16.522, Space Propulsion Prof. manuel martinez-sanchez Lecture 23-25: COLLOIDAL ENGINES APPENDIX Al. 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 l 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 F charge limited current density (as V ) and leads to a thrust density, 4=23d) (d=grid spacing), which is larger in proportion to V2, and therefore to(m/q).In addition to the higher thrust density the higher voltage also increases efficiency since any cost-of-ion voltage VLoss becomes then less significant n In a sense, this succeeded too well. Values of droplet m/q that could be generated with the technology of the 60s 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 l An. 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-Sanche
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 V3/2), 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 V2 , 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 VLOSS becomes then less significant η = V 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 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 -I by increasing ne, 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 A2I SURFACE CHARGE Consider first a flat liquid surface subjected to a strong normal electric field, En. If the liquid contains free ions(from a dissolved electrolyte), those of the attracted polarity will concentrate on the surface. Let p be this charge, per unit area; we can determine it by applying Gauss law VE=Pc/E, in integral form to the"pill box "control volume shown in the figure 只=EEn E n A similar effect(change concentration)occurs in a Dielectric liquid as well, even the ere are ++++|+ E=0 no free charges. The appropriate law is then Conductive Liquid VD=pch,where Figure 1 D= EE E and E is the relative dielectric constant, which can be fairly large for good solvent fluids(E=80 for water at 20oC). There is now a non-zero normal field in the liquid and w E E.,Ent =o(no free charges)(A2) and in addition E(Eng -End)=P, dipole (A3) Ege Eliminating Em,t between these expressions, 1--)En 16.522, Space Pre Prof. Manuel mar ropelsinnchez Lecture 23-25
thrusters (ion engines, Hall thrusters, arcjets) lead to the need to reduce the ionization mean free path 1 σ ionne ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ by increasing ne, 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, En. If the liquid contains free ions (from a dissolved electrolyte), those of the attracted polarity will concentrate on the surface. Let ρs 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 = ρch / ε o ρs = ε oEn (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 = ρchfree , 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 oEn, g − εε oEn,l = o (no free charges) (A2) and, in addition, ε o (En,g − En,l )= ρs, dipoles (A3) Eliminating En,l between these expressions, ρs, dip. = 1− 1 ε ⎛ ⎝ ⎞ ⎠ En,g (A4) 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 2 of 36
which, if a>>1 is nearly the same as for a conducting liquid (eq. A1). The field inside the liquid follows now from(A2) E E and is very small if E>>1(zero in a conductor) 16.522, Space Pr Lecture 23-25 Prof. Manuel martinez-Sanchez Page 3 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): En,l = 1 ε En, 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
A2. 11 CHARGE RELAXATION Consider a conductive liquid with a conductivity K normally due to the motion of ions of both polarities If their concentration is and their mobilities are u,μ((m/s)/(v/m), then n(*+μ)⑤m) (1) Suppose there is a normal field eg 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 eg points away from the liquid), so a free charge density o builds up over time. at a rate =KE The charge is related to the two fields, Eg, EI from the"pillbox"version of E。E-EEE= From(3), E and substituting in(2) (4) The quantity Eo=t is the Relaxation Time of the liquid. In terms of it, the solution of (4)that satisfies a(0)=0(for a constant En at t>O)is 16.522, Space P pessan Lecture 23-25 Prof. Manuel martinez Page 4 of
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 + - µ µ, ((m/s)/ (V/m)), then ( ) ( ) + - K = n + Si m µ µ (1) Suppose there is a normal field g En 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 En points away from the liquid), so a free charge density σf builds up over time, at a rate f l n d = KE dt σ (2) The charge is related to the two fields, g l E , E n n from the “pillbox” version of free ∇ . D = ρ JG JG g l ε εε σ 0n 0n f E- E= (3) From (3), g l n f n 0 E E= - σ ε εε , and substituting in (2), f g f n 0 d K K + =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 σ 0 =0 (for a constant l En at t>0) is A2.1.1 CHARGE RELAXATION
E The surface charge approaches the equilibrium value E(at which point, from(3) En=0)but it takes a time of the order of t=ok to reach this equilibrium. For a concentrated ionic solution, with K-1 Si/m and e-100, this time is about T=10-5=1 ns, 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, K-104 Si/m, and t-10s= 10 us which can be directly measured in the lab The math can be generalized to a gradual variation of the field, E9=En(t). Using the method of variation of the constant o,=c(t)e/i dtdt t and substituting into(4) dc_e/KEg(t); C=Co+e/E(t)dt Since o(0)=0,c(0)=0 And So Co=0 o,=e En(t )dt 16.522, Space Propulsion ure23-25 Prof. Manuel martinez-Sanchez
16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 5 of 36 g t - 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 E =0 n ) but it takes a time of the order of = 0 K εε τ to reach this equilibrium. For a concentrated ionic solution, with K 1 Si m ∼ and ε ∼ 100 , this time is about -9 τ = 10 s = 1 ns , 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 K 10 Si m ∼ , and -5 τ ∼ 10 s = 10 sµ which can be directly measured in the lab. The math can be generalized to a gradual variation of the field, ( ) g g E Et n n = . Using the method of “variation of the constant” ( ) t - f =c t e τ σ ; t - d f 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 0 n 0 K c = c + e E t' dt' τ ε ∫ Since σf (0 =0 ) , c(0) = 0 And so 0 c =0 : ( ) t t-t' - g f n 0 K = e E t' dt' τ σ ε ∫ (6)
A22 SURFACE STABILITY E: 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 (e)=o En for a conductor(the 1/2 accounts for the variation of En from its outside value toO inside the liquid ). This traction then intensifies on the protruding parts, and the process can become unstable if surface tension, r, 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 potental, which ple is assumed sinusoidal, and small (initially at least), then the outside If the surface ripp ch obeys Vo=o with p=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")is a harmonic function(z-X+iy), 中≡-Ey+ The surface is where d=0, and this, when ay<< l, is approximately given by O≡-Ey+中cosa,or (A7) The surface has a curvature 1/R cosax. which is maximum at crests R=2 and gives rise to a surface tension restoring force(perpendicular to the surface)of (cylindrical surface) 16.522 spel m artipezssanch Lecture
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 ρs ( )En 2 = ε o 2 En 2 for a conductor (the 1/2 accounts for the variation of En 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 + φ1e−αy cosαx (A6) The surface is where φ = o , and this, when αy << 1, is approximately given by o ≅ −E∞y +φ1 cosαx , or y ≅ φ1 En cosαx (A7) The surface has a curvature 1/ Rc ≅ d2 y dx2 = φ1 α2 E∞ cosαx , which is maximum at crests (cosα x=1): Rc = E∞ φ1α 2 (A8) and gives rise to a surface tension restoring force (perpendicular to the surface) of γ Rc (cylindrical surface). 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 6 of 36
The normal field, from(A6), is E=ssp E +ap,e cos ax and at ayy E (A9) The quantity a is 2/2, where A is the wavelength of the ripple. Thus, if long-wave ripples are possible, a small field is sufficient to produce instability. We will later b interested in drawing liquid from small capillaries; if the capillary diameter is D, the largest wavelength will be 2D, or o-I which gives the instability condition E> (A10) E。D For example, say D-0 1mm, and y=0.05N/m(Formamide, CHBON) The minimum 丌×0.05 field to produce an instability is then V885×10-2×10 1.33x10'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 x10x2x10-=2660 Volt to generate it. a more nearly correct estimate for this will be given next 16.522, Space Pre Lecture 23-25 Prof. Manuel martinez
The normal field, from (A6), is Ey = −∂φ ∂y = E∞ +αφ1e−αy cosαx and at αy γ φ1α2 E∞ or E∞ > γα εo (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 oD (A10) For example, say D=0.1mm, and γ = 0.05N /m (Formamide, CH3ON). The minimum field to produce an instability is then π × 0.05 8.85 ×10−12 ×10−4 = 1.33 ×107 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 ×107 × 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 coordinates called m=const "Prolate Spheroidal and is an angle about the line Ff Fig. 4a Here and so, lines of n= const are confocal hyperboloids( foci at F, F)while 5= const. lines are confocal ellipsoids with the same foci. The surface n=o is the symmetry plane, S and one of the n-surfaces, n=no, can be chosen to represent(at least near its tip) the protruding liquid surface from a capillary as in Fig 4a If the potential o is assumed to be constant(V)on n=no, and zero on the plane s, then the entire solution for o will depend on n alone. The n part of Laplaces equation in these coordinates is which, with the stated boundary conditions, integrates easily to th-ln (Al2) Let R=x2+y(cylindrical radius). From n=2, the(z, R)relationship for an n= const. hyp 16.522 spel m artipezssanch Lecture 23-25
A2.3 STARTING VOLTAGE FOR A CAPILLARY Fig. 4b shows an orthogonal system of coordinates called “Prolate Spheroidal Coordinates”, in which η = r 1 − r2 a ; ξ = r1 + r2 a and ϕ is an angle about the line FF’. Here r1 = x 2 + y 2 + z + a 2 ⎛ ⎝ ⎞ ⎠ 2 r2 = 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=ηo, 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=ηo , 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 (A11) which, with the stated boundary conditions, integrates easily to φ = V th−1 η th−1 ηo (A12) Let R (cylindrical radius). From 2 = x 2 + y 2 η = r 1 − r2 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
which, for z>o, can be simplified to z The radius of curvature R of this surface Is given by、 72, which yields R2/a2 R 1+4 A13) 2 n) Also, from Fig. 4b, the tip-to-plane distance is d==(R=0,n=n2)=3m (A14) Eqs.(Al3),(A14)give the parameters a and no if Re and d are specified R a= d 7= R The electric field at the tip is E. dn do 2,and using Eq(A12) R=a7= 2/a -2)hn (A16) which can be expressed in terms of Rc, d, when r<<d, as (Al7) 4d 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-Sanche
aη = R2 + z + a 2 ⎛ ⎝ ⎞ ⎠ 2 − R2 + z − a 2 ⎛ ⎝ ⎞ ⎠ 2 which, for z>o, can be simplified to z = η a2 4 + R2 1 −η 2 . The radius of curvature Rc of this surface is given by 1 Rc = zRR 1 + zR 2 ( )3 / 2 , which yields, Rc = 1 −η2 2η a 1+ 4 R2 / a2 1− η 2 ( )2 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 3/ 2 (A13) Also, from Fig. 4b, the tip-to-plane distance is d = z R = o,η = ηo ( ) = a 2 ηo (A14) Eqs. (A13), (A14) give the parameters a and ηo if Rc and d are specified: a = 2d 1 + Rc d ; ηo = 1 1 + Rc d (A15) The electric field at the tip is Ez = − ∂φ ∂z ⎛ ⎝ ⎞ ⎠ TIP = − dφ dη dη dz ⎛ ⎝ ⎜ ⎞ ⎠ TIP . Now ∂z ∂η ⎛ ⎝ ⎜ ⎞ ⎠ TIP = ∂z ∂η ⎛ ⎝ ⎜ ⎞ ⎠ R= o,η= ηo = a 2 , and using Eq. (A12), ETIP = − 2V / a 1 −ηo 2 ( )th−1 ηo (A16) which can be expressed in terms of Rc, d, when Rc<<d, as ETIP = − 2V / Rc ln 4d Rc ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ (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
2PR (2y /R, because there are two equal curvatures in an axisymmetric tip Substituting(a18), the starting voltage" is farr A19) eturning to the example with re=0.05mm, y=0.05 N/m, and assuming an attractor plane at d=5mm, the required voltage 0.05×5×10- rRV8.85×10 n(400=3184Vols whereas if the attractor is brought in to d=0.5mm, vSTarT1960 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 agr trong feld perimental observations(Zeleny, 1914-1917)5,6, it was known that when From early is applied to the liquid issuing from the end of a tin tube, the liquid surface lopts 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, GI Taylor! 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'8e-/2 due to the electric field must be balanced everywhere or the conical surface by the pull of the surface tension. The curvatures of the surface. In a cone, 1/Rc is zero along the generator, while the curvature of the normal section is 16.522, Space Pre Prof. Manuel mar ropelsinnchez Lecture 23-25 Page 10 of 36
ε o 2 ETIP 2 > 2γ Rc (A18) (2γ / Rc , because there are two equal curvatures in an axisymmetric tip). Substituting (A18), the “starting voltage” is VStart = γRc ε o ln 4d Rc ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ (A19) Returning to the example with Rc=0.05mm, γ=0.05 N/m, and assuming an attractor plane at d=5mm, the required voltage is VSTART = 0.05 × 5 ×10−5 8.85 ×10 −12 ln( ) 400 = 3184 Volts whereas if the attractor is brought in to d=0.5mm, VSTART=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 oEn 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 Rc1 + 1 Rc 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ , where 1/ Rc1 ,1/ Rc 2 are the two principal curvatures of the surface. In a cone, 1/Rc 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
