flux of secondary electrons from the walls, i.e.,

�� e · n =1

4ne�8kBTe

me�1/2

− ��� Ar+ · n , �7�

where n is the unit normal vector pointed toward the wall.The solid walls include the electrode and dielectric surfaces.

The first term in Eq. �7� is the Maxwellian flux of electronsto the surface and the second term is the secondary electronemission flux from the surface. In this study, we assume thatthe secondary emission coefficient, �, includes contributionsfrom ion impact, fast atoms, metastable atoms, dimers, andultraviolet photons.

For the ions’ continuity and ions’ momentum equations,the flux of positive ions is set to 0 for all solid boundaries forwhich the electric field points away from the wall �E� · n�0�. If the electric field points toward the wall �E� · n 0�,the ion flux is extrapolated from the interior by imposing azero gradient of the ion flux at the boundary.

The Maxwellian flux condition is imposed for the neutralspecies, given by

�� n · n =1

4nn�8kBTg

mn�1/2

. �8�

For the electron energy equation, the following energy flux isimposed at the solid walls:

QeW = 5

2kBTe�eW, �9�

where �eW is the electron wall number flux.

The potential on dielectric surfaces is determined usingGauss’ law for the dielectric-gas interface boundary condi-tion, which depends on the total accumulated surface chargedensity at the surface.9 The equation for evolution of the netsurface charge density �s is given by

��s

�t= �

k=1

Kg

eZk�� k · n . �10�

4. Plasma chemistry and surface processes

A pure argon plasma gas chemistry is used, which com-prises six species: electrons �e�, atomic argon ions �Ar+�,molecular argon ions �Ar2

+�, electronically excited atoms�Ar��, electronically excited molecules �Ar2

��, and the back-ground argon atoms �Ar�. Dimer species are included be-cause of the relatively high pressures in the upstream sectionof the micronozzle ��100 Torr�. The list of reactions con-sidered in the study is given in Table I and comprises elec-tron impact ionization and excitation reactions, Penning ion-ization reactions, three-body reactions for dimer excitedspecies and ion formation, quenching, and de-excitation re-actions. As in the case of electron transport properties, theelectron impact reaction rates are determined a priori bysolving the zero-dimensional electron Boltzmann equation�BOLSIG+ 12� with appropriate electron energy-dependentreaction cross sections. These rates are tabulated as a func-tion of the electron temperature using the same procedure asfor the electron transport properties.

At solid surfaces all excited species and charged speciesare assumed to get quenched with unity sticking coefficient.Upon quenching at surfaces, each dimer ion and excited spe-cies is assumed to return to plasma as a pair of ground stateneutral Ar atoms, while the monomer species return as singleAr atoms.

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B. Flow module

Conservation equations for the gas mass density, mass-averaged gas velocity, and the gas energy are solved usingcompressible Navier–Stokes equations in axisymmetricform. These equations can be written as

dU

dt+ �� · F� inviscid = �� · F� visc + S , �11�

where U represents the conservative variables ��, �Vx, �Vr,and �et� �� is the background density, Vx is the axial velocity,Vr is the radial velocity, and et is the total internal energy ofthe gas�, F� inviscid represents the inviscid flux terms, F� visc rep-resents the viscous flux terms, and S represents the sourceterms.

The flow solution influences the plasma dischargethrough the pressure, temperature, and velocity fields appear-ing in the plasma governing equations �Eqs. �1�–�5��. On theother hand, electrostatic forces and electrothermal heating actas external body forces and external heat source, respec-tively, in the Navier–Stokes equations. These source termsare defined as

f� = �i

mini�u� i − V� ��i − eneE� �12�

for the electrostatic force, and

Sheat = − �J�e�h

Zk�� k · �� �� +3

2kBne

2me

mkb

�Te − Tg��e

− e�j=1

Ig

�Ejgrj �13�

for the electrothermal heating, such that S= �0, fx , fr ,Sheat�T.The first term on the RHS of Eq. �13� represents the effectiveion Joule heating, taken here to be a fraction �J �0��J

�1� of the local ion Joule heating �J�ion ·E� , where J�ion is the

total ion current density and E� is the local electric field�. Thisfraction is �1 for cases where the ion mean free path iscomparable or larger than the characteristic length scale forthe plasma, which results in incomplete conversion of thelocal kinetic energy gained by the ions in the electric field toion/heavy species thermal energy. For microdischarge condi-tions, we use a fixed value of �J=0.25 as suggested in Ref.15. The Joule heating terms appearing in the electron energyequation �Eq. �3�� and in Eq. �13� are evaluated using a spe-cies flux reconstruction approach.16

C. Numerical approach

Both the plasma governing equation and the compress-ible flow Navier–Stokes equations are spatially discretizedusing a cell-centered finite volume approach on an unstruc-tured mesh with mixed mesh cell types. A steady state solu-tion is sought in all cases. Both the plasma and flow govern-ing equations are solved as transient problems with timestepping of the solution to a steady state. The plasma gov-erning equations are solved using an implicit time-discretization approach with local linearization of the gov-erning equations at each time step. For the flow governingequations, the inviscid flux terms are evaluated with the ad-vection upstream splitting methods �AUSM� �Ref. 17� andthe viscous flux terms are evaluated using the Haselbacherapproach.18 The flow equations are also solved using implicittime discretization, with a dual-time stepping approach toiteratively solve for the solution at each time step. The vis-cous terms in the flow equations require computation of thesolution variable gradients at cell centers. We use a Green–Gauss method to reconstruct these gradients based oncell-centered values of these variables. Finally, the com-pressible flow can develop shock discontinuities. TheVenkatakrishnan19 flux limiter approach is used to stabilizeand produce monotone solutions in the presence of such dis-continuities.

D. Physical operating conditions

Figure 1 shows the geometry of the MPT used for thesimulations. The geometry consists of an axisymmetric con-stant area “pipe” section of 500 �m length, followed by adiverging section that is 200 �m in length, which is termi-nated by a 150 �m long constant area section. The radius ofthe upstream constant area pipe section is 50 �m, and theexit section is 150 �m in radius. The ring shaped electrodeshave an axial thickness of 150 �m, while the dielectric layer

TABLE I. High pressure argon plasma gas-phase chemistry used in thisstudy.

No. Reactions Reaction ratea Ref.

G1 e+Ar→e+Ar b 12G2 e+Ar→e+Ar� b 12G3 e+Ar→2e+Ar+ b 12G4 e+Ar�→2e+Ar+ b 12G5 e+Ar�→e+Ar b 12G6 e+Ar+→Ar� 4.0�10−13Te

−0.5 21G7 2e+Ar+→Ar�+e 5.0�10−27Te

−4.7 cm6 s−1 21G8 e+Ar2

+→Ar�+Ar 5.38�10−8Te−0.66 21

G9 2Ar�→Ar++Ar+e 5�10−10 21G10 2Ar2

�→Ar2++2Ar+e 5�10−10 21

G11 Ar�+2Ar→Ar2�+Ar 1.14�10−32 cm6 s−1 21

G12 Ar++2Ar→Ar2++Ar 2.5�10−31 cm6 s−1 21

G13 Ar2�→2Ar 6.0�107 s−1 21

G14 e+Ar2�→2e+Ar2

+ 9�10−8Te0.7 exp�−3.66 /Te� 21

G15 e+Ar2�→e+2Ar 10−7 21

aRate coefficients have units of cm3 s−1, unless mentioned otherwise. Elec-tron temperatures, Te, are in eV.bTabulated rate coefficient as a function of mean electron temperature wasobtained by the Boltzmann equation solver BOLSIG+ �Ref. 12�.

Cathode

Anode Dielectric150mm

Inlet:P

0= 100 Torr

T = 300K

Symmetry BC (axisymmetric)

Outlet:P= 0.01 Torr

550mm 150mm150mm

50mm

FIG. 1. Schematic of the MPT device and computational mesh. The geom-etry is cylindrically symmetric.

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has an axial thickness of 550 �m. The mesh consists ofabout 3600 cells, which includes a combination of trianglesand quadrilaterals.

The flow direction is from left to right in Fig. 1. Theflow enters the constant area pipe section on the left and exitsthe domain along the curved �arc-shaped� boundary on theright. Four boundary sections are not modeled as solid wallsfor the gas discharge governing equations. Zero-flux bound-ary conditions are imposed at the boundary section formedby the symmetry axis. For numerical stability reasons, zero-flux boundary conditions are also used for the gas dischargegoverning equations at the inlet and outer cathode surfaceboundaries. These boundaries are sufficiently away from themain discharge region and do not influence the results. Thefarfield boundary on the right side of the computational do-main is modeled as an “outflow” boundary, where the plasmavariables �species number densities and electron energy den-sity� are convected away by the gas flow, while zero-fluxboundary conditions are imposed for Poisson’s equation. Thevalue of the plasma variable from the adjacent interior cell isused to interpolate the flux at this farfield boundary. Thepower input is provided by applying a fixed positive dc volt-age at the anode �without ballast resistance�, while the cath-ode is grounded. For the base case, the secondary electronemission coefficient is set at a value of 0.03, correspondingto a nickel surface interacting with an argon plasma.20

The inlet total �stagnation� pressure is 100 Torr �for thebase case�, and a small but nonzero outlet pressure �0.05Torr� is required to stabilize the numerical scheme in the“vacuum” part of the domain. The inlet gas static tempera-ture is fixed at 300 K. The solid wall temperatures are fixedat 300 K. The inlet flow velocity is computed self-consistently using the inlet flow total conditions, the inletstatic pressure extrapolated from the interior of the domain,and isentropic expansion relations. No-slip boundary condi-tions are applied at the solid walls for the momentum equa-tion. The Knudsen numbers for the flow are sufficiently lowover much of the MPT device length �about 0.01 at the inletto about 0.08 at the exit plane� that slip-flow and wall-temperature jump boundary conditions are not necessary.This has been verified by running simulations with the ap-propriate jump boundary conditions.

III. RESULTS AND DISCUSSION

A. Base case

Table II lists discharge conditions for which results arepresented. For the base case, the discharge voltage is set at avalue of 750 V, and the computed discharge current is 0.87mA. The total stagnation pressure �100 Torr� and static tem-perature �300 K� imposed at the inlet determine the inletvelocity, which is computed at about 100 m/s correspondingto a flow rate of 5.2 SCCM. Plasma properties at steady stateare shown in Fig. 2 for the base case. The space chargedensity is high enough that the bulk plasma extends well intothe diverging part of the micronozzle creating a hollow-cathode-like annular cathode sheath. Under these operatingconditions, the cathode sheath is about 100 �m thick �seeFig. 2�a��, occupying a significant fraction of the total dis-charge volume. In the cathode fall, the potential drops bynearly 750 V over �100 �m producing a characteristic fieldstrength of �75 kV cm−1 and a reduced electric field �elec-tric field/gas number density� of �105 Td. Electron densitycontours are presented in Fig. 2�b� and show two localmaxima. The first peak is inside the constant area pipe sec-tion ��7�1019 # /m3�, and the second peak is in the diverg-ing section of the nozzle ��3�1020 # /m3�. The well-defined cathode sheath structure observed in Fig. 2�a� isapparent in the density contours with an abrupt drop in theelectron densities in the diverging micronozzle section. TheMPT discharge clearly operates in the glow discharge moderather than the Townsend/predischarge mode. The electrontemperature contour is shown in Fig. 2�c�. The electron tem-perature is only shown in regions of the discharge where theelectron number density is greater than 3�1017 # /m3, i.e.,10−3 of the peak value of the electron number density. Theelectron energy content is negligible in the rest of the do-main, although the electron temperatures can increase expo-nentially in these regions of vanishingly small electron den-sities owing to a numerical artifact of the fluid model. Theelectron temperature remains nearly uniform �2 eV� in thebulk plasma over most of the constant area pipe section, andgradually increases in the diverging section of the devices totemperatures around 3 eV at the cathode sheath edge.

The generation rate of electrons in the discharge throughgas-phase reactions is shown in Fig. 2�d�. Significant genera-tion of electrons is observed over the entire micronozzle withmaximum generation observed in the diverging section. This

TABLE II. Operating conditions considered for this study. The characteristic discharge dimension D is taken tobe the thickness of the dielectric layer located between the electrodes �550 �m� and the characteristic pressureis taken to be the inlet total pressure.

Case

Imposed inlettotal pressure

�Torr�PD

�Torr cm�Flow rate�SCCM�

Imposed voltage�V�

Current�mA�

Power�mW�

Base case 100 5.5 5.2 750 0.87 650Larger power input 100 5.5 5.2 1000 1.8 1800Larger pressure/flow rate 200 11.0 13.0 750 0.75 560Larger cathode temperature 100 5.5 5.2 750 0.72 540Larger �J 100 5.5 5.2 750 0.72 540Lower �eff 100 5.5 5.2 750 0.33 250

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location corresponds to a relatively higher electron tempera-ture �3 eV� and reasonably high background densities inthe expanding gas flow. The argon monomer ion �Ar+� anddimer ion �Ar2

+� number densities are shown in Figs. 2�e�and 2�f�, respectively. Atomic argon ions constitute the domi-nant ion species in the microdischarge. High backgrounddensities favor the three-body reaction G12 �see Table I� thatforms dimer species. Therefore, most dimer ions are locatedin the constant area pipe section, where the pressure is rela-tively high. The dielectric surfaces support a net negativecharge owing to electron trapping, which in turn supports apositive sheath over the entire dielectric length between theanode and the cathode. This is apparent from the excess of

positive ions compared to electrons, just above the dielectricsurfaces. The net negative charge at the dielectric enforcesthe electron wall flux to equal the total positive ion wall flux�at steady state�, so no net current is drawn through the di-electric surfaces.

The Ar+ ion number flux, computed using the ion mo-mentum equation �Eq. �5��, is shown in Fig. 3. Most of theions are generated in a localized region close to the axis inthe diverging portion of the device and subsequently driftand diffuse away from this region. Most of the ions eventu-ally drift toward the cathode and the dielectric surface in thevicinity of the cathode, where they are quenched. A negligi-bly small fraction of the ions ��1%� leave the nozzle. Con-

(a) Electrostatic potential (V) (b) Electron number density (#/m�)

(c) Electron temperature (eV) (d) Electron generation rate (#/m�-s)

(e) Ar+ number density (#/m�) (f ) Ar2+ number density (#/m�)

FIG. 2. �Color online� Plasma properties in the MPT. The inlet total pressure is 100 Torr �13.3 kPa� and the mass flow rate is 5.2 SCCM. The applied potentialdifference between the electrodes is 750 V and the power input is 650 mW.

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sequently, the electrostatic ion thrust contribution to theoverall thrust produced by the MPT device is negligible.

The base case simulation was also run with the drift-diffusion approximation �Eq. �6�� used for ion transport, in-stead of the ion momentum equation �Eq. �5��. Comparisonof the discharge results for the drift-diffusion ion simulationcase �not shown� with the ion momentum case indicates thatthe results can be significantly different �e.g., the peak valueof the monomer ion number densities differs by about 20%between the two cases�, confirming the non-negligible role ofthe ion inertia in the MPT configuration.

The flow field solution for the base case is shown in Fig.4. The major influence of the plasma on the flow field isthrough the heat addition that is shown in Fig. 4�a�. We haveconfirmed through additional simulations that the electro-static forcing source term in the flow momentum equation�Eq. �12�� has a negligible effect on the overall flow anddischarge solutions. The thermal power deposition to gas�Eq. �13�� originates in large part from ion Joule heating inthe cathode sheath. A peak in the gas heating is observedclose to the MPT surface at the corner between the dielectricand the cathode where the electric fields are the highest. Thegas temperature contours are shown in Fig. 4�b�. The gastemperature contours show a peak value of about 950 K. Thelocation of the peak temperature is slightly downstream andcloser to the axis compared to the location of peak thermalpower deposition. The localized power deposition in the im-mediate vicinity of the cathode surface leads to significantheat loss to the solid wall, indicating that thruster perfor-mance could be improved in the future by optimizing thedesign to minimize this loss.

The axial and radial velocity components, Mach number,and pressure fields are shown in Figs. 4�c�–4�f�. The axialvelocity increases rapidly at the start of the diverging sectionowing to gas dynamic expansion, followed by a small de-crease owing to compression waves launched from thestraight cathode section. Subsequently, the axial velocity in-creases once it expands out the exit plane. Peak axial veloci-ties greater than 550 m/s are seen outside the exit plane. TheMach number contours follow the trends in the axial veloc-ity, with an increase at the start of the diverging section,

followed by a decrease near the cathode section �owing toaxial velocity decrease as well as an increase in the soundspeed due to high temperatures�, and a subsequent increasebeyond the exit plane. The radial velocity profiles reflect themass continuity requirements for the gas expansion process.The pressure profile shows a gradual, almost linear Poi-seuille, pressure drop in the straight pipe section of the mi-cronozzle, followed by a rapid decrease in the diverging sec-tion and out the exit plane. We must note that no sharp shockdiscontinuities are observed in the gas flow solution, owingmainly to the very low Reynolds number viscous dominatednature of the flow, despite the high Mach numbers. For thesebase case operating conditions, the computed thrust is100 �N �compared to 67 �N for the cold gas micronozzle�,which corresponds to a specific impulse of 74 s. The electro-static component of thrust �owing to ion momentum leavingthe exit plane� is found to be negligible and the thrust isalmost entirely due to neutral momentum transport out theexit plane.

Figure 5 shows a schematic of the energy flow pathwaysin the MPT. The electrical power input �the discharge currenttimes the discharge voltage� for the base case is 650 mW. Ofthis, most of the power �about 550 mW� is attributed to ionJoule heating integrated over the entire domain �VJ�ion ·E� dV�,while the electron Joule heating �VJ�e− ·E� dV� is about 45mW. The mismatch between the input electrical power andthe integrated Joule heating in the domain �ion and electron�is attributed to numerical discretization error. According toour model a large fraction of the ion Joule heating is lost tothe wall while the remaining is thermalized to provide asource term to the gas energy equation. This fraction is set bythe parameter �J, which equals 0.25 in these simulations.Consequently, 75% of the intrinsic ion Joule heating is lostdirectly to the wall surfaces. A small fraction �less than 1%�of the power deposited in the heavy species thermal pool�gas energy� is lost through inelastic collisions, while themajority is lost at solid walls and the rest at the outflowsection. The rate of energy transfer from the electron thermalpool to the heavy species �gas� thermal pool through inelasticcollisions amounts to a small fraction of the electron Jouleheating �less than 1%�. A significant amount of power �35mW or �80% of the electron Joule heating� is lost throughinelastic collisions, while the rest of the power deposited inthe electron thermal pool is lost through transport at solidwalls and at the outflow section.

B. Effect of power input

Figure 6 shows the effects of power input on dischargecharacteristics. The results are presented for the base case�where the discharge voltage is fixed at 750 V� and for a casewhere the discharge voltage is fixed at 1000 V. For bothcases, the inlet total pressure is 100 Torr and the flow rate is5.2 SCCM. The plasma is more intense �i.e., the ionizedspecies number densities are higher� for higher power inputs.Electron and dominant ion Ar+ number densities increase bya factor of �2 as the power input is increased from 650 �basecase� to 1800 mW.

Cathode

Anode Dielectric

2x1022

Cathode

6x1022 1x1023 1.4x1023

FIG. 3. �Color online� Monomer �Ar+� ion number flux �# / �m2 s�� in theMPT. The contours indicate the magnitude of the ion number flux. Thevectors indicate the direction of ion flux, with the length of arrows beingproportional to the flux magnitude. The operating conditions correspond tothe base case indicated in Fig. 2 caption.

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Figure 7 shows the effect of power input on flow prop-erties. Contours of the power deposition in the MPT areshown in Fig. 7�i�. The net power deposition into neutral gasscales almost linearly with the power input as the ratio be-tween these two quantities remains equal to about 20% forboth cases. Gas temperature contours are shown in Fig. 7�ii�.A peak temperature of 1850 K is reached for the high powercase �compared to 950 K for the base case�. Importantly,these results show that changing the external power input isan effective method to control the level of gas heating in thedischarge, which, in turn, affects the gas temperature and thethrust produced by the device. This constitutes a key feature

of our proposed MPT concept. The computed thrust in-creases to a value of 128 �N for the high power case, whichcorresponds to a specific impulse of 95 s.

C. Effect of pressure/flow rate

We now vary the inlet total �stagnation� pressure to 200Torr �the base case inlet total pressure is 100 Torr�. Otherconditions such as the inlet static temperature �300 K� andthe discharge voltage drop �750 V� are kept the same as thebase case. The increase in the inlet total pressure causes anincrease in flow rate through the micronozzle to a value of 13

(a) Power deposition (W/m3) (b) Gas temperature (K)

(c) Axial velocity (m/s) (d) Radial velocity (m/s)

(e) Mach number (f ) Pressure (Pa)

FIG. 4. �Color online� Flow properties in the MPT. The operating conditions are indicated in Fig. 2 caption.

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SCCM �compared to 5.2 SCCM for the base case�. Contoursof the axial velocity are shown in Fig. 8�i�. Higher flowvelocities and gas mass densities �not shown� are observedthroughout the nozzle for the high pressure case. The changein flow field has a profound effect on the microdischargecharacteristics. The high velocities combined with decreaseddiffusive transport properties lead to a further downstreamconvection of the ionized species. The discharge also appearsmore constricted for the high pressure case with a slightly

lower peak charge density for the high pressure case. Thehigher pressures decrease the plasma conductivity resultingin a decrease in the discharge current to 0.75 mA �comparedto 0.87 mA for the base case�. The resulting input powerdecreases for the high pressure case to about 560 mW �com-pared to 650 mW for the base case�. The peak gas tempera-ture also decreases compared to the base case �600 K com-pared to 950 K, not shown� owing to lower power depositionand also because of the increased heat capacity of the higherpressure gas. Case studies were also performed with a lowertotal inlet pressure of 60 Torr. For these conditions, the mi-crodischarge could not be sustained for discharge voltagesranging from 750 to 1000 V.

D. Sensitivity to model parameters

1. Cathode temperature

Since a large fraction of the input power is lost to thewalls, the cathode temperature can be expected to be signifi-cantly higher than the constant 300 K assumed in the aboveresults. An accurate estimate of the cathode temperaturewould require solving an energy balance equation for theentire MPT device �including solid materials�.

The effect of the cathode temperature on the MPT dis-charge parameters and performance is studied by comparingthe base case solution to a case where the cathode tempera-

Power deposition (J · E )

Ion Pool Electron Pool

Direct walllosses

Heavy species pool

Inelasticcollisions Wall losses Outflow

Inelasticcollisions

Outflow

WalllossesαJ

(1-αJ )

J ion · E J e− · E

Elasticcollisions

FIG. 5. Energy flow pathways in the MPT.

(i) Electron number density (#/m�)

(ii) Ar+ number density (#/m�)

(a) Power input: 650 mW (b) Power input: 1800 mW

FIG. 6. �Color online� Electron and dominant ion �Ar+� number density contours for different values of the power input. The inlet total pressure is 100 Torr�13.3 kPa� and the flow rate is 5.2 SCCM for both cases.

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ture is fixed at 1000 K. All other conditions are kept the sameas the base case. Figure 9�i� plots the gas temperature con-tours for the two cases. The spatial distribution of gas tem-perature appears strongly dependent on the cathode tempera-ture. For the case where the cathode temperature is fixed at1000 K, the peak gas temperature reaches 1150 K �comparedto 950 K for the base case� and the region of high gas tem-peratures occupies a larger fraction of the MPT volume thanfor the base case. Increasing the cathode temperature reducesthe temperature gradients at the cathode surface, thereby de-creasing the net thermal loss.

The axial velocity contours are shown for the two casesin Fig. 9�ii�. As the cathode temperature is increased, thereduced heat losses allow for an increased expansion of thegas in the diverging section of the micronozzle. The com-puted thrust therefore increases from a value of 100 �N forthe base case to a value of 112 �N for the 1000 K cathodetemperature case. It should be noted that, for this highercathode temperature case, the power input has decreased to avalue of 540 mW, compared to 650 mW for the base case.The higher gas temperatures �and therefore lower back-ground Ar densities� result in a decrease in ionization. Thelower power/current can be attributed to the lower chargedensities and hence to the lower plasma conductivity.

2. Fraction of ion Joule energy thermalized with thegas

For high pressure microdischarges and conventional dis-charges at very low pressures, a large fraction of the kineticenergy of the ions is deposited directly onto dischargebounding surfaces, particularly the cathode. This phenom-enon has been simulated by Revel et al.22 using a particle-in-cell Monte Carlo approach for a one-dimensional argon dis-charge. For current densities of �20 A /m2, an interelec-trode distance of 1.5 cm, and a background pressure of 1Torr, they observed that �25% of the ion Joule energy isconverted to gas thermal energy, with the remainder beingtransferred directly to the cathode. Boeuf et al.15 used anominal fixed value of �J=0.25 for simulation studies of a100 Torr xenon microhollow cathode discharge, which is thesame value used in our study.

The sensitivity of our model results to this parameter isdetermined by varying �J to 0.5. The power deposition andgas temperature contours are shown in Fig. 10 for both cases.Since most of the thermal �gas� power deposition in the MPTis attributed to ion Joule heating, the thermal power deposi-tion increases significantly from a value of 140 mW for thebase case to 260 mW when �J=0.5. At the same time, thetotal input power to the MPT decreased to 540 mW �com-

(i) Power deposition (W/m�)

(ii) Gas temperature (K)

(a) Power input: 650 mW (b) Power input: 1800 mW

FIG. 7. �Color online� Power deposition and gas temperature contours for different values of the power input. The inlet total pressure is 100 Torr �13.3 kPa�and the flow rate is 5.2 SCCM for both cases.

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pared to 650 mW for the base case�. As described in Sec.III D 1, the increase in the gas temperature �to a peak valueof 1350 K� results in decreased neutral density, with a result-ing decrease in ionization rate and hence the charge density.Consequently, the plasma conductivity decreases resulting indecreased current and total power deposition into the dis-charge. The increased fraction of power deposited into gasimproves the efficiency of the device. The computed thrustincreases to 116 �N �compared to 100 �N for the basecase�.

Our model assumes a constant value of �J. In reality, thisparameter will depend on the local background pressure. In-deed, as the mean free path of ions decreases with increasingpressure, ions will deposit a larger fraction of their kineticenergy to the background gas. In future studies we will ex-plore more accurate strategies �e.g., hybrid models� to modelthis effect consistently.

3. Secondary electron emission coefficient

The value of the secondary electron emission coefficient,�, constitutes an important uncertain parameter of the plasmamodel. This coefficient is sensitive to surface conditions andits value can vary by several orders of magnitude dependingon the surface material and the cleanliness of the surface. Inthe above simulation results, we chose a constant value of

�=0.03 �corresponding to a nickel surface interacting withan argon plasma�.20 The sensitivity of our results to this pa-rameter is analyzed by comparing our base case to a casewith �=0.01. The discharge current is significantly reducedfor the model with the lower secondary electron coefficient�to a value of 0.33 mA�. Lower � results in fewer electronsreleased from the surface through Auger processes per ionimpact, which in turn results in a much weaker discharge.This behavior is characteristic of all gamma-mode dischargessuch as the direct-current microdischarge presented in thisstudy.23,24

Figure 11 shows the ion number density and gas tem-perature contours obtained with the two models for a dis-charge voltage of 750 V. While the peak ion number densityhas about the same value ��3�1020 # /m3� for both cases,the discharge activity appears more constricted for the lower� case. Since the ion Joule heating scales with the current, asignificant decrease in power deposition into the neutral gasis observed �not shown�. This results in a decrease in the gastemperature �shown in Fig. 11�ii�� and a decrease in the com-puted thrust to a value of 87 �N, compared to 100 �N forthe base case.

IV. CONCLUSION

Microdischarge and flow interaction phenomena for mi-cropropulsion applications have been studied using a detailed

(i) Axial velocity (m/s)

(ii) Electron number density (#/m3)

(a) Total inlet pressure: 100 Torr (b) Total inlet pressure: 200 Torr

FIG. 8. �Color online� Axial velocity and electron number density contours for different values of the inlet total pressure/flow rate. Other conditions are thesame as the base case. The discharge voltage is fixed at 750 V for both cases.

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self-consistent computational model. The model consists of aplasma module coupled to a flow module and is solved on anunstructured mesh framework. The plasma module providesa self-consistent, multispecies, multitemperature descriptionof the microdischarge phenomena while the flow moduleprovides a description of the low Reynolds number com-pressible flow through the micropropulsion system.

The MPT concept consists of a direct-current microdis-charge in a geometry comprising a constant area flow sectionfollowed by a diverging exit nozzle. For a discharge voltageof 750 V, a power input of 650 mW, and an argon mass flowrate of 5.2 SCCM, the computed thrust is 100 �N �com-pared to 67 �N for an equivalent cold gas micronozzleflow�, resulting in a specific impulse of the device is 74 s, afactor of �1.5 increase compared to the cold gas micron-ozzle. For these conditions, charged species densities on theorder of 1020 m−3 and peak gas temperatures of �1000 Kare predicted. The microdischarge remains mostly confinedinside the micronozzle and operates in an abnormal regime.The electrostatic component of thrust is found to be negli-gible for our current MPT configuration. Additional simula-tion studies on the MPT indicate that the power input has astrong influence on overall discharge properties. The netpower deposition into the neutral gas scales with the power

(i) Gas temperature (K)

(ii) Axial velocity (m/s)

(a) Cathode temperature: 300 K (b) Cathode temperature: 1000 K

FIG. 9. �Color online� Gas temperature and axial velocity contours for different values of the imposed temperature at the cathode. The flow rate is 5.2 SCCMand the applied potential difference between the electrodes is 750 V.

(i) Power deposition (W/m�)

(ii) Gas temperature (K)

(a)aJ= 0.25 (b)a

J= 0.5

FIG. 10. �Color online� Power deposition and gas temperature contours fordifferent fractions of ion Joule energy converted to gas thermal energy. Theflow rate is 5.2 SCCM and the applied potential difference between theelectrodes is 750 V.

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input, providing a method of controlling the gas temperatureand the thrust level of the MPT. This feature constitutes animportant advantage over traditional cold gas thrusters. Thesensitivity of the above results to important uncertain param-eters of the model has been studied. These parameters are thecathode temperature, the fraction of ion Joule heating that islocally converted to gas thermal energy, and the secondaryelectron coefficient. A higher cathode temperature is found toreduce the heat losses at solid walls and is beneficial tothruster performance, though in practice there will be an up-per limit to how high the cathode temperature can get. In-creased efficiency in the ion Joule heating conversion to lo-cal gas heating results in improved thrust and specificimpulse as does an increase in the secondary electron emis-sion coefficient.

ACKNOWLEDGMENTS

This work was supported by the Air Force Office ofScientific Research under Grant No. FA 9550-06-1-0176with Dr. Mitat Birkan as program manager.

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(i) Ar+ number density (#/m�)

(ii) Gas temperature (K)

(a)g= 0.03 (b)g= 0.01

FIG. 11. �Color online� Dominant ion �Ar+� number density and gas tem-perature contours for different values of the secondary electron emissioncoefficient. The flow rate is 5.2 SCCM and the applied potential differencebetween the electrodes is 750 V.

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