The Effect of Using Finite-Width Electrostatic Gates in Time-of-Flight Measurements for Colloid Thrusters *

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1 The Effect of Using Finite-Width Electrostatic Gates in Tie-of-Flight Measureents for Colloid Thrusters * Paulo Lozano and Manuel Martínez-Sánchez Massachusetts Institute of Technology Roo Vassar St Cabridge, MA plozano@it.edu and art@it.edu IEPC--286 In this paper we present the latest results in our tie-of-flight (TOF) spectroeter progra for analyzing colloid thruster plues. Preious results, particularly i n relatiely slow droplet beas, showed an inconsistency between the easureents and the theory of electrosprays operating in the cone-jet ode. This inconsistency was due to the type of electrostatic gating used. A atheatical analysis is deeloped in which the effect of Finite-Width gates is described and used to recoer as uch inforation as possible fro the data. This ethod could be extended to other analytical systes that reuire the use of electrostatic gates. Introduction Colloid thrusters, working on the principle of electrostatic extraction and acceleration of highly charged liuid droplets and/or ions, are one of seeral icrothruster technologies currently receiing attention for applications ranging fro ain propulsion of icrosatellites to high-precision altitude control and station keeping of constellation ebers. An introduction to the technology and a reiew of its earlier ipleentation as a ainline propulsion concept (in the 96 s and 97 s) can be found in []. One of the ost useful techniues to characterize these plues is by the use of Tie-of-Flight (TOF) spectroetry in which a bea of charged particles is forced to drift towards a detector in a pulse ode. The tie of trael fro the eitter to the detector gies enough inforation to estiate the specific charge of the different particles that copose the bea if its ean energy alue and spread are known. Specific charge and bea coposition are ery iportant to deterine the perforance of a Colloid Thruster as the propulsie efficiency and specific ipulse depend directly on those paraeters. A coon part of all TOF spectroeters is at least one electrostatic gate, which allows or stops the flow of particles towards the detector. The gating echanis introduces electric fields that could odify the energy leel of the particles. This is an iportant topic for ass spectroeter specialists, who hae created gates that present syetric potentials upstrea and downstrea along the particle s path when closed and therefore are alost free fro these effects. Soe of these deices are constructed fro ery thin conducting filaents closely spaced together haing alternating positie and negatie potentials (interleaed cobs). The ost eleentary type of syetric gate would be a set of parallel plates. Interleaed cob gates ay be unsuitable for colloid experientation since droplets could deposit in the filaents, possibly short-circuiting the. If finitewidth deices need to be used, gate effects that would odify the easureents can be expected. In this paper we discuss such effects and elaborate on a odel to describe the and to deterine to what extent useful inforation can be recoered fro affected data. Colloid Plues: Ion-Droplet Regie The physics of colloidal electrosprays, as presently understood [], offers the possibility of achieing high specific ipulses (> s.) with pure droplets * Presented as Paper IEPC--286 at the 27 th International Electric Propulsion Conference, Pasadena, CA, 5-9 October, 2.

2 only, but only at fairly high acceleration oltages. This is because, een with ionic conductiities as high as seeral Si/, the droplet specific charge achieable is liited to roughly that which would result fro all the ions of one polarity in the initial solution, for a droplet at the electrostatic breakup liit (Rayleigh liit). On the other hand, it is known that, under soe conditions, pure ions can be extracted fro liuids (FEEP being the extree case of liuid etals). So far only strong acids hae been shown to yield ions with no accopanying droplets [2], but there are preliinary indications [3] that ionic liuids ay in fact be ade to operate in the pure ion ode as well. For ions with single charge and olecular ass in the range of -4 g/ole, oltages of the order of -3kV, as reuired t o oercoe surface tension, tend to yield ery high specific ipulses, generally aboe the optiu range. An alternatie, which we are pursuing, is the use of ixed streas of ions and droplets in order to tailor the specific ipulse to the ission reuireents. The efficiency penalty due to the strea s polydispersity [] ay be acceptable for any issions. To characterize the colloid plues in this regie we reuire to perfor TOF easureents of both species. Most TOF spectroeters are specifically designed to detect ionic species, not droplets. In ost cases, they ake use of icro channel plates (MCP) as the detector eleent. The extree sensitiity of MCPs akes the ideal for species identification. They howeer, gie no direct indication of particle fluxes. In addition, it is not clear how the flow of droplets would affect the functioning of a MCP since there would also be unaoidable liuid deposition oer the detector surfaces. An alternatie to the use of a MCP is to substitute it by a sensitie electroeter capable of easuring extreely low current leels. We hae designed a TOF apparatus that separates droplets and ions by perforing easureents for each possible charged species present in the bea, allowing one to construct the spectru. This is accoplished by the use of two electrostatic finitewidth gates, one at the eission point and one near the detector. The gates open and close in such a way that they allow a single species through the TOF path for a gien tie during each cycle until the electroeter picks up enough charges to deterine the flux. The delay between the pulses is then odified to detect the next bea coponent. The process is repeated until the spectru is copleted. A coprehensie description of the custo designed TOF spectroeter can be found in [4]. Preliinary easureents done with this syste [4] showed that the principle works, but there was an inconsistency between the obserations and what can be expected theoretically for the fluid used. As it turns out, such discrepancies appear to be related to the type of electrostatic gate installed in the experiental setup. It was found that the droplet elocities easured were larger than expected; soehow the closing of the gates added energy to the bea, increasing its speed. Electrostatic Gate Modeling As discussed aboe, the electrostatic finite-width gates used in our experients present a potential barrier that could be either open or closed for a particle to reach the current collector or be stopped and bounced back. The siplest odel (which could be also the correct one in a charge-free spatial region) of the gate is a linear barrier grounded at its extrees as shown in figure. The paraeters that copletely describe this barrier are the position of the axiu oltage (x ), the position of the second ground (x ), the axiu oltage (φ ) and the tie (T) during which the gate oltage goes t o zero and particles are allowed to go through it. Shown also is the potential leel (V) of the particle bea. It is clear that, since V < φ, all particles are stopped when the barrier is up. φ x Figure Electrostatic gate potential barrier In our experients, the barrier is norally closed and opens only for a sall tie T. Defining x as the position of a particle at the tie when the barrier is φ V x x x

3 opened, the position of the particle at the oent when the barrier closes is x xt = x + T 2 V φ for x > x x = x + T 2 V for x < T () If the particle has enough energy to pass the peak potential of the barrier by the tie it closes, it will in general leae with a different energy than the one it had before entering the barrier influence zone. This is eident since the fields generated by the potential distribution are axially directed and they accelerate or decelerate charged particles depending on their positions. The different regions that share an energy leel in the exit plane are shown in table in ters of the non-diensional paraeters: x x V ξ = ξ = = x x φ 2 T = φ( T x) Table Regions with different energy characteristics Region ξ ξ T A ξ T < A2 ξ < < ξ T < A3 < ξt < ξ A4 ξ T ξ B ξ < ξt < B2 ξ > < ξt < ξ B3 ξ ξ T > (2) For each of these regions the non-diensional energy gain (or loss) is gien by ε = ( φt φ) V (3) Table 2 shows the energy leel for each region and whether it contains particles that pass or bounce fro the potential barrier. Table 2 Energy change of particles R ε Pass the barrier? A No A2 If ξ ξ + 2 is larger than T 2T A3 ξ ξ 2T ( ξ ) Yes A4 Yes B If ξ is less than 2T( ξ) 2 2 ( ) ( ) B2 ξ ξ 2T ξ ξ B3 ξ ξ Yes Yes The elocity at which particles leae the gate is then = s + ε with s = 2 V (4) s being the source particle elocity. Plotted in figure 2 are lines of / s with respect to the initial position ξ. Three cases are shown. In the first one (/ = 27 C/kg), all particles exit the gate with a elocity > s. The net effect that would be expected is a shift and broadening of the easured spectru for these conditions. In the second and third cases the region A4 appears as the region in which the elocity reains unchanged. For these cases a fraction of the particles leae the gate with elocities lower than the source elocity. For these plots the following paraeters were used: T = 3µsec, x =, x = 5, V = 7V and φ = 95V. T

4 inf / s inf / s inf / s.5.5 / = 27 C/kg / = 2 C/kg / = 3 C/kg Particle initial position x /x Figure 2 Velocity aps for different specific charges Data Analysis and Recoery There are two iportant applications that can result fro the analysis of the ideal finite-width electrostatic gate. The first is a detailed understanding of the effects that such gates hae on experiental easureents and the second is the possibility of recoering useful inforation fro degraded data. If we write the probability density function for haing a elocity between and + d as depending on the indiidual probability density functions f x x we hae ( ) and f x ( ) with x = ( ) ( ( )) = ( ), f d f x dx f, x d (5) and for a line of constant x d x dx = + d = x d (6) We obtain for the elocity distribution function ( )= x ( ) [ ( )], f f f x x d (, ) x (7) ( ) and x ( ) as the nuber of Defining nx particles per unit length and particle elocity at x, then the fraction of passing particles per unit length dx in the interal between x and x + dx is f x ( x )= x in ( ) nx nx dx x ax (8) ( ) [ ] defines the interal in which ( ) where xin, xax particles pass the barrier. Since the particle flux Γ ust be consered Γ= n = n( x ) ( x ) (9) Substituting (9) into (8) and using tables and 2, the passing particle fraction is f ( x )= Q x for ξ < ( ) > ξ 2 for ξ in which the coefficient Q is gien by ( )( + ) () Q= x + 2 T 2 T () Euations (-4, -) and tables and 2 can be used to copute the eleents of (7) for a gien set of physical paraeters, with the exception of those particles contained in region A4 for which a special treatent is reuired. Since all particles in region A4 pass the barrier without energy change, then the function fx ( x) is a constant and the partial deriatie in e. (7) is zero. This eans that f [ x( ) ]( ) (2) x x

5 and this uantity is a Dirac delta whose contribution can be added to the rest of the ters in e. (7). The delta in (2) can be written as [ ( )]( ) = ( ) x x A4 f x C δ (3) Using () and perforing the integral in the A4 region we copute the alue of C C = 2 T ξ ( T )( + T ) s (4) The additie contribution of region A4 into (7) is then ( f ) = f C A [ ( ) ] ( ) 4 [ ] 2 (5) V If the spectru of the easureent is known (known f ( )), then we can copute the integral in (7) directly to obtain the distribution function for the gate-leaing elocity. In the experiental results presented in [4], the working fluid was Tributyl Phosphate (TBP) doped with an ionic liuid. The dielectric constant of TBP solutions is relatiely low, and therefore the specific charge is liited to low alues. TOF easureents done by Gaero [5] for the sae fluid yielded / alues close to those predicted by the cone-jet electrospray theory of F. de la Mora [6] I f = = () ε ρq ρ γk εq (6) For the experiental conditions used in [4] (shown in table 3) a / alue of about 27 C/kg was expected, which corresponds to a tie of flight of 355µsec. Table 3 Experiental paraeters used in [4] Qty. Value Qty. Value f(ε) 7 T 3-6 s ε 8.9 x ρ 976 kg/ 3 x 5 γ.28 N/ V 7 V K.5 Si/ φ 95 V Q /s L drift 5 c As shown in figure 2, for a specific charge of / = 27 C/kg, the elocity at the exit plane is odified for all particles, in particular all of the suffer a net acceleration which depends on the point they were located at the tie the gate opened. The continuous line in figure 4 shows the experiental data fro [4]. The theory predicted a peak near 355µsec. Instead, a broad peak appeared close to 26µsec, significantly faster than expected. We can now perfor the exercise of calculating the effect the gate has on a pure spectral line. Assue that we know the ideal distribution function for specific charge, and suppose that it has a single peak at / = 27 C/kg, in the sae way as the expected experiental easureents of [4]. This function is shown as the finite width (Gaussian shaped) peak in figure 3. Distribution function f / Single peak centered at / = 27 C/kg Specific charge (/) Figure 3 Single peak at / = 27 C/kg The elocity distribution function is coputed directly fro (7) and the result is shown as the dashed line in figure 4. There are two iportant points that can be extracted fro this calculation: (a) there is no eidence of the noinal peak at 355µsec, which is in agreeent with the discussion aboe which lead t o figure 2. And (b) the peak was shifted to higher elocities and broadened as predicted. It is also interesting to see that the calculated peak resebles the one that was obtained experientally.

6 s pulse width Experiental data Nuerical calculation If the kernel atrix K is suare, then it can be decoposed in two unitary atrices U and V and a third diagonal atrix W.5 Noinal peak at t = 355usec K = UWV T (9) Current (na) Tie (icrosec) Figure 4 Coparison of experiental data with results fro the electrostatic gate odel This exercise is useful for erifying the assuptions ade in the electrostatic gate odel, but of course it would be uch ore interesting to sole the inerse proble: gien the elocity distribution function (experiental data) obtain the specific charge distribution. For this purpose, euation (7) can be iewed as Fredhol integral euation of the first kind for the unknown input spectru f. Discretization of the uantities inside the integral sybol by using a nuerical ethod, such as the trapezoidal rule n h f( x) dx = f( a) + f( b) + 2 f( xk) +... (7) b 2 a k= b a where h = n and xk = a+ kh, allow us to write (7) as a atrix euation of the for r r f = K f (8) where K is coonly known as the euation kernel. In principle one could find the solution by inerting the kernel. Howeer, for this kind of euations, the kernel is alost always singular to soe degree and inersion by coon ethods usually fails. Singular alue decoposition (SVD) can be used t o obtain an approxiation to the atrix inerse in a least-suares sense. Since an exact solution is not possible, soe inforation is always lost in the physical process triggered by the electrostatic gates. The inerse of (9) can be easily coputed, since the inerse of a unitary atrix is its transpose and the inerse of a diagonal atrix is also diagonal, fored by the reciprocals its eleents, T = [ ( j) ] (2) - K V diag /w U The proble appears when soe of these w j s are ery sall or zero. This iplies that there is no physical process in the odel that would yield a ode with such singular alues and therefore they would contribute only to noise in the syste. Inforation would be lost, as we predicted, if we reoe the, but we ust do so to be able to hae a good approxiation to the kernel inerse. Applying this procedure to the experiental data obtained in [4] yields the result plotted in figure 5. Distribution function f / Specific charge (C/kg) Figure 5 Results fro inerting the atrix A proinent peak can be seen close to / = 2 C/kg. Not exactly the 27 C/kg that was expected theoretically, but by far closer to it than the alue of 23 C/kg that can be estiated by only looking at the easured spectra. This is encouraging in the sense that the ethod was able to recoer soe useful inforation fro data taken in ery aderse conditions; naely, when the elocity of all particles was odified by the effect of the electrostatic gate.

7 Sensitiity of the gate effect These results could be used to iproe the design of an electrostatically gated spectroeter in two ways: (a) by recoering inforation fro otherwise corrupted data, and (b) by assessing how odifications in the hardware paraeters could odify the outcoe of the easureent. The first iproeent was already discussed in the section aboe. For the second, we could try different cobinations of paraeters to see their effects on the distribution functions. Two cases were analyzed, in the first one the distances between the gated electrode and the grounds were changed to half the size described in table 3 leaing eerything else constant. The results of this are shown in figure 6. Fro de la Mora s theory [3] we copute / for a solution with K = 2.32 Si/, γ =.58 N/, ρ = 33.4 kg/ 3, ε =, f(ε) = 2 and Q = /s, fro (6) we obtain / = 674 C/kg, which corresponds to a elocity of 2 /s if the acceleration potential is V = 774 V. A tie of flight of 54µsec is calculated if the drift distance between gates is L drift = 5.7 c. The results of applying the odel are shown in figure 7. Distribution function inf Exit spectru Input spectru Exit spectru Input Spectru Velocity (/s) Distribution function f inf Velocity (/s) Figure 6 Effect of geoetry change in a single peak at / = 27 C/kg This siple change decreases the tie the particles spend inside the gate, and therefore soe of the fall in region A4. Besides soe distribution of particles at higher elocities, we can obsere the appearance of a distinctie peak at about 42 /s, corresponding to / = 27 C/kg. In the second case two changes are ade; the electrode s geoetry is odified such that x = 3 and x = 9 and also the fluid is changed to one with higher /, like Foraide (CH 3 NO) doped with NaI salt. Figure 7 Effect of fluid and geoetry change (Foraide, / = 674 C/kg) We can see that the effect of using a faster solution is siilar to the one we obtained by reducing the size of the gate. Physically speaking both cases iniize the tie the particles spend inside the gate, so the results are what we could expect. Experiental Results To erify the second alternatie presented aboe, experients were perfored using a Foraide solution with the sae characteristics already described. The easureents are shown in figure 8, including error bars. In contrast with the results reported in [4], the peak obsered near 6µsec agrees uite well with what can be expected fro the electrospray theory. Fro our discussion aboe we deterine that the finite-width gate effect is sall enough to draatically change the spectru characteristics.

8 CH3NO + NaI, K = 2.32 Si/ References [] M. Martinez-Sanchez, J. Fernandez de la Mora, V. Hruby, M. Gaero-Castaño and V. Khays, Research on Colloid Thrusters. Paper IEPC 99-4, 26 th IEPC Kitakyushu, Japan, Oct Current (pa) Tie of flight (icrosec) Figure 8 Experiental results with Foraide Conclusions A atheatical odel for finite-width electrostatic gates was presented to analyze the effects that such deices hae on TOF easureents for Colloid Thrusters. It was found that such gates introduce elocity ariations, which are stronger for slow charged particles or for large gates. Zero-width gates are ore difficult to work with since the icro interleaed cob assebly could get short circuited by liuid deposition. Deflection plates introduce other effects, aong the a reduction of the current easured. The electrostatic gate odel was used also to recoer useful inforation fro degraded data. A new set of experiental results showed that odel predictions are accurate and that they could be used to design TOF hardware. [2] J. Perel, T. Bates, J. Mahoney, R.D. Moore, Research on a charged Particle Bipolar Thruster. Paper No , AIAA Electric Propulsion and Plasa Dynaics Conference, Colorado Springs, CO, Septeber 967 [3] M. Gaero-Castaño, priate counication (April 2) [4] P. Lozano, M. Martinez-Sanchez, Experiental Study of Colloid Plues, 37 th JPC, Salt Lake City, Utah, AIAA [5] M. Gaero-Castaño and V. Hruby, Electric Measureents of Charged Sprays Eitted by Cone- Jets, New York: Subitted for publication. [6] J. Fernandez de la Mora, The Current Eitted by Highly Conducting Taylor Cones. J. Fluid Mech. (994), Vol. 26, pp