16.522, Space Propulsion Prof. Manuel Martinez-Sanchez Lecture 23-25: COLLOIDAL ENGINES APPENDIX A1. INTRODUCTION. Colloidal thrusters are electrostatic accelerators of charged liquid droplets. They were first proposed and then 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 thrust density of an ion engine. This is because the accelerating voltage is V = mc 2 2q , where 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 charge limited current density (as V 3/2 ), and leads to a thrust density, F A = ε o 2 4 3 V d ? ? ? ? 2 , (d=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 voltage V LOSS becomes then less significant η = V V + V LOSS ? ? ? ? ? ? . 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 over 20 years, there is now a resurgence of interest in colloid 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 charged colloids, 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 (1-5KV). With regard to point (a), one essential advantage of colloid engines for very small thrust levels is the fact that no gas phase ionization is involved. Attempts to miniaturize other 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 1 of 36 thrusters (ion engines, Hall thrusters, arcjets) lead to the need to reduce the ionization mean free path 1 σ ion n e ? ? ? ? ? ? by increasing n e , and therefore the heat flux and energetic ion flux to walls. This leads inevitably to life reductions. In the colloidal 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. A2. BASIC PHYSICS A2.1 SURFACE CHARGE Consider first a flat liquid surface subjected to a strong normal electric field, E n . If the liquid contains free ions (from a dissolved electrolyte), those of the attracted polarity will concentrate on the surface. Let ρ s be this charge, per unit area; we can determine it by applying Gauss’ law in integral form to the “pill box” control volume shown in the figure: ?. r E = ρ ch /ε o ρ s = ε o E n (A1) A similar effect (change concentration) occurs in a Dielectric liquid as well, even though there are no free charges. The appropriate law is then ?. r D = ρ ch free , where r D = εε o r E and ε is the 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, and we have ε o E n, g ?εε o E n,l = o (no free charges) (A2) and, in addition, ε o E n,g ?E n,l ( ) = ρ s, dipoles (A3) Eliminating E n,l between these expressions, ρ s, dip. = 1? 1 ε ? ? ? ? E n,g (A4) 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 2 of 36 which, if ε >>1 is nearly the same as for a conducting liquid (Eq. A1). The field inside the liquid follows now from (A2): E n,l = 1 ε E n, g (A5) and is very small if ε >>1 (zero in a conductor). 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 3 of 36 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 4 of 36 Consider a conductive liquid with a conductivity K, normally due to the motion of ions of both polarities. If their concentration is +- 3 n=n=n (m/s) and their mobilities are +- , μμ ((m/s)/ (V/m)), then ()() +- K=n + Simμμ (1) Suppose there is a normal field g n E 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 g n E points away from the liquid), so a free charge density f σ builds up over time, at a rate lf n d =KE dt σ (2) The charge is related to the two fields, g l nn E, E from the “pillbox” version of free . D =? ρ nullnull nullnull gl 0n 0n f E- E=εεε σ (3) From (3), g l nf n 0 E E= - σ εεε , and substituting in (2), gf fn 0 d KK +=E dt σ σ εε ε (4) The quantity 0 = K εε τ is the Relaxation Time of the liquid. In terms of it, the solution of (4) that satisfies ( ) f 0=0σ (for a constant l n E at t>0) is A2.1.1 CHARGE RELAXATION 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 5 of 36 gt - n f 0 E =1-e τ ?? σ ?? ε ?? (5) The surface charge approaches the equilibrium value g n 0 E ε (at which point, from (3), l n E=0) but it takes a time of the order of 0 = K εε τ to reach this equilibrium. For a concentrated ionic solution, with K1 Sim～ and 100ε ～ , this time is about -9 =10 s=1 nsτ , which is difficult to measure directly, but has measurable consequences in the dynamics of very small liquid flows, as we will see. For normal “clean” water, -4 K10 Sim～ , and -5 10 s = 10 sτ μ～ which can be directly measured in the lab. The math can be generalized to a gradual variation of the field, () gg nn =EEt. Using the method of “variation of the constant” () t - f =c t e τ σ ; t - f d dc c =-e dt dt τ σ ?? ?? τ ?? and substituting into (4), () t g n dc K =e E t dt τ ε ; () t t' g 0n 0 K c=c + e E t' dt' τ ε ∫ Since ( ) f 0=0σ , c(0) = 0 And so 0 c=0: () t t-t' - g fn 0 K =eEt'dt' τ σ ε ∫ (6) A2.2 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 ρ s () E n 2 = ε o 2 E n 2 for a conductor (the 1/2 accounts for the variation of E n from its outside value 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 (initially at least), then the outside potential, which obeys ? 2 φ= o with φ =o on the surface, can be represented approximately by the superposition of that due to the applied field E ∞ , plus a small perturbation. Using the fact that Re e iαz ( ) is a harmonic function (z=x+iy), φ ??E ∞ y +φ 1 e ?αy cosαx (A6) The surface is where φ =o , and this, when αy <<1, is approximately given by o ??E ∞ y +φ 1 cosαx , or y ? φ 1 E n cosαx (A7) The surface has a curvature 1/ R c ? d 2 y dx 2 = φ 1 α 2 E ∞ cosαx , which is maximum at crests (cosα x=1): R c = E ∞ φ 1 α 2 (A8) and gives rise to a surface tension restoring force (perpendicular to the surface) of γ R c (cylindrical surface). 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 6 of 36 The normal field, from (A6), is E y = ??φ ?y = E ∞ +αφ 1 e ?αy cosαx and at αy <<1 and on the crests, this is E y = E ∞ +αφ 1 . The perturbation of electric traction is then (per unit area) δ ε o 2 E y 2 ? ? ? ? =ε o E ∞ αφ 1 . Instability will occur if this exceeds the restoring surface tension effect: ε o E ∞ αφ 1 >γ φ 1 α 2 E ∞ or E ∞ > γα ε o (A9) The quantity α is 2π /λ , where λ is the wavelength of the ripple. Thus, if long-wave 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 E ∞ > πγ ε o D (A10) For example, say D=0.1mm, and γ = 0.05N / m (Formamide, CH 3 ON). The minimum field to produce an instability is then π ×0.05 8.85×10 ?12 ×10 ?4 =1.33×10 7 V / m. This is high, but since the capillary tip is thin (say, about twice its inner diameter, or 0.2mm), it may take only about 1.33×10 7 ×2×10 ?4 = 2660 Volt to generate it. A more nearly correct estimate for this will be given next. 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 7 of 36 A2.3 STARTING VOLTAGE FOR A CAPILLARY Fig. 4b shows an orthogonal system of coordinates called “Prolate Spheroidal Coordinates”, in which η= r 1 ? r 2 a ; ξ = r 1 + r 2 a and ? is an angle about the line FF’. Here r 1 = x 2 + y 2 + z + a 2 ? ? ? ? 2 r 2 = x 2 + y 2 + z? a 2 ? ? ? ? 2 and so, lines of η= const.are confocal hyperboloids (foci at F, F’) while ξ= const. lines are confocal ellipsoids with the same foci. The surface η= o is the symmetry plane, S, and one of the η-surfaces, η=η o , can be chosen to represent (at least near its tip) the protruding liquid surface from a capillary as in Fig. 4a. If the potential φ is assumed to be constant (V) on η=η o , 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 1?η 2 () ?φ ?η ? ? ? ? ? ? = o (A1) which, with the stated boundary conditions, integrates easily to φ = V th ?1 η th ?1 η o (A12) Let (cylindrical radius). From R 2 = x 2 + y 2 η= r 1 ? r 2 a , the (z,R) relationship for an η= const. hyperboloid is 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 8 of 36 aη = R 2 + z + a 2 ? ? ? ? 2 ? R 2 + z? a 2 ? ? ? ? 2 which, for z>o, can be simplified to z=η a 2 4 + R 2 1?η 2 . The radius of curvature R c of this surface is given by 1 R c = z RR 1+ z R 2 () 3/2 , which yields, R c = 1?η 2 2η a 1+ 4 R 2 / a 2 1?η 2 () 2 ? ? ? ? ? ? 3/ 2 (A13) Also, from Fig. 4b, the tip-to-plane distance is d = zR= o,η=η o ()= a 2 η o (A14) Eqs. (A13), (A14) give the parameters a and η o if R c and d are specified: a = 2d 1+ R c d ; η o = 1 1+ R c d (A15) The electric field at the tip is E z =? ?φ ?z ? ? ? ? TIP =? dφ dη dη dz ? ? ? ? ? TIP . Now ?z ?η ? ? ? ? ? TIP = ?z ?η ? ? ? ? ? R=o,η=η o = a 2 , and using Eq. (A12), E TIP =? 2V / a 1?η o 2 () th ?1 η o (A16) which can be expressed in terms of R c , d, when R c <<d, as E TIP =? 2V / R c ln 4d R c ? ? ? ? ? ? (A17) 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 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 9 of 36 ε o 2 E TIP 2 > 2γ R c (A18) (2γ / R c , because there are two equal curvatures in an axisymmetric tip). Substituting (A18), the “starting voltage” is V Start = γR c ε o ln 4d R c ? ? ? ? ? ? (A19) Returning to the example with R c =0.05mm, γ=0.05 N/m, and assuming an attractor plane at d=5mm, the required voltage is V START = 0.05× 5×10 ?5 8.85×10 ?12 ln 400()= 3184 Volts whereas if the attractor is brought in to d=0.5mm, V START =1960 V. These values are to be compared to the estimate at the end of Sec. A2.2. 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 in A2.2. As Eq. (A19) 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. A2.4 The Taylor Cone From early experimental observations (Zeleny, 1914-1917) [1,5,6] , 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 (See Figs. 5,6, from J. Fernandez de la Mora and I. Loscertales, 1994) [26] . In 1965, G.I. Taylor [7] explained analytically (and verified experimentally) this behavior, and the conical tip often (but not always!) seen in electrospray emitters is now called a “Taylor Cone”. The basic idea is that the surface “traction” ε o E n 2 /2 due to the electric field must be balanced everywhere or the conical surface by the pull of the surface tension. The latter is per unit of area, γ 1 R c 1 + 1 R c 2 ? ? ? ? ? ? , where 1/ R c 1 ,1/ R c 2 are the two principal curvatures of the surface. In a cone, 1/R c is zero along the generator, while the curvature of the normal section is 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 10 of 36 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 11 of 36 the projection on it of that of the circular section through the same point (Meusnier’s theorem): 1 R c = 1 R ? ? ? ? cosα = cosα rsinα = cotα r (A20) This means that 1 2 ε o E n 2 = γ cotα r E n = 2γ cotα ε o r (A21) The question then is to find an external electrostatic field such that the cone is an equipotential (say, φ =o ), with a normal field varying as in (A21), i.e., proportional to 1/ r . Notice that the spheroids of Sec. A2.3 do generate cones in the limit when r>>a (with η o = cosα) , but this type of electrostatic field has E n ≈ 1/ r , and cannot be the desired equilibrium solution. If we adopt a spherical system of coordinates (Fig. 8), it is known that Laplace’s equation admit axi-symmetric “product” solutions of the type φ = AP ν cos?( )r ν (A22a) or φ = A Q ν cos?( )r ν (A22b) where P ν ,Q ν are Legendre functions of the 1 st and 2 nd kind, respectively. Of the two, P ν has a singularity when ? =180 o , and Q ν has one at ? = o. The latter is acceptable, because ? = o is inside the liquid cone, and we only need the solution outside. The normal field, from (A22b) is then 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 12 of 36 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 13 of 36 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 14 of 36 E n = E ? =? 1 r ?φ ?? =+A dQ ν d cos?() sin? 1 r 1?ν and, in order to have E n ≈ 1 r 1/ 2 , we need ν = 1 2 . Thus, φ = A r 1/2 Q 1/2 cos?() (A23) The function Q 1/2 cos?() is shown in Fig. (9) ? . The essential point is that this function has a single zero, at ? =α = 49.290 o (A24) which can therefore be taken as the equipotential liquid surface. Notice that this angle is 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, and as long as the electrode geometry is “reasonably similar” to what is implied by Eq. (A27). This latter point is clarified by Fig. 10, where one generic equipotential of (A23) is shown together with the Taylor cone; notice that all other equipotentials have shapes which can be simply scaled from the one shown, according to r 2 / r 1 = φ 2 /φ 1 ( ) 2 for a given angle ? . The experimental fact that stable Taylor cones do form even when the electrodes applying the voltage are substantially different from the shape in Fig. 10 apparently indicates that the external potential distribution near the cone is dictated by the 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 15 of 36 equilibrium condition (A21), 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 non-ideal 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 (r=o) will produce various physical absurdities. Something must yield before that point, and that is explored next. A3. CURRENT AND FLOW FROM TAYLOR CONES As the photograph in Fig. 6 shows, a jet is seen to issue from the cone’s tip, implying the need for a flow rate, say Q (m 3 /s). Since the surface being ejected is charged, this also implies a net current, I. It will be seen that these flows and currents are (in the regime of interest) extremely small: Q ≈10 ?13 m ?13 / s, ≈ 10 ?8 A per needle. The tip jet is likewise extremely thin (of the order of 20-50 nm). Not very near the cone’s tip, the current is mostly carried by ionic conduction in the electrolytic solution. In a good, highly polar solvent (i.e., one with ε >>1), the salt in solution is highly dissociated, at least at low concentration. For example, LiCl in Formamide dissociates into L and i + Cl ? , and each of these ions, probably “solvated” (i.e., with several molecules of formamide attached), will drift at some terminal velocity (in opposite directions) in response to an electric field. At high concentrations (several molar) the degree of dissociation decreases. Following are the measured electrical conductivities K, of solutions of LiCl in Formamide ( ε ? 100( ) 1/ 2 x()= K 1 + x 2 ? ? ? ? ? 2E 1+ x 2 ? ? ? ? P 1/2 x()= 2 ? NOTE: Use is made of Q , where K and E are the complete elliptic integrals of the 1 st and 2 nd kind, respectively. It is also noteworthy that π Q 1/2 ?x() P 1/2 cos 180 o ?? () [ , so ] could equivalently be used as the angular part in (A23). 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 16 of 36 Concentration mol / l() K(Siemens/m) 1.47×10 ?3 1.47×10 ?2 0.147 1.0 3 3.16×10 ?2 5.49×10 ?2 0.27 1.12 2.2 This finite conductivity implies that, under current, there will be some electric field directed radially . This contradicts the assumption made that the Taylor cone’s surface is an equipotential, especially near the tip, where the current density must be strongest. Hopefully, E E r ≠o() r is at least much less than E ? over most of the cone. The area of a spherical cap of radius r bounded by the cone is A = 2πr 2 1? cosα( ), and the radial field must be E r = I / A K = I c 2π 1? cosα()Kr 2 (A25) where I c is that part of I which is due to conduction. Compared to the azimuthal surface field E ? = E n , given by (A21), we see that E r will indeed decrease much more rapidly as r increases. The two become comparable inside the liquid (assumed to behave near the tip as a dielectric) when 1 ε 2γ cotα ε o r = I c 2π 1?cosα()Kr 2 or r = r 1 = ε o 8π 2 cotα 1?cosα() 2 ? ? ? ? ? ? 1/3 ε 2 I c 2 K 2 γ ? ? ? ? ? 1/3 (A26) 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 17 of 36 and this provides a first indication of the size of the tip jet, since the cone solution becomes untenable for r<r 1 . The surface current, associated with a fluid velocity u, will be I s = 2π r sinα()ρ s u (A27) where u = Q 2π 1? cosα( r 2 ) (A28) and ρ s = ε o E ? , with E ? given by (21). We then find I s r()= ε o Qsinα 1?cosα 2γ cotα ε o 1 r 3/2 (A29) This shows clearly that the surface current must be insignificant at large distances from the apex, but may become dominant near it. It is interesting to speculate that at r=r 1 (where E r becomes comparable to E ? , the surface current may also become comparable to the conduction current. Using r=r 1 (from A26) in (A29) yields I s r 1 ()= 4πγ cosα KQ εI c (A30) where I cond instead of the total current has been used in (A26). If we now say that at r=r 1 , I s = βI I c = 1? β () I (A31) where β is some unknown fraction, we obtain from (A30) 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 18 of 36 I = 4πcosα β 1?β() γKQ ε ? ? ? ? ? ? 1/2 (A32) Using a slightly different argument, de la Mora (1994) [26] concluded (and verified experimentally) that I ? f ε() ε γKQ() 1/2 (A3) where f ε()≈18? 25 for ε ≥ 40. Using f=25, ε = 100 (Formamide) yields . This differs from Eq. A33 only in the numerical factor, and we see that (A33) and (A32) coincide if I ≈ 2.5 γKQ() 1/2 4π cosα β 1? β() = f 2 ε(). If f ε( )= 25, this yields β =0.0133 . Eqs. (A32) or (A33) are remarkable in several respects: (a) Current is independent of applied voltage (b) Current is independent of electrode shape (c) Current is independent of fluid viscosity (even though some of the fluids tested are very viscous. The degree of experimental validity of (A33) in shown in Fig. 11. Here, non-dimensional parameters are defined as follows: ξ= I γε o / ρ () 1/2 (A34) η= ρKQ γεε o ? ? ? ? ? ? 1/2 (A35) so that Eq. (A33) becomes ξ= f ε( )×η (A36) 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 19 of 36 For six different fluids, and over a wide range of flows, the correlation in Fig. 11 is remarkable. A4. Droplet Size and Charge From the nature of the Taylor cone, the liquid, as it progresses towards the tip jet, maintains an equilibrium on its surface between electrostatic and capillary forces. This equilibrium is disturbed near the tip, but it is reasonable to conjecture that something close to it will be sustained into the jet, and even after jet break-up, into the droplets which result. If we postulated this for a droplet of radius R and charge q (hence with a surface field E n = q /4πε o R 2 ( ) , we must have 1 2 ε o q 4πε o R 2 ? ? ? ? ? ? 2 = 2γ R (A37) q (A38) = 8πε o γ() 1/2 R 3/2 The droplet mass is m = 4 3 πR 3 ρ , so that q m = 6 ε o γ () 1/2 ρ R 3/ 2 (A39) 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 20 of 36 Eqs. (A38)-(A39) represent the so-called “Rayleigh limit”, namely, they give the largest charge (or charge per unit mass) which can be supported by a drop of a given radius. An interesting related expression can be obtained if we ask what the least-energy subdivision of a given total mass m TOT and charge q is. If this subdivision is into N equal drops of radius R, we have TOT N = m TOT / ρ 4 3 πR 3 ? ? ? ? , and each drop will carry a charge q = q TOT N =ρ 4 3 πR 3 q TOT / m TOT . The energy per drop comprises an electrostatic part 1 2 qφ, and a surface part 4πR 2 γ : E = N 1 2 q 2 4πε o R +4πR 2 γ ? ? ? ? ? ? = ρq TOT 2 R 2 6m TOT ε o + 3m TOT γ R (A40) Differentiating and equating to zero, we obtain R = 9 m TOT / q TOT () 2 ε o γ 1/3 or q m ? ? ? ? MIN .E = 3 ε o γ () 1/2 ρ R 3/ 2 (A41) So, the minimum-energy assembly of drops has a specific charge exactly 1/2 the maximum possible. Experimental observations (Fig. 12) tend to fall on the line given by (A41), although some difficulty of interpretation arises with polydisperse clouds (many sizes present), and Eq. (A39) is also supported in more recent data with monodisperse sprays (de Juan and de la Mora, 1996) [30] . If the droplet size R is assumed known, we can deduce the radius R jet of the jet from whose breakdown they originate. Several experiments confirm that this jet breakup conforms closely to the classical Rayleigh-Taylor stability theory for un-charged jets, which predicts a ratio R / R jet =1.89 (A42) What is the relationship between R jet and the distance r 1 defined in Eq. (26)? To see this, let us first substitute I c = 1? β( I) (with I from (A32)) into (A26): 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 21 of 36 r 1 = ε o 8π 2 cotα 1?cosα() 2 ε 2 K 2 γ ? ? ? ? ? ? 1/3 1?β() 2/3 4πcosα β 1?β() γKQ ε ? ? ? ? ? ? 1/3 or r 1 = 1? β β sinα 2π 1?cosα() 2 ? ? ? ? ? ? 1/ 3 εε o Q K ? ? ? ? 1/3 = 0.999 1?β β ? ? ? ? ? 1/ 3 r * (A43) with . (A43b) r * ≡ εε o α / K( 1/3 ) We similarly express R jet as a function of flow and fluid quantities by assuming Rayleigh- limited drops (Eq. 39), and using q m /ρ() = I Q : 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 22 of 36 R jet = 1 1.89 6 ε o γ () 1/ 2 I / Q ? ? ? ? ? ? 2/3 or R jet = 1 1.89 6 2/3 ε o γ() 1/ 3 β 1? β( ) 4π cosα εQ γK ? ? ? ? ? ? 1/3 (A4) Both, r 1 and R jet are seen to scale with the group εε o Q K ? ? ? ? 1/3 (which was called r* by de la Mora, 1994) [26] . By division of (A44) and (A43), R jet r 1 = 1 1.89 18β 2 1?cosα() 2 sinαcosα ? ? ? ? ? ? 1/3 = 0.868β 2/3 (A45) For f ε()= 25, this gives . R jet / r * ? 0.205 This value is in the range of the data shown in Fig. 13 (from de la Mora, 1994) ) [26] which strongly supports the validity of the arguments used. It can be also Fig. 13: Dimensionless terminal radius R j = 1/2d j from photographs of ethylene glycol jets (K = 5 x 10 -5 S m -1 ; 1.64 < Re < 10.2) showing that d j scales with r*. observed that f ε() is known to fall for ε less than about 40, and Eq. (A47) constitutes a prediction for a corresponding increase in the jet diameter. No direct data appear to be available on this point. To conclude this discussion, we observe that, from (A33), q m = I ρQ = f ε() ρε γK Q ? ? ? ? ? 1/ 2 (A48) 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 23 of 36 which means that the highest charge per unit mass is obtained with the smallest flow rate. A high q/m can be important in order to reduce the needed accelerating voltage V for a prescribed specific impulse Isp= c g : V = c 2 2 q / m() (A49) For concreteness, suppose we desire c=8000 m/s Isp ? 800sec( ) without exceeding V=5KV. From (49), we need q m > 8000 2 2 × 5000 = 6, 400Coul / Kg. Suppose we use a Formamide solution with K=1Si/m, ε = 100, f ε( )= 25,ρ = 1130Kg / m 3 ,γ = 0.059N / m : 25 1130 100 0.059× 1 Q > 6400 ; Q < 7.1× 10 ?15 m 3 / s Y m < 8.0ng / s ; I < 5.1×10 -8 A ( ) These are really small flow rates and currents. The input power per emitter is then less than 5000 × 5.1×10 ?8 = 2.6×10 ?4 W = 0.26mW, and the thrust is less than 8×10 ?12 ×8000 = 6.4×10 ?8 N = 0.064μN.. For this example, we also calculate r * = 100 ×8.85 ×10 ?12 × 7.1×10 ?15 1 ? ? ? ? ? 1/3 = 1.85×10 ?8 m , which gives a jet diameter 2R jet = 2× 0.202r * = 7.5×10 ?9 m, and a droplet radius R = 1.89R jet = 7.0 ×10 ?9 m. The drop charge is q = 6400 4π 3 7.0×10 ?9 () 3 ×1130 = 1.04× 10 ?17 Coul (65 elementary charges for about 21,000 Formamide molecules). Notice also the scalings: Q ≈ K V 2 c 4 , F ≈ K V 2 c 3 . The required flow rate is quite sensitive to the prescribed specific impulse. For small ?V missions, where high specific impulse is not imperative, the design can be facilitated by both, reducing V and increasing Q. A5. LIMITATIONS TO THE DROPLET CHARGE/MASS As we have seen, high q/m can be obtained (Eq. (A48)) by increasing the conductivity K of the liquid (more concentrated solutions), and, for a given conductivity, by reducing the flow rate Q. As Q K is reduced, the jet becomes thinner (as r * = εε o Q / K( ) 1/3 ), the droplets become smaller in the same proportion, and their specific charge increases as . It would appear then that q/m can be indefinitely increased through flow reduction. Two phenomena have been identified, however, which limit this increase. γK / Q() 1/2 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 24 of 36 A5.1 TAYLOR CONE INSTABILITY It has been noted that the Taylor cone becomes intermittently disrupted when the non- dimensional group introduced in Eq. (A35) becomes less than some lower limit, of the order of 0.5. The nature of this instability is not currently well understood, and so there is some uncertainty as to its generality. One likely explanation is the fact that q/m cannot exceed the specific charge that would result from full separation of the positive and negative ions of the salt used: η= ρKQ /γεε o ( 1/ 2 ) q V ? ? ? ? MAX = q m/ ρ ? ? ? ? ? ? MAX = F ×1000c d (Coul / m 3 ) (A50) where F=96500 Coul/mol is Faraday’s constant, and c d is the dissociated part of the solution’s equivalent normality (mole equivalents/ ). The dissociated concentration c l d is linearly related to the conductivity K through a “mobility parameter” Λ o : K ?Λ o c d (A51) For aqueous solutions Λ is 15 (Si/m)/(mol/ ) if there are no H o l + ions, in which case Λ o is ≈40 (Si/m)/(mol/ ). We therefore can write, from (48) l f ε() ε γK Q MIN ? ? ? ? ? ? 1/2 ?1000F c d ? 1000F Λ o K or η MIN ? ρ ε o Λ o 1000F f ε( ) ε (A52) 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 25 of 36 Assuming Λ o = 20, ρ=1130Kg / m 3 , f =25, ε =100, Eq. (52) yields η MIN = 0.59 , which is of the right order. The mobility factor Λ o would, however, be expected to depend on viscosity, so the argument is incomplete. Using this criterion, Q MIN = γεε o ρK η MIN 2 (A52a) and so (A48) gives a maximum droplet specific charge q m ? ? ? ? MAX = f ε() εη MIN K ε o ρ (A53) which reduces, as the argument above implies, to q m ? ? ? ? MAX = 1000F ρ c d (A54) For formamide, K can be raised to about 2 Si/m, and using η MIN = 0.5, (A53) yields q m ? ? ? ? MAX ? 10,000Coul / Kg . This implies a relationship Voltage-Isp V = gIsp() 2 2×10 4 (5000V for Isp?1000s.). A5.2 ION EMISSION FROM THE TIP The normal field E ? increases towards the cone’s tip, and will be maximum more or less at the start of the jet. We can estimate this maximum using Eq. (A21) and r ? R jet / cosα ? 2.68 f 2/3 r * . This gives E TIP ? 2γ cotα ε o r = 2γ cotα ε o f 1/ 3 2.68() 1/2 K εε o Q ? ? ? ? ? ? 1/6 =1.87 ×10 7 f 1/3 ε()γ 1/2 K εQ ? ? ? ? ? 1/ 6 (A55) This field can be very high at low flow rates and with highly conductive fluids. It is of interest to evaluate E TIP at the lowest stable flow rate, as given by Q MIN = γε o εη MIN 2 ρK . The result is then E TIP () MAX ? 1.30 ×10 9 η MIN 1/3 ρ 1/6 f ε( ) ε γK ? ? ? ? ? ? 1/ 3 (A56) Using data for formamide and ε = 100, f ? 25, γ = 0.059N / m, ρ =1130Kg / m 3 , assuming η MIN = 0.5, K = 2Si / m, this gives E TIP ( ) MAX ?1.63×10 9 V / m =1.63V / nm.. 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 26 of 36 It is known experimentally that at normal fields in the range 1-2 V/nm individual ions begin to be extracted from the liquid by the process called field emission (for N a I solutions in formamide, the threshold is E TIP ≈1.1V / nm , from Gamero, 1999) [35] . Once the threshold field is reached, field emission increases rapidly with field. Assume the liquid used has a large enough conductivity (and surface tension) that the peak field given by (A55) reaches 1-2 V/nm as the flow Q is decreased before the minimum stable flow is reached (in other words, the field given by (A56) is more than 1-2 V/nm). In that case, further reductions in flow, which increase E, will result in copious emission of ions from the tip, and the emitted current (droplets plus ions) will increase instead of decreasing as Q 1/2 . Fig. 14 (from Gamero and de la Mora, 1999) [35] shows this behavior for formamide solutions of NaI, which do satisfy the above conditions. Notice that very small reductions in flow are required for very large ion currents to be extracted, once Q drops below the value where ion emission begins. From Iribarne and Thompson (1976) ) [32] , the field current emitted per unit area is given by j =ε o E kT h e ? ?G?GE ( ) kT (A57) where k is Boltzmann’s constant, h = 6.625×10 ?34 (J) (sec) is Plank’s constant, ?G is the free energy of solvation (of the order of 2 eV for many ion/solvent pairs, known separately), and G(E) is the reduction of this free energy due to the normal field E. A good model for G(E), as shown experimentally by Loscertales (1995) ) [34] is the so- called “image charge model”, analogous to Schottky’s theory for electron emission: GE()= e 3 E 4πε o ? ? ? ? ? ? 1/2 (A58) Since, at room temperature, T ? 0.025eV , while ?G and G(E) are ≈1-2eV, Eq. (A57) shows the very strong sensitivity to E also evidenced in the data. Whether or not ion emission is a desirable feature for colloid propulsion is still a matter of some debate. This question is discussed next, in connection with propulsive efficiency. A6. Propulsive Efficiency. Effect of Polydispersity As in any propulsive device, an exhaust stream containing more than a single speed is a less than optimum arrangement, because the energy spent to accelerate the faster constituents is larger in proportion than the extra thrust derived from them. Suppose our colloidal stream contains a mixture of droplets of various sizes and charges, including single ions. Let Y N be the number of droplets of type j or ions emitted per j 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 27 of 36 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 28 of 36 second, and assume they are all accelerated through a voltage V, to a velocity c j = 2q j V m j . The total mass flow rate is Y m = Y N j j ∑ m j (A59) and the total current is I = Y N j j ∑ q j (A60) The thrust is F = Y N j j ∑ m j c j = Y N j j ∑ 2Vm j q j (A61) The propulsive efficiency (propulsive power/(input power) is η p = F 2 2 Y m IV = Y N j / 2 / V m j q i j ∑ ? ? ? ? ? ? 2 / 2 / V Y N j j ∑ m j ? ? ? ? ? ? Y N j q j j ∑ ? ? ? ? ? ? (A62) Restricting attention now to only modisperse drops (m d , q d ) and ions (m i , q i ), η p = Y N d m d q d + Y N i m i q i () 2 Y N d q d + Y N i q i Y N d m d + Y N i m i () (A63) The current carried by drops is I d = Y N d q d , and that carried by ions is I i = Y N i q i . If we let β i = I i I ; β d =1?β i = I d I (A64) then (A63) can be written as η p = β d m d q d + β i m i q i ? ? ? ? ? ? 2 β d m d q d +β i m i q i = 1? 1? ε () β i[] 2 1? 1?ε()β i (A65) where ε = q / m () d q / m() i (A6) 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 29 of 36 Alternatively, if we work with the mass fractions α i = Y m i Y m , etc, then, η p = α d q d m d +α i q i m i ? ? ? ? ? ? 2 α d q d m d +α i q i m i (A67) It is easy to check that η p is less than unity, unless β i = o or β i = 1 (or, alternatively α i = o or α i = 1). Minimum efficiency occurs at an ion current fraction β i () η p MIN = 1 1+ ε (A68) or an ion mass fraction α i () η p MIN = ε 1+ ε (A69) and this minimum is η p () MIN = 4 ε 1+ ε () 2 (A70) From this discussion, it is clear that two high-efficiency regimes exist: (a) Pure or near pure ions β i ≈1( ). Since ε <<1 typically, this requires 1?β i to be fairly small ? ε () . (b) Mainly droplets β i <<1(). If neither of these conditions is feasible, Eq. (A70) shows that it is important to keep ε from being too small. This, in turn, implies drops being as small as possible (large q/m) d ) and ions being as heavy as possible (not too large (q/m) i ). As an example, probably close to the limit of what is now possible, consider a highly conductive formamide solution of the heavy organic ion Tethra-heptyl ammonium (4(C 7 H 15 )N + , molecular mass 410 g/mol. For this type of solution, the threshold tip field for ion emission (Gamero, 1999) ) [35] is about 1.28V/nm. This field is achieved, according to Eq. (A55), when Q = 2.5×10 ?14 m 3 / s, while the minimum stable flow rate (Eq. A52a) is 5.8 about four times smaller. ×10 ?15 For the threshold condition 2.5×10 ?14 m 3 / s ( ) , Eq. (A33) gives q / m() d = I / ρQ= 4800 Coul Kg , whereas, if ions were never emitted, we found before that (q/m)d could be raised to about 10,000 Coul/Kg. 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 30 of 36 If we wish to operate with mostly droplets, and stay above Q = 2.5×10 ?14 m 3 / s, while also wanting to avoid V>5000 Volts, the specific impulse will be limited to Isp = 1 9.8 2 × 4800× 5000 = 707sec. If we restrict the flow further, we enter a mixed mode where both, ions and droplets are being emitted. This regime has not been studied much, so only rough extrapolations can be made at this time. The main difficulty is in understanding to what extent the depletion of ions caused by ion emission will modify the simultaneous emission of droplets. For the purpose of making some performance estimations, we will assume for now that (a) Q is essentially limited to a minimum dictated by ion onset, and (b) (q/m) for the droplets peaks at a value 10-20% higher than that for the ion onset flow. The critical flow (at E TIP =E cr ) is found from (A55) Q cr = f 2 ε() ε γ 3 K 1.87×10 7 E cr ? ? ? ? ? ? 6 (A71) and using (A33), this gives q m ? ? ? ? cr = 1 γρ E cr 1.87×10 7 ? ? ? ? 3 (A72) We saw (Eq. A70) that polydispersity inefficiencies are minimized if ε = q / m () d q / m() i is not very small. If we assume q / m() d ?1.2 q / m( ) cr and q i = e (singly charged ions), then ε ? 1.2 γρ ε cr 1.87×10 7 ? ? ? ? 3 m i e (A73) The dependencies of E cr on fluid properties are not well known, as only a few experimental data points exist. However, some trends can be theoretically predicted. First, since the basic free energy of ion evaporation, ?G, is proportional to the 1/3 power of the surface tension, γ 1/3 , while, from Eq. (58), its field-induced reduction is G(E)≈E 1/2 , we expect E cr ≈γ 2/3 , which yields a proportionality to γ for both (q/m) and cr ε . Secondly except for very small ions, the image field of the ion during detachment scales as 1/d 2 , where d is the ionic diameter, which itself scales as m . It is to be noted here that this m , as well as that in (q/m) , i 1/ 3 i i is the mass of the solvated ion, in whatever solvation state it leaves the liquid. Thus, from (A73) it would appear that ε ≈γ / m i (A74) 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 31 of 36 and so the ideal fluid for operation in the mixed regime should have both, high surface tension, and small (solvated) ion mass. Unfortunately, the factors which determine degree of solvation are complex, and it is not possible therefore to conclude that lighter ions (say or ) are necessarily better than heavier ones (say, Tetra-heptyl ammonium or related). N a + L i + It is noteworthy that the performance parameter ε (Eq. A73) turns out to be independent of the conductivity K of the solution used. It must be remembered, however, that this mixed regime is only possible if ion emission begins before minimum flow is reached for stability of the Taylor cone, i.e., if Q . Using (A71) and (A52a), this condition can be recast as cr > Q MIN f ε() ε γK E cr 3 > 4.6×10 ?28 η MIN ρ (A75) which implies a minimum conductivity level. For formamide, with , and levels of this order are necessary in general. E cr =1.28×10 9 V / m, K MIN ? 0.96Si / m A6.1 EXAMPLE OF MIXED REGIME PERFORMANCE Assume a solution of Tetra-heptyl ammonium in formamide, for which we calculated and increase this by 20% to 5770 Coul/Kg to account for some additional tip thinning as we push into the mixed regime. Assume also that the ions are emitted with a single charge and q / m() cr = 4800Coul/ Kg no solvation, giving q m ? ? ? ? i = 96500/ 0.41= 235,000Coul / Kg, or ε = 5.770 / 235,000 = 0.0245. The specific impulse can be expressed, following the development in Sec. A6, as gI sp = 2V q m ? ? ? ? d 1? 1? ε ( ) β i 1? 1?ε()β i (A76) and the efficiency is as given by (A65). Under these conditions, Fig. 15 shows the variation of η p with β i , and Fig. 16 shows the accelerating voltage vs. β i for various desired specific impulses. 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 32 of 36 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 33 of 36 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 34 of 36 Fig. 15 clearly illustrates the two efficient regimes at low and very high ion fraction, with poor performance in between. If η p = 0.6 is arbitrarily chosen as the minimum acceptable efficiency, the ion current fraction should be below 58% or above 97%. From Fig. 16 then, we see that the low β i regime (β i <0.58) requires voltages greater than 1500V, 6000V or 13,200V for Isp of 500, 1000, 1500s, respectively, and is therefore probably acceptable for Isp less than about 700s. The high β i regime, on the other hand, requires very low voltages (under 2KV even for Isp=1500s), which is very desirable, provided a stable Taylor cone can still be maintained. This probably requires an accel- decel electrode structure, where the inner (accel) electrode acts as an extractor. Alternatively, if the emitter capillary is thin enough the starting voltage may be low enough to obviate this need. Using Eq. (A19), with d/R c =200, V start =1KV requires a diameter of 6.7μm. Diameters of this order may also be required to ensure a low enough evaporative loss of formamide (see Sec. A7). If accel-decel geometries are used, one concern would be the resulting increase of the beam spreading angle, in analogy with the similar effect known from ion engine work ) [35] . A7. Other Design and Operational Considerations A7.1 Evaporation from emitter tip. As noted in Section A3, the current emitted in droplet form by a Taylor cone depends on the flow rate Q, but not on the emitter’s diameter. The same flow rate, and hence the same current, can be produced using a thin capillary under a high supply pressure or a wider one with correspondingly reduced supply pressure. For liquids of moderate to high volatility, it is then advantageous to reduce the emitter diameter, because the loss due to evaporation from the exposed liquid surface does scale as the square of this diameter (this is in addition to the advantage in starting voltage). The cone’s surface area, for a tube diameter D, is πD 2 / 4sinα(), so that the evaporated mass flow rate is Y m v = πD 2 rsinα P v T() 2πm v kT m v (A7) where m v is the mass of a vapor molecule, and P v (T) is the vapor pressure at the tip temperature. If a design constraint is imposed that Y m v ≤ f v ρQ MIN , with some prescribed fraction f v of minimum emitter flow, we obtain the condition (using (A52a) for Q MIN ) D≤ 2 f v sinαη MIN 2 c v γεε o KP v ? ? ? ? ? ? 1/ 2 (A78) where c v = 8 π kT m v ? ? ? ? ? ? 1/2 is the vapor’s mean thermal speed. Consider the case of formamide. Ignoring for this calculation the reduction in P v due to the solute, we have 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 35 of 36 P v Pa()= 3.31×10 12 e ? 8258 T (K ) (A79) Taking T=293K, η MIN =0.5, m v =0.045Kg/mol, and K=1 Si/m, we calculate from (A78) a minimum diameter for f v =0.01 of D MIN =6.2μm. This is of the same order as the diameter required for start-up at 1 KV voltage. Both of these results point clearly to the desirability of thruster architectures with large numbers of very small emitters, which motivates research into microfabrication techniques for their production. 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 36 of 36