Session 20: Electrospray Propulsion

Session : Electrospray Propulsion Electrospray thrusters are electrostatic accelerators of charged particles produced from elec­ trified liquid surfaces. There are three types of electrospray thruster technologies: 1. Colloid thrusters, which are accelerators of charged droplets and, under special circum­ stances, ions and use solvents such as doped glycerol and formamide as propellant. 2. FEEP (field emission electric propulsion) makes use of liquid metals (typically Cs and In) and produce positively charged metallic ions. 3. ILIS (ionic liquid ion sources), that use room-temperature molten salts, also known as ionic liquids and produce salt ion beams, or mixtures of ions and droplets. The first form of electrospray propulsion (which can be tracked back to the beginning of the 20th century) came in the form of colloid thrusters. They were intensively studied from around 1960 to 1975 as an alternative to normal ion engines. Their appeal at that time rested with the large “molecular mass” of the droplets, which was known to increase the mc2 thrust density of an ion engine. This is because the accelerating voltage is V = , where 2q m is the mass of the ion or droplet, and q its charge, and c is the final speed. If c is pre-defined (by the mission), then V can be increased as m/q increases. This, in turn, increases the space 2 F ε0 4 V = , (d is charge limited current density (as V 3/2 ), and leads to a thrust density, A 2 3d the grid spacing), which is larger in proportion to V 2 , and therefore to (m/q)2 . In addition to the higher thrust density, the higher voltage also increases efficiency, since any cost-of-ion V . voltage Vloss becomes then less significant, η = V + Vloss In a sense, this succeeded too well. Values of droplet m/q that could be generated with the technology of the 60’s were so large that they led to voltages from 10 to 100 kV (for typical Isp ≈ 1000 s). This created very difficult insulation and packaging problems, making the device unattractive, despite its demonstrated good performance. In addition, the droplet generators were usually composed of arrays of a large number of individual liquid-dispensing capillaries, each providing a thrust of the order of 1 µN. For the missions then anticipated, this required fairly massive arrays, further discouraging implementation. After lying dormant for many years, there is now a resurgence of interest in electrospray engine technology. This is motivated by: (a) The new emphasis on miniaturization of spacecraft. The very small thrust per emitter now becomes a positive feature, allowing designs with both, fine controllability and high performance. (b) The advances made by electrospray science in the intervening years. These have been motivated by other applications of electrospraying, especially in recent years, for the extraction of charged biological macromolecules from liquid samples, for very detailed mass spectroscopy. These advances now offer the potential for overcoming previous limitations on the specific charge q/m of droplets, and therefore may allow operation at more comfortable voltages (1 − 5 kV). 1 (c) The advances in micro-manufacturing technologies allow for efficient clustering of a large number of emitter tips on a small surface, potentially to the point of competing with plasma thrusters (ion or even Hall) in achievable current density. With regard to point (a), one essential advantage of electrospray engines for very small thrust levels is the fact that no gas phase ionization is involved. Attempts to miniaturize other thrusters (ion engines, Hall thrusters, arcjets) lead to the need to reduce the ionization mean free path (σion ne )−1 by increasing ne , and therefore the heat flux and energetic ion flux to walls. This leads inevitably to life reductions. In the electrospray case, as we will see, the charging mechanisms are variations of “field ionization” on the surface of a liquid; small sizes naturally enhance local electric fields and facilitate this effect. Basic Physics Consider first a flat liquid surface subjected to a strong normal electric field, E0 . If the liquid contains free ions (from a dissolved electrolyte), those of the attracted polarity will concentrate on the surface. Let σ be this charge, per unit area; we can determine it by ~ = ρ/ε0 in integral form to a “pill box” control volume, applying Gauss’ law ∇ · E σ = ε 0 E0 (1) A similar effect (change concentration) occurs in a dielectric liquid as well, even though there ~ = 0 where D ~ = εε0 E ~ and ε is the are no free charges. The appropriate law is then ∇ · D relative dielectric constant, which can be fairly large for good solvent fluids (ε = 80 for water at 20◦ C). There is now a non-zero normal field in the liquid (i), and we have, ε0 E0 − εε0 Ei = 0 (no free charges) (2) ~ = ρtotal /ε0 = 0 + ρd /ε0 , and, in addition from ∇ · E ε 0 E0 − ε 0 Ei = σ d (dipolar charge) (3) Eliminating Ei between these expressions,  σd = 1 1− ε  ε 0 E0 (4) which, if ε  1 is nearly the same as for a conducting liquid, however, is important to recall that only in a perfect conductor both normal and tangential fields are required to be zero. In a very high ε material the normal field, Ei = 1 E0 ε (5) may vanish, but the tangential component, if any, will be continuous across the interface. Consider a conductive liquid with a conductivity K, normally due to the motion of ions of both polarities. If their concentration is n+ = n− = n (m−3 ), and their mobilities are, µ+ and µ− (m/s)/(V/m), then, 2 K = ne Z + µ+ + Z − µ− (Si/m) (6) Suppose there is a normal field E0 applied suddenly to the gas side of the liquid surface. The liquid surface side is initially un-charged, but the field draws ions to it (positive if E0 points away from the liquid), so a free charge density σ builds up over time, at a rate, dσ = KEi dt (7) E = ρ, The charge is related to the two fields E0 , Ei from the “pillbox” version of \ · D ε0 E0 − εε0 Ei = σ From here we find a differential equation for the free charge density, dσ K K + σ = E0 ε dt εε0 (8) εε0 is the relaxation time of charges in the liquid. In terms of it, the K solution of Eq. (8) that satisfies σ(0) = 0 (for a constant Ei at t > 0) is, The quantity τ = σ = ε0 E0 1 − e−t/τ (9) Surface Stability If the liquid surface deforms slightly, the field becomes stronger on the protruding parts, and more charge concentrates there. The traction of the surface field on this charge is 1 σ E = 12 ε0 E 2 for a conductor (the 12 accounts for the variation of E from its outside value 2 s to 0 inside the liquid). This traction then intensifies on the protruding parts, and the process can become unstable if surface tension γ is not strong enough to counteract the traction. In that case, the protuberance will grow rapidly into some sort of large-scale deformation, the shape of which depends on field shape, container size, etc. If the surface ripple is assumed sinusoidal, and small (ini­ tially at least), then the outside potential, which obeys \2 φ = 0 with φ = 0 on the surface, can be represented approximately by the superposition of that due to the ap­ plied field E∞ , plus a small perturbation. Using the fact that Re{eiαz } is a harmonic function (z = x + iy), φ ≈ −E∞ y + φ1 e−αy cos αx (10) The surface is where φ = 0, and this, when αy « 1, is approximately given by 0 ≈ −E∞ y + φ1 cos αx, or, 3 y ≈ The surface has a curvature 1 ≈ Rc φ1 cos αx E∞ d2 y dx2 (11) φ1 α2 cos αx, which is maximum at crests E∞ = (cos αx = 1), E∞ φ1 α 2 Rc = (12) and gives rise to a surface tension restoring force (perpendicular to the surface) of γ/Rc (cylindrical surface). ∂φ The normal field, from Eq. (10), is Ey = − = E∞ + αφ1 e−αy cos αx and at αy  1 and ∂y on the crests, this
is Ey = E∞ + αφ1 . The perturbation of electric traction (per unit area) is then δ 12 ε0 Ey2 = ε0 E∞ αφ1 . Instability will occur if this exceeds the restoring surface tension effect, φ1 α 2 ε0 E∞ αφ1 > γ E∞ r or E∞ > γα ε0 (13) The quantity α is 2π/λ , where λ is the wavelength of the ripple. Thus, if longwave ripples are possible, a small field is sufficient to produce instability. We will later be interested in drawing liquid from small capillaries; if the capillary diameter is D, the largest wavelength will be 2D , or α = π/D , which gives the instability condition, r E∞ > πγ ε0 D (14) For example, say D = 0.1 mm, and γ = 0.05 N/m (Formamide, CH3 NO). The minimum field to produce an instability is then 1.33 × 107 V/m. This is high, but since the capillary tip is thin (say, about twice its inner diameter, or 0.2 mm), it may take only about 2500 V to generate it. A more nearly correct estimate for this will be given next. Starting Voltage The figure shows an orthogonal system of coordinates called “Prolate Spheroidal Coordir1 − r2 r1 + r2 nates”, in which η = ,ζ= and φ is an angle about the line FF’. a a The cartesian-prolate transformation is, r r1 = a 2 2 2 x +y + z+ 2  r and r2 = x2 + y2 a 2 + z− 2  and so, lines of constant η are confocal hyperboloids (foci at F, F’) while lines of constant ζ are confocal ellipsoids with the same foci. The surface η = 0 is the symmetry plane, S, and one of the η-surfaces, η = η0 , can be chosen to represent (at least near its tip) the protruding liquid surface from a capillary. 4 If the potential φ is assumed to be constant (V ) on η − η0 , and zero on the plane S, then the entire solution for φ will depend on η alone. The η part of Laplace’s equation in these coordinates is, ∂ ∂η 
∂φ 1−η ∂η 2  = 0 (15) which, with the stated boundary conditions, integrates easily to, φ = V tanh−1 η tanh−1 η0 Let R2 = x2 +y 2 (cylindrical radius). From η = η hyperboloid is, r aη = R2 r1 − r2 , the (z,R) relationship for a constant a a 2 + z+ − 2  (16) r  a 2 R2 + z − 2 which, for z > 0, can be simplified to, s z = η a2 R2 + 4 1 − η2 The radius of curvature Rc of this surface is given by, 1 zRR = 3/2 Rc (1 + zR2 ) which yields,   1 − η2 R2 /a2 Rc = a 1+4 2η (1 − η 2 )2 5 (17) a Also, the tip-to-plane distance is d = z(R = 0, η = η0 ) = η0 , so we find the parameters a 2 and η0 if Rc and d are specified, r a = 2d Rc 1+ d The electric field at the tip is Ez = − and η = η0 , so that ∂η ∂z tip and ∂φ ∂z =− tip  −1/2 Rc η0 = 1 + d ∂φ ∂η ∂η ∂z (18) . Now, at the tip we have R = 0 tip 2 = , and using Eq. (16), a Etip = − 2V /a (1 − η02 ) tanh−1 η0 (19) which can be expressed in terms of Rc and d, when Rc  d, as, Etip = − 2V /Rc ln (4d/Rc ) (20) Now, in order for the liquid to be electrostatically able to overcome the surface tension forces and start flowing, even with no applied pressure, one needs to have, 1 2γ 2 ε0 Etip > 2 Rc (21) (2γ/Rc , because there are two equal curvatures in an axisymmetric tip). Substituting Eq. (20), the “starting voltage” is, r Vstart = γRc ln ε0  4d Rc  (22) Returning to the example with Rc = 0.05 mm, γ = 0.05 N/m, and assuming an attractor plane at d = 5 mm, the required voltage is 3184 V, whereas if the attractor is brought in to d = 0.5 mm, Vstart = 1960 V. These values are to be compared to the previous estimate. They still ignore the effect of space charge in the space between the tip and the plane, which would act to reduce the field at the liquid surface. But we have also ignored the effect of an applied pressure, which can be used to start the flow as well. What an applied pressure cannot do, however, is to trigger the surface instability described before. As Eq. (22) shows, if the radius of curvature at the tip is reduced, so is the required voltage to balance surface tension. One can then expect that, once electrostatics dominates, the liquid surface will rapidly deform from a near-spherical cap to some other shape, with a progressively sharper tip. The limit of this process will be discussed next. The Taylor Cone From early experimental observations, it was known that when a strong field is applied to the liquid issuing from the end of a tin tube, the liquid surface adopts a conical shape, with a very thin, fast-moving jet being emitted from it apex. In 1965, G.I. Taylor explained analytically 6 ⎛ 1 1 ⎞ surface tension. The latter is per unitand of area, 1 Relectrospray γ⎜ + (andthe verified experimentally) this behavior, the conical tip often seen in c1 ,1 Rc2 ⎟⎠ , where R R ⎝ c1 c2 emitters is now called a “Taylor Cone”. The basic idea is that the surface “traction” ε0 E 2 /2 due are to the field must be balanced conical the the pull of the electric two principal curvatures of theeverywhere surface. In cone, is zero by along 1i Rc surface e ora−1the −1 the generator, surface tension. latter is per unitnormal of area,section γ Rc1 is+ the Rc2 projection , where 1/R 1/Rc2 while The the curvature of the on cit1 and of that are the two principal curvatures of the surface. In a cone, 1/R is zero along the generator, c theorem), of the circular section through the same point (Meusnier’s while the curvature of the normal section is the projection on it of that of the circular section through the same point (Meusnier’s theorem), 1 cos α cos α cot α (23) = = = Rc R r sin α r 1 cos θ cos θ cot θ = = = (23) Rc R r sin θ r This means that, This means that, 1 γ cot α 2 12 ε 0 E2n = γ cot θ ε 0 En = r 2 r Electrified Menisci as Field Amplifiers 2γ cot α or or En = 2γ cot θ En = ε 0 r ε0 r (24) (24) Balance: question to find an external electrostatic the cone an TheThe question thenthen is toisfind an external electrostatic fieldfield suchsuch that that the cone isForce anisequipo­ equipotential, with a normal field varying as in Eq. (24). Notice that the spheroids tential, with a normal field varying as in Eq. (24). Notice that the spheroids of the previous V of the generate cones in the limit when >>electrostatic a but this type of has section do previous generate section cones indothe limit when r » a but this typer of field Surface tension electrostatic field has cannot besolution. the desired equilibrium solution. En ∝desired 1 r , and En ∝ 1/r , and cannot be the equilibrium f st = γ∇ ⋅ nˆ f e = 12 ε o E n 2 θT r En Electric pressure we adopt a spherical system of Laplace’s coordinates Laplace’s admits If weIfadopt a spherical system of coordinates, equation admitsequation axisymmetric “prod­ axisymmetric “product” solutions of the type, uct” solutions of the type, φ = AQν ( cosθ )νr ν φ ∝ Qν (cos θ)r or or φ = APν ( cosθ ) r νν φ ∝ Pν (cos θ)r (25) (25) Qνν are are Legendre where Pν and where Legendrefunctions functionsofofthe the1st 1stand and2nd 2ndkind, kind,respectively. respectively.OfOfthe thetwo, Pν ,Q Pν has when θ = π,when and Q 0. The because two,a singularity andatQθν =has onelatter at θ =is0acceptable, . The latter is Pν has a singularity θ ν=has π , one θ = 0 is inside the liquid cone, and we only need the solution outside. The normal field is acceptable, because θ = 0 is inside the liquid cone, and we only need the then, solution outside. The normal field is then, 1 ∂φ dQν En = E θ = − =A sin θrν−1 r ∂θ d cos θ √ where A is a constant. In order to have En ∝ 1/ r, we need ν = 1/2. Thus, 16.522 Space Propulsion Martinez-Sanchez and Lozano φ = AQ1/2 (cos θ)r 7 1/2 Lecture 20 Page 9 of 10 (26) 1.5 5 1 4 0.5 3 0 2 Q1/2 P 1/2 -0.5 -1 zero at α = 49.29° zero at α = 130.71° -1.5 1 0 -2 -1 -2.5 -3 0 50 100 alpha (deg) -2 0 150 50 100 alpha (deg) 150 The function Q θ) has a singlepolynomials zero at, 1/2 (cos Figure 3.1.3. Legendre with ν = 1/2 as functions of α θT = 49.29◦ (27) which can therefore be taken as the equipotential liquid surface. Notice that this angle is 42 universal (independent of fluid properties, applied voltage, etc). Taylor (and others) have verified experimentally this value, as long as no strong space charge effects are present, no flow, and as long as the electrode geometry is “reasonably similar” to what is implied by Eq. (26). The experimental fact that stable Taylor cones do form even when the electrodes applying the voltage are substantially different from the shape given by Eq. (26) apparently indicates that the external potential distribution near the cone is dictated by the equilibrium condition Eq. (24), and that the transition to some other potential distribution capable of matching the real electrode shape takes place far enough from the liquid to be of little consequence. We should expect, however, that the Taylor cone solution will be disturbed to some extent by nonideal conditions, and will eventually disappear. In one respect at least, the Taylor cone cannot be an exact solution: the infinite electric field predicted at the apex will produce various physical absurdities. Something must yield before that point, and that is explored next. 8