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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 =mc2

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, FA

=εo

243

Vd

⎛ ⎝

⎞ ⎠

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 of 36

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 / εo

ρs = εoEn (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 = εεo

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

εoEn, g − εεoEn ,l = o (no free charges) (A2)

and, in addition,

εo En,g − En,l( )= ρs, dipoles (A3)

Eliminating En ,l between these expressions,

ρs, dip. = 1−1ε

⎛ ⎝

⎞ ⎠ En,g (A4)


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 =

1ε

En , g (A5)

and is very small if ε >> 1 (zero in a conductor).



Consider a conductive liquid with a conductivity K, normally due to the motion of ions of both polarities. If their concentration is

+ - 3n = n = n (m /s) and their mobilities are

+ -, µ µ ((m/s)/ (V/m)), then

( ) ( )+ -K = n + Si mµ µ (1)

Suppose there is a normal field g

nE 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

nE points away from the liquid), so a free charge density fσ builds up over time, at a rate

lfn

d= KE

dtσ

(2)

The charge is related to the two fields, g l

n nE , E from the “pillbox” version of

free . D =∇ ρ

g l0 n 0 n fE - E =ε εε σ (3)

From (3),

gl n fn

0

EE = -

σε εε

,

and substituting in (2),

gff n

0

d K K+ = E

dtσ

σεε ε

(4)

The quantity 0 =Kεε

τ is the Relaxation Time of the liquid. In terms of it, the solution

of (4) that satisfies ( )f 0 = 0σ (for a constant lnE at t>0) is

A2.1.1 CHARGE RELAXATION


g t-

nf

0

E= 1- e τ

⎛ ⎞σ ⎜ ⎟

ε ⎝ ⎠ (5)

The surface charge approaches the equilibrium value gn

0

Eε (at which point, from (3),

lnE = 0 ) but it takes a time of the order of 0= K

εετ to reach this equilibrium. For a

concentrated ionic solution, with K 1 Si m∼ 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, -4K 10 Si m∼ , and -510 s = 10 sτ µ∼ which can be directly measured in the lab. The math can be generalized to a gradual variation of the field, ( )g g

n n=E E t . Using the

method of “variation of the constant”

( )t-

f = c t e τσ ; t-fd dc c

= - edt dt

τσ ⎛ ⎞

⎜ ⎟τ⎝ ⎠

and substituting into (4),

( )t g

n

dc K= e E t

dtτ

ε; ( )

tt' g

0 n0

Kc = c + e E t' dt'τ

ε ∫

Since ( )f 0 = 0σ , c(0) = 0

And so 0c = 0 :

( )t t-t'

- gf n

0

K= e E t' dt'τσ

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

ρs( )En

2=

εo

2En

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 ∇2φ = 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 eiαz( ) 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 ≅d 2 ydx2 =

φ1α2

E∞

cosαx , which is maximum at crests

(cosα x=1): Rc =

E∞

φ1α2 (A8)

and gives rise to a surface tension restoring force (perpendicular to the surface) of γ

Rc

(cylindrical surface).


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) δεo

2Ey

2⎛ ⎝

⎞ ⎠ = εoE∞αφ1 . Instability will occur if this exceeds the restoring surface

tension effect:

εoE∞αφ1 > γφ1α

2

E∞

or E∞ >γαεo

(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 =

πD

, which gives the instability condition

E∞ >πγεo D

(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 ×107V / 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.


A2.3 STARTING VOLTAGE FOR A CAPILLARY

Fig. 4b shows an

orthogonal system of

coordinates called

“Prolate Spheroidal

Coordinates”, in which

η =r1 − r2

a ; ξ =

r1 + r2

aand ϕ is an angle

about the line FF’.

Here

r1 = x2 + y2 + z +a2

⎛ ⎝

⎞ ⎠

2

r2 = x2 + y2 + z −a2

⎛ ⎝

⎞ ⎠

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 −η2( )∂φ∂η

⎡ ⎣ ⎢

⎤ ⎦ ⎥ = o (A11)

which, with the stated boundary conditions, integrates easily to

φ = Vth−1ηth−1ηo

(A12)

Let (cylindrical radius). From R2 = x 2 + y2 η =

r1 − r2

a, the (z,R) relationship for an

η = const. hyperboloid is


aη = R2 + z +a2

⎛ ⎝

⎞ ⎠

2

− R2 + z −a2

⎛ ⎝

⎞ ⎠

2

which, for z>o, can be simplified to z = ηa2

4+

R2

1 −η2 . The radius of curvature Rc of

this surface is given by 1Rc

=zRR

1 + zR2( )3 / 2 , which yields,

Rc =1 −η2

2ηa 1+ 4

R2 / a2

1− η2( )2

⎡

⎣ ⎢

⎤

⎦ ⎥

3/ 2

(A13)

Also, from Fig. 4b, the tip-to-plane distance is d = z R = o,η = ηo( )=

a2

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

= −dφdη

dηdz

⎛ ⎝ ⎜ ⎞

⎠ TIP

.

Now ∂z∂η

⎛ ⎝ ⎜ ⎞

⎠ TIP

=∂z∂η

⎛ ⎝ ⎜ ⎞

⎠ R= o,η= ηo

=a2

, and using Eq. (A12),

ETIP = −2V / a

1 −ηo2( )th−1ηo

(A16)

which can be expressed in terms of Rc, d, when Rc<<d, as

ETIP = −2V / Rc

ln 4dRc

⎛ ⎝ ⎜ ⎞

⎠ ⎟

(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


εo

2ETIP

2 >2γRc

(A18)

(2γ / Rc , because there are two equal curvatures in an axisymmetric tip). Substituting (A18), the “starting voltage” is

VStart =

γRc

εo

ln4dRc

⎛

⎝ ⎜ ⎞

⎠ ⎟ (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” εoEn

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, γ1Rc1

+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


the projection on it of that

of the circular section through the same point

(Meusnier’s theorem):

1Rc

=1R

⎛ ⎝

⎞ ⎠ cosα =

cosαrsinα

=cotα

r (A20)

This means that 1

2εo En

2 =γ cotα

r

En =2γ cotα

εor (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 ηo = cosα) , but this type of electrostatic field has En ≈ 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 = APν cosϑ( )rν (A22a) or φ = A Qν cosϑ( )rν (A22b) where Pν ,Qν are Legendre functions of the 1st and 2nd kind, respectively. Of the two, Pν has a singularity when ϑ = 180o , and Qν has one at ϑ = o . The latter is acceptable, because ϑ = o is inside the

liquid cone, and we only need the solution outside. The normal field, from (A22b) is then


En = Eϑ = −1r

∂φ∂ϑ

= +AdQν

d cosϑ( )sinϑ

1r1−ν

and, in order to have En ≈1

r1/ 2 , we need ν =12

. Thus,

φ = A r1 / 2Q1 / 2 cosϑ( ) (A23)

The function Q1 / 2 cosϑ( ) is shown in Fig. (9)•. The essential point is that this function

has a single zero, at

ϑ = α = 49.290o (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 r2 / r1 = φ2 / φ1( )2 for a

given angle ϑ .

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


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 (m3/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( ) = K1 + x

2⎛ ⎝

⎞ ⎠ − 2E

1 + x

2⎛ ⎝

⎞ ⎠

P1/2 x( ) =2

• NOTE: Use is made of Q , where K and E are the complete elliptic

integrals of the 1st and 2nd kind, respectively. It is also noteworthy that π

Q1/2 −x( )

P1 / 2 cos 180o − ϑ( )[

, so

] could equivalently be used as the angular part in (A23).


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

Er ≠ o( )

r is at least much less than Eϑ 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

Er =I / AK

=Ic

2π 1 − cosα( )Kr2 (A25)

where Ic is that part of I which is due to conduction.

Compared to the azimuthal surface field Eϑ = En , given by (A21), we see that Er 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

1ε

2γ cotαεor

=Ic

2π 1 − cosα( )Kr 2

or r = r1 =εo

8π 2 cotα 1 − cosα( )2

⎡

⎣ ⎢ ⎤

⎦ ⎥

1 / 3

ε 2Ic

2

K2γ⎛ ⎝ ⎜ ⎞

⎠

1 / 3

(A26)


and this provides a first indication of the size of the tip jet, since the cone solution

becomes untenable for r<r1.

The surface current, associated with a fluid velocity u, will be

Is = 2π r sinα( )ρsu (A27)

where u =Q

2π 1− cosα( r2) (A28)

and ρs = εoEϑ , with Eϑ given by (21). We then find

Is r( ) =εoQsinα1− cosα

2γ cotαεo

1r3 / 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=r1

(where Er becomes comparable to Eϑ , the surface current may also become comparable

to the conduction current. Using r=r1 (from A26) in (A29) yields

Is r1( ) = 4πγ cosαKQεIc

(A30)

where Icond instead of the total current has been used in (A26). If we now say that at

r=r1,

Is = βI

Ic = 1− β( )I (A31)

where β is some unknown fraction, we obtain from (A30)


I =4π cosαβ 1− β( )

γKQε

⎡

⎣ ⎢ ⎤

⎦ ⎥

1/ 2

(A32)

Using a slightly different argument, de la Mora (1994)[26] concluded (and verified

experimentally) that

I ≅f ε( )

εγKQ( )1 / 2 (A33)

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 cosαβ 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

γ εo / ρ( )1 / 2 (A34)

η =ρKQγεεo

⎛

⎝ ⎜ ⎞

⎠ ⎟

1 / 2

(A35)

so that Eq. (A33) becomes ξ = f ε( )×η (A36)


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 En = q / 4πεoR

2( ), we must have

12

εoq

4πεo R2

⎛

⎝ ⎜ ⎞

⎠ ⎟

2

=2γR

(A37)

q (A38) = 8π εoγ( )1/ 2

R3/ 2

The droplet mass is m =

43

πR3ρ , so that

qm

=6 εoγ( )1 / 2

ρ R3/ 2 (A39)


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 mTOT and charge q is. If this subdivision is into N

equal drops of radius R, we have

TOT

N = mTOT / ρ43

πR3⎛ ⎝

⎞ ⎠ , and each drop will carry a charge

q =qTOT

N= ρ

43

πR3qTOT / mTOT . The energy per drop comprises an electrostatic part 12

qφ ,

and a surface part 4πR2γ :

E = N

12

q2

4πεo R+ 4πR

2

γ⎛ ⎝ ⎜ ⎞

⎠ ⎟

= ρqTOT2 R2

6mTOTεo

+ 3mTOTγR

(A40)

Differentiating and equating to zero, we obtain R = 9 mTOT / qTOT( )2

εoγ( )1/ 3

or qm

⎛ ⎝

⎞ ⎠

MIN .E=

3 εoγ( )1 / 2

ρ R3/ 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 Rjet 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 / Rjet =1.89 (A42) What is the relationship between Rjet and the distance r1 defined in Eq. (26)? To see this, let us first substitute Ic = 1− β( I) (with I from (A32)) into (A26):


r1 =εo

8π 2 cotα 1− cosα( )2ε2

K 2γ⎡

⎣ ⎢ ⎤

⎦ ⎥

1 / 3

1− β( )2 / 3 4π cosαβ 1− β( )

γKQε

⎡

⎣ ⎢ ⎤

⎦ ⎥

1/ 3

or r1 =1− β

βsinα

2π 1 − cosα( )2

⎡

⎣ ⎢ ⎤

⎦ ⎥

1/ 3εεoQ

K⎛ ⎝

⎞ ⎠

1/ 3

= 0.9991 − β

β⎛ ⎝ ⎜ ⎞

⎠

1/ 3

r * (A43)

with . (A43b) r* ≡ εεoα / K( 1 / 3) We similarly express Rjet as a function of flow and fluid quantities by assuming Rayleigh-limited drops (Eq. 39), and using q

m / ρ( ) =IQ

:


Rjet =1

1.896 εoγ( )1/ 2

I / Q

⎡

⎣ ⎢

⎤

⎦ ⎥

2 / 3

or Rjet =1

1.8962 / 3 εoγ( )1/ 3 β 1− β( )

4π cosαεQγK

⎡

⎣ ⎢ ⎤

⎦ ⎥

1/ 3

(A44)

Both, r1 and Rjet are seen to scale with the group εεoQ

K⎛ ⎝

⎞ ⎠

1/ 3

(which was called r* by de la

Mora, 1994)[26]. By division of (A44) and (A43),

Rjet

r1

=1

1.8918β 2 1 − cosα( )2

sinα cosα⎡

⎣ ⎢ ⎤

⎦ ⎥

1 / 3

= 0.868β 2 / 3 (A45)

For f ε( ) = 25, this gives

. Rjet / 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 Rj = 1/2dj from photographs of ethylene glycol jets (K = 5 x 10-5 S m-1; 1.64 < Re < 10.2) showing that dj 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),

qm

=I

ρQ=

f ε( )ρ ε

γKQ

⎛ ⎝ ⎜ ⎞

⎠

1/ 2

(A48)


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 =

cg

:

V =c2

2 q / m( ) (A49)

For concreteness, suppose we desire c=8000 m/s Isp ≅ 800sec( ) without exceeding

V=5KV. From (49), we need qm

>80002

2 × 5000= 6, 400Coul / Kg. Suppose we use a

Formamide solution with K=1Si/m, ε = 100, f ε( ) = 25,ρ = 1130Kg / m3,γ = 0.059N / m :

251130 100

0.059 × 1Q

> 6400 ; Q < 7.1× 10−15m3 / s

Ý 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 ×10−4W = 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 ×10−8m , which gives a jet diameter

2Rjet = 2 × 0.202r* = 7.5 ×10−9 m , and a droplet radius R = 1.89Rjet = 7.0 ×10−9 m . The

drop charge is q = 64004π3

7.0 ×10−9( )3×1130 = 1.04 × 10−17Coul (65 elementary

charges for about 21,000 Formamide molecules).

Notice also the scalings: Q ≈ KV 2

c4 , F ≈ KV 2

c3 . 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* = εεoQ / 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


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 /γεεo( 1/ 2)

qV

⎛ ⎝

⎞ ⎠

MAX=

qm / ρ

⎛ ⎝ ⎜ ⎞

⎠ ⎟

MAX

= F ×1000cd (Coul / m3 ) (A50)

where F=96500 Coul/mol is Faraday’s constant, and cd 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” Λo : K ≅ Λocd (A51) For aqueous solutions Λ is 15 (Si/m)/(mol/ ) if there are no Ho l

+ ions, in which case Λo is ≈40 (Si/m)/(mol/ ). We therefore can write, from (48) l

f ε( )

εγK

QMIN

⎛

⎝ ⎜ ⎞

⎠ ⎟

1/ 2

≅ 1000F cd ≅1000F

Λo

K

or ηMIN ≅ρεo

Λo

1000Ff ε( )ε

(A52)


Assuming Λo = 20, ρ = 1130Kg / m3, f = 25, ε = 100, Eq. (52) yields ηMIN = 0.59 , which is of the right order. The mobility factor Λo would, however, be expected to depend on viscosity, so the argument is incomplete. Using this criterion,

QMIN =γεεo

ρKηMIN

2 (A52a)

and so (A48) gives a maximum droplet specific charge

qm

⎛ ⎝

⎞ ⎠

MAX

=f ε( )

εηMIN

Kεoρ

(A53)

which reduces, as the argument above implies, to

qm

⎛ ⎝

⎞ ⎠

MAX=

1000Fρ

cd (A54)

For formamide, K can be raised to about 2 Si/m, and using ηMIN = 0.5, (A53) yields

qm

⎛ ⎝

⎞ ⎠

MAX≅ 10,000Coul / Kg . This implies a relationship Voltage-Isp V =

gIsp( )2

2 ×104

(5000V for Isp≅ 1000s.). A5.2 ION EMISSION FROM THE TIP The normal field Eϑ 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 ≅ Rjet / cosα ≅2.68f 2 / 3 r *. This gives

ETIP ≅2γ cotα

εor=

2γ cotαεo

f 1/ 3

2.68( )1 / 2K

εεoQ⎛

⎝ ⎜ ⎞

⎠ ⎟

1/ 6

=1.87 ×107 f 1 / 3 ε( )γ 1 / 2 KεQ

⎛ ⎝ ⎜ ⎞

⎠

1/ 6

(A55)

This field can be very high at low flow rates and with highly conductive fluids. It is of

interest to evaluate ETIP at the lowest stable flow rate, as given by QMIN =γεoεηMIN

2

ρK. The

result is then

ETIP( )MAX ≅1.30 ×109

ηMIN1 / 3 ρ1/ 6 f ε( )

εγK⎡

⎣ ⎢ ⎤ ⎦ ⎥

1/ 3

(A56)

Using data for formamide and ε = 100, f ≅ 25, γ = 0.059N / m, ρ = 1130Kg / m3( ),assuming ηMIN = 0.5, K = 2Si / m, this gives ETIP( )MAX ≅1.63 ×109 V / m =1.63V / nm. .


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 NaI solutions in formamide, the threshold is ETIP ≈ 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 Q1/2 . 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 = εoEkTh

e−

∆G −G E( )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:

G E( ) =e3E4πεo

⎛

⎝ ⎜ ⎞

⎠ ⎟

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 Ý N be the number of droplets of type j or ions emitted per j


second, and assume they are all accelerated through a voltage V, to a velocity

cj =2qjV

mj

. The total mass flow rate is

Ý m = Ý N jj

∑ mj (A59)

and the total current is I = Ý N j

j∑ qj (A60)

The thrust is F = Ý N j

j∑ mjcj = Ý N j

j∑ 2Vmjqj (A61)

The propulsive efficiency (propulsive power/(input power) is

ηp =F2

2 Ý m IV=

Ý N j / 2 / V mjqij

∑⎛

⎝ ⎜ ⎞

⎠ ⎟

2

/ 2 / V Ý N jj

∑ mj

⎛

⎝ ⎜ ⎞

⎠ ⎟ Ý N jq j

j∑

⎛

⎝ ⎜ ⎞

⎠ ⎟

(A62)

Restricting attention now to only modisperse drops (md, qd) and ions (mi, qi),

ηp =Ý N d mdqd + Ý N i miqi( )2

Ý N dqd + Ý N iqi( ) Ý N d md + Ý N imi( ) (A63)

The current carried by drops is Id = Ý N dqd , and that carried by ions is Ii = Ý N iqi . If we let βi =

Ii

I ; βd = 1 − βi =

Id

I (A64)

then (A63) can be written as

ηp =

βdmd

qd

+ βimi

qi

⎛

⎝ ⎜ ⎞

⎠ ⎟

2

βdmd

qd

+ βimi

qi

=1− 1 − ε( )βi[ ]2

1− 1 − ε( )βi

(A65)

where ε =q / m( )d

q / m( )i

(A66)


Alternatively, if we work with the mass fractions αi =

Ý m iÝ m

, etc, then,

ηp =

αdqd

md

+α iqi

mi

⎛

⎝ ⎜ ⎞

⎠ ⎟

2

α dqd

md

+αiqi

mi

(A67)

It is easy to check that ηp is less than unity, unless βi = o or βi = 1 (or, alternatively αi = o or αi = 1). Minimum efficiency occurs at an ion current fraction

βi( )ηpMIN=

11+ ε

(A68)

or an ion mass fraction α i( )η pMIN=

ε1 + ε

(A69)

and this minimum is ηp( )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 βi ≈ 1( ). Since ε << 1 typically, this requires 1 − βi to be fairly small ≅ ε( ).

(b) Mainly droplets βi << 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(C7H15)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 −14m3 / 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 −14m3 / s( ), Eq. (A33) gives

q / m( )d = I / ρQ = 4800CoulKg

, whereas, if ions were never emitted, we found before that

(q/m)d could be raised to about 10,000 Coul/Kg.


If we wish to operate with mostly droplets, and stay above Q = 2.5 × 10 −14m3 / s , while also wanting to avoid V>5000 Volts, the specific impulse will be limited to Isp =

19.8

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 ETIP=Ecr) is found from (A55)

Qcr =f 2 ε( )

εγ 3K

1.87 ×107

Ecr

⎛

⎝ ⎜ ⎞

⎠ ⎟

6

(A71)

and using (A33), this gives

qm

⎛ ⎝

⎞ ⎠

cr=

1γρ

Ecr

1.87 ×107⎛ ⎝

⎞ ⎠

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 qi = e (singly charged ions), then

ε ≅1.2γρ

εcr

1.87 ×107⎛ ⎝

⎞ ⎠

3 mi

e (A73)

The dependencies of Ecr 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)≈E1/2 , we expect Ecr ≈ 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/d2, 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) ,

i1/ 3

i iis the mass of the solvated ion, in whatever solvation state it leaves the liquid. Thus, from (A73) it would appear that ε ≈ γ / mi (A74)


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).

Na+ Li

+

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 > QMIN

f ε( )ε

γKE

cr

3 >4.6 × 10−28ηMIN

ρ (A75)

which implies a minimum conductivity level. For formamide, with

, and levels of this order are necessary in general. Ecr = 1.28 ×109 V / m, KMIN ≅ 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 qm

⎛ ⎝

⎞ ⎠

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

gIsp = 2Vqm

⎛ ⎝

⎞ ⎠

d

1 − 1 − ε( )βi

1 − 1− ε( )βi

(A76)

and the efficiency is as given by (A65). Under these conditions, Fig. 15 shows the variation of ηp with βi , and Fig. 16 shows the accelerating voltage vs. βi for various desired specific impulses.


Fig. 15 clearly illustrates the two efficient regimes at low and very high ion fraction, with poor performance in between. If ηp = 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 βi regime (βi <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 βi 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/Rc=200, Vstart=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 πD2 / 4sinα( ), so that the evaporated mass flow rate is

Ý m v =πD2

rsinαPv T( )

2πmvkTmv (A77)

where mv is the mass of a vapor molecule, and Pv(T) is the vapor pressure at the tip temperature. If a design constraint is imposed that Ý m v ≤ fvρQMIN , with some prescribed fraction fv of minimum emitter flow, we obtain the condition (using (A52a) for QMIN)

D ≤ 2 fv sinαηMIN2 c v

γεεo

KPv

⎡

⎣ ⎢ ⎤

⎦ ⎥

1/ 2

(A78)

where c v =8π

kTmv

⎛

⎝ ⎜ ⎞

⎠ ⎟

1 / 2

is the vapor’s mean thermal speed.

Consider the case of formamide. Ignoring for this calculation the reduction in Pv due to the solute, we have


Pv Pa( )= 3.31×1012 e−

8258T (K ) (A79)

Taking T=293K, ηMIN=0.5, mv=0.045Kg/mol, and K=1 Si/m, we calculate from (A78) a minimum diameter for fv=0.01 of DMIN=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 prof. manuel martinez-sanchez ... · prof. manuel martinez-sanchez lecture...

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