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 36ε 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