18.086 Final Project: Finite Element Modellingof Ionic Liquid Flow Through Porous

Electrospray EmittersRobert S. Legge Jr.∗

∗Massachusetts Institute of Technology/Department of Aeronautics and Astronautics, Boston, USA

Abstract—A numerical simulation based on finitedifferences was created to model fluid flow through thebulk of a porous metal electrospray emitter. The modelcalculates the maximum fluid flow, and correspondingionic current, for a given pressure head at the emittertip. Numerical results are given for variations in:emitter geometry, electric field strength at the emittertip, fluid properties, and finally pore diameter and theintroduction of porosity gradients of differing shapes.

I. INTRODUCTION

Traditional colloid thrusters utilize pressure fedcapillary emitter geometry to transport liquid to thetips of emitter structures, where Taylor cones areformed. This requires pressurization systems on-board the spacecraft which add both mass and com-plexity. The difficulties in fabricating small, uniformcapillaries also pose problems in terms of minia-turization of needle arrays. One way to avoid theseissues is to use externally wetted emitter geometrieswhere the liquid is drawn from its reservoir by capil-lary forces alone. Such passively fed systems supplyliquid at the rate established by the electrosprayemission process. The use of externally fed emittersin vacuum is possible with ionic liquids such asEMI-BF4[1], [2], [3]. These liquids are molten saltsat room temperature and exhibit extremely low vaporpressures making them ideal for use in vacuum.They are formed by positive and negative ions whichboth can be directly extracted and accelerated toproduce thrust when used in bipolar operation[4].

A. Porous Metal Emitters

A simple capillary electrospray emitter setup isshown in Figure 1. An applied potential betweenan extractor electrode and conductive liquid insidea capillary causes a liquid meniscus at the end ofthe capillary to deform into a conical shape. Thisstructure is called a taylor cone in honor of G.I.Taylor who first derived a mathematical model to

describe the phenomenon in 1964[5] for certain con-ditions. The apex of the cone emits a jet of chargedliquid which breaks up into charged droplets. Thesedroplets are then accelerated by means of an elec-trostatic force to a final exit velocity. In thrusterapplications, the electrostatic force applied to thecharged droplets is equal but opposite to the resultingthrust felt by the thruster device. In cases where thelocal electric field at the tip of the taylor cone ishigh enough, the electrostatic pull overcomes the ionsurface energy and ions are extracted directly fromthe liquid. This causes the ions to be acceleratedthrough the potential and achieve a much higherexit velocity, which is a more efficient propulsionsystem in terms of propellant consumption, albeit atthe expense of lower thrust.

Fig. 1. Capillary Electrospray Emitter Setup

B. Importance of Fluid Flow on Emitter Operation

Since the amount of thrust of an electrosprayemitter is rate limited by the number of ionic liquidions that are able to reach the emission region, thephysics behind fluid flow might greatly effect overallperformance. The fluid transport in porous metalelectrospray emitters is entirely passive which placeseven more importance on accurate flow models. Inthis paper, a first pass attempt will be made tocharacterize the maximum amount of current thatcan be extracted from a porous emitter in steady

state operation. The paper is presented as follows:first the numerical modelling is explained and var-ious discretization derivations are presented. Next,flow results are shown for a number of variationalparameters including emitter substrate properties,ionic liquid properties and the the effect of emittersize and the addition of a porosity gradient. Finally,conclusions are drawn as to what the results meanfor the operation of electrospray emitters.

II. PROBLEM SETUP

The numerical scheme will implement a finitedifference simulation of the two-dimensional flowequation (Equation 1) using Darcy’s Law. The sim-ulation will be solved in two dimensions and willassume axial symmetry for the emitter. Inputs forthe model will be a pressure head dirichlet boundarycondition on the emitter tip as well as various geo-metrical constants which detail the emitter porosityas well as fluid properties.

∂

∂x

(Kx (x)

∂

∂x(h)

)+

∂

∂y

(Ky (x)

∂

∂y(h)

)= 0

(1)Where Kx and Ky are the effective fluid con-

ductivity of the liquid in the porous media. For thisapplication, K will only vary with x so that in thesecond term, it can be considered a constant andtaken out of the expression.

A. Darcy’s Law and the Effective Fluid Conductivity

The fluid conductivity for the system is dependanton both parameters of the porous media, but alsoof some parameters which characterize the fluidproperties. The general relation which explains whyK is a conductivity is given below:

q = K∇h (2)

Which states that the flow rate per unit areais equal to the conductivity times the hydraulicpotential gradient. The general analytical expressionfor finding the value of the conductivity is given as:

K =κρg

µ(3)

Where κ is the intrinsic permeability of the poroussubstrate, ρ is the density of the fluid, g is thegravitational constant at sea level, and µ is thedynamic viscosity of the fluid.

The intrinsic permeability of the porous mediadepends on many factors including the average porediameter, the complexity of flow channels, the rel-ative reynolds number of the flow, and etc... It is a

quantity that is best found experimentally, althoughsome analytical solutions exist. One analytical modelfor the fluid conductivity is given by the Kozeny-Carman equation of the following form[6]:

K =ρg

µκ =

ρg

µ

d2φ3

180 (1− φ)2 (4)

Where φ is the total porosity and d is the represen-tive particle diameter. Of course, for our application,we only know the pore size and would need a wayto connect the pore size to the effective particlediameter. Figure 2 shows the fluid conductivity plot-ted as a function of particle diameter assuming atotal porosity of 0.75, which has been experimentallyfound for our samples.

Fig. 2. Fluid Conductivity given as a function of Pore Diameterfor the Kozeny-Carman Equation

A better way to find the effective fluid conduc-tivity is to use experimental data. Data providedby Mott Corporation, a porous metal supplier, wasused to generate an analytical approximation forthe intrinsic permeability as a function of bothdynamic viscosity and average pore diameter. Thepredictions of the model were based on curve fits tothe experimental data provided and are:

κ = 1.9734× 10−13 (µ)−2.0021 (DP )1.2542 (5)

Where DP is the average pore diameter in µm.Figure 3 shows the model predictions (solid lines)and the corresponding experimental values. Themodel provides a rough estimate for the intrinsic per-meability that will be utilized in the finite differencescheme.

B. Descritizing Darcy’s Law

The dicretization scheme used is a standard cen-tral difference scheme in the non varying porositydirection and a central difference scheme using a

Fig. 3. Intrinsic Permeability vs Pore Size and Viscosity as:predicted by the model (solid) and given by experimental data(dotted)

staggered grid to approximate the porosity gradientin the directions of varying porosity. The flow equa-tion can be re-written as:

0 =∂

∂x(K (x)Ux) +K (x)Uyy (6)

The term in the y direction is trivial, and aswas said before, will be approximated using thefollowing central difference scheme:

Uyy =Ui,j+1 + 2Ui,j + Ui,j−1

(∆y)2 (7)

The term in the x direction is a little morecomplicated. If we say that:

V = K (x)Ux = Ki+ 12,j

(Ui+1,j − Ui,j

∆x

)(8)

And then the derivative of this expression in thex direction is approximated by:

V′

=Vi,j − Vi−1,j

∆x(9)

V′

=K+Ui+1,j − (K+ −K−)Ui,j +K−Ui−1,j

(∆x)2

(10)Where K+ = Ki+ 1

2and K− = Ki− 1

2. Figure 4

shows a graphical description of the stencil used inthe above numerical approximation.

C. Simulation Geometry and Boundary Conditions

In order to model the emitter geometry effectively,the geometry to be modeled is a triangular regionshown in Figure 5. The right side, as well as the flattop piece, are modelled with homogeneous dirichletboundary conditions to model the constant head ofthe fluid reservior. This sets the zero head reference

Fig. 4. Numerical stencil used to model flow

point corresponding to atmospheric pressure condi-tions. The left side is modelled with a heterogenousdirichlet boundary that serves to set the electrostatictraction pressure, which drives the flow. The diag-onal side of the emitter as well as the bottom aremodelled as neumann conditions with zero-flow. Thebottom is an axi-symmetric line if we assume thatthe emitter is a cone.

Fig. 5. Geometry used in numerical scheme

The numerical descretization was acheived byusing a modified version of the MatLab numgridfunction. Simulations were carried out using threegeometries with different dimensions correspondingto those shown in Figure 6. The important geometricparameters include: the overall emitter length (L),the emitter width (w), the tip diameter (d), andthe emitter cone half angle (α). Table I shows thegeometry that was tested. The parameters that werevaried were L and W while d and α were keptconstant.

D. Run Parameters

Current flow results will be obtained as a functionof the applied voltage between the extractor elec-trode and the emitters. This voltage will be linked

Fig. 6. Geometrical Measurements

Sample # L W d θ[µm] [µm] [µm] [deg]

1 839 593 15 19.52 622 445 15 19.73 395 297 15 20.5

TABLE IEMITTER GEOMETRY

to a pressure head at the emitter tip through anelectrostatic model. The head will consist of thedifference between the enhanced electric field at theemitter tip and the electric field required to overcomesurface tension and produce flow from the tip. Theexcess pressure will be the driving pressure for theflow through the bulk material. The excess electricfield can be written as:

∆P =12ε0E

2tip −

2γrc

(11)

Where γ is the surface tension of the fluid and rcis the average radius of curvature of the emitter tip.The pressure head can be then written as:

h =∆Pρg

(12)

Results will be obtained for: the three differentemitter geometries and four different liquids as wellas three different porosity configurations. The ionicliquids to be tested each have different physicalproperties (e.g. density and viscosity) that will affectperformance. Table II shows the properties of thefour liquids. In addition, three different porosityconfigurations will be tested, including: a constantporosity with a pore diameter of 0.5µm, an 5:1linear porosity gradient with 2.5mum at the baseand 0.5µm at the emitter tip, and a 5:1 non-linearporosity gradient with 2.5mum at the base and0.5µm at the emitter tip. The porosity gradients areplotted in Figure 7.

Fig. 7. Different Porosity Functions

Ionic Liquid # Density Viscosity[kgm−3] [cP ]

EMI-BF4 1271 37EMI-IM 1517 41

C5MI-(C2F5)3PF3 1600 61EMI-(CF5SO2)2N 1590 140

TABLE IIIONIC LIQUID PROPERTIES

III. DATA PROCESSING

The numerical simulation solved for the pressurehead in the porous emitter. In order to solve for theflowrate at the emitter tip, the following equationmust be used:

q = K∂h

∂x(13)

Where we obtain the pressure gradient at the tipthrough the following approximate formula:

∂h

∂x≈hi(tip)−1,j − hi(tip),j

∆x(14)

This gives the flowrate per unit area passingthrough the emitter tip as:

q = K∂h

∂x(15)

And then the actual flowrate passing through theemitter tip is then:

q = qπ (rc)2 (16)

By knowing the limiting flowrate, we can thensolve for the maximum current that can be emittedfrom the emitter tip, as this is what ultimately effectsperformance in our application. The relationshipbetween current and liquid flowrate is given as:

I = qρ

(q

m

)(17)

Where ρ is the fluid density and qm is the average

charge to mass ratio of the emitted ions, which forthis analysis will be set at 440000 C/kg whichcorresponds to the liquid EMI-BF4 operating in thepositive polarity. In a more detailed analysis, thecharge to mass ratios for the different ionic liquidswould have to be experimentally found.

IV. RESULTS AND DISCUSSION

A. Effect of Increasing Pore Size

As the average pore diameter d is increased, wewould expect that the amount of emitted currentwould increase due to an increase in the hydraulicconductivity. Figure 8 shows results from the numer-ical model. Shown is the maximum current plottedas a function of emitter tip electric field for variousvalues of average pore diameter. We see that as theelectric field increases, the current increases. A moreinteresting result is that as the average pore sizeincreases, the amount of current that increases perunit increase in electric field increases.

Fig. 8. Current vs. Electric Field for Different Average PoreSizes

B. Effect of a Porosity Gradient

One way that has been postulated to increase thefluid flow to the emitter tip would be to introducea porosity gradient to actively pump fluid to theemitter tip. Figure 9 shows a resulting pressurehead profile along the length of the emitter forthe different porosity gradient configurations. Sincethe amount current that can be extracted is directlyproportional to the slope of the head at the emittertip, we want the pressure drop to be largest here.In terms of this criterion, we would expect that the

most current would be produced by the non-lineargradient case, and the least to be produced by theconstant porosity case.

Fig. 9. Pressure Head as a Function of Emitter Length forDifferent Porosity Gradients

Figure 10 shows the emitted current as a functionof average pore diameter at the emitter tip forthe four porosity configurations. We see that asexpected, the non-linear porosity gradient has thelargest emitted current while the constant porositycase has the least. We see that by introducing aporosity gradient we can significantly increase themaximum emitted current, while at the same time,we can make sure that the fluid preferentially travelsto the emitter tip.

Fig. 10. Current vs. Pore Size for Different Porosity Gradients

C. Effect of a Emitter Size

Another aspect of electrospray emitters that canbe modelled using this technique is what happens asthe whole emitter is miniaturized. On first pass, it isobvious that as the emitter’s size is reduced, and itstip geometry and aspect ratio remain constant, that

the amount of current that can be extracted shouldincrease, since the fluid has less distance to travelthrough the porous emitter material. Figure 11 showsthe results of the numerical simulation for emittersof three different sizes. We see that as the emittersize is reduced, the maximum amount of emittedcurrent increases significantly. Figure 12 shows themaximum amount of current plotted as a functionof emitter porosity at the tip, which also shows thesame trend.

Fig. 11. Current vs. Tip Electric Field for Emitters of DifferentSizes

Fig. 12. Current vs. Pore Size for Emitters of Different Sizes

D. Effect of Different Ionic Liquids

One final variation that was calculated was howliquids with different physical properties affect themaximum emitted current. Four liquids were runwith the simulation and we see, in Figure 13, that thedifferent liquids have a large effect on fluid transportthrough the emitters. The results are primarity afunction of the liquid’s viscosity and surface tension,and that EMI-IM will emit the most current.

Fig. 13. Current vs. Pore Size for Different Ionic Liquids

V. CONCLUSIONS

The results of this project have shown that theflow of ionic liquids through the porous emittersis not the limiting factor in the magnitude of theemitted current. Additionally, the model can be usedin the future to explore better emitter geometries,for increasing or decreasing the amount of emittedcurrent for a particular application. The model willalso be useful in the future to explore how emittersin an array will impact each other in terms ofperformance, and what the resulting flow nets looklike. Lastly, the code could also be implemented toexplore the effects of pore blocking and clogging onperformance.

REFERENCES

[1] I. Romero-Sanz and R. Bocanegra and M. Gamero-Castanoand J. Fernandez de la Mora, “Source of Heavy MolecularIons Based on Taylor Cones of Ionic Liquids Operatingin the Pure Ion Evaporation Regime,” Journal of AppliedPhysics, vol. 94, No. 94, pp. 3599-3601, 2003.

[2] Paulo Lozano and Manuel Martinez-Sanchez, “Ionic LiquidIon Sources: characterization of externally wetted emitters,”Journal of Colloid and Interface Science, No. 282, pp. 415-421, 2005.

[3] Robert S. Legge, Paulo C. Lozano and Manuel Martinez-Sanchez, “Fabrication and Characterization of Porous MetalEmitters for Electrospray Thrusters,” 30th InternationalElectric Propulsion Conference, No. 145, 2007.

[4] Paulo Lozano and Manuel Martinez-Sanchez, “Ionic LiquidIon Sources: Suppression of Electrochemical Reactions Us-ing Voltage Alternation,” Journal of Colloid and InterfaceScience, No. 280, pp. 149-154, 2004.

[5] G.I. Taylor, “Disintegration of Water Drops in an ElectricField,” Proceedings of the Royal Society of London, Vol.280, pp. 383-397, 1964.

[6] Charles Harvey, “MIT 1.72 Groundwater Hydrology ClassLecture Notes,”