《航天推进 Space Propulsion》（英文版）Lecture 23-25: COLLOIDAL ENGINES

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)