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

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 ay<<I and on the crests, this is E,= E+ ao,. The perturbation of electric traction is then(per unit area)d sae)=6,E,ae,. Instabilit will ocer f this excees the restoring surfac tension effect. E。E2Qφ1>y 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 << 1 and on the crests, this is Ey = E∞ + αφ1 . The perturbation of electric traction is then (per unit area) δ ε o 2 Ey ⎛ 2 ⎝ ⎞ ⎠ = ε oE∞αφ1 . Instability will occur if this exceeds the restoring surface tension effect: ε oE∞αφ1 > 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