A theoretical model for a contactor, collecting electrons from an ambient, unmagnetized plasma and emitting a current Iiis discussed. The relation between Ii and the potential

bias of the contactor is found to be crucial for the formation of a quasineutral core around the anode and, consequently, for the current colleted. Approximate analytical

laws and charts for the current-voltage response are provided

Ahedo Galilea, Eduardo

Martínez-Sánchez, M.

Sanmartín Losada, Juan Ramón

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

C U R R E N T - VOLTAGE R E S P O N S E 
OF A S P H E R I C A L P L A S M A C O N T A C T O R 
E. Ahedo and J.R. Sanmartin 
E.T.S.I. Aeronauticos, Universjdad Politecnica, 28040-Madrid. SPAIN 
M. Martinez-Sanchez 
Dept. of Aeronautics, M.I.T., Cambridge, MA 02139,USA 
A theoretical model for a contactor, collecting electrons from an ambient, unmagnetized 
plasma and emitting a current /,• is discussed. The relation between J,- and the potential 
bias of the contactor is found to be crucial for the formation of a quasineutral core 
around the anode and, consequently, for the current colleted. Approximate analytical 
laws and charts for the current-voltage response are provided. 
1. Introduct ion 
The use of hollow cathodes as plasma contactors appears as a promising solu-
tion for current collection in electrodynamic tethers but both technology and theory 
still present gross uncertainties; on the one hand, several difficult phenomena are si-
multaneously present and, on the other hand, no real tests have yet been performed. 
Meanwhile, theoretical analysis of simplified models can be illuminating. Recently, 
we completed work on the steady-state response of a spherical anode immersed in an 
unmagnetized, quiescent plasma [1]. Compared with models found in the literature, 
we offer (i) an accurate kinetic treatment of the ambient plasma, taken from pas-
sive probe theories [2,3], (ii) a consistent use of asymptotic tools to find the electric 
potential profile in the different regions, (iii) the computation of the dimensionless 
C-V response, and (iv) the identification of, at least, two modes of operation. Sec-
tion 2 briefly summarizes all the model hypotheses and equations, and the way we 
solved them. Section 3 discusses the contactor response in terms of dimensionless 
parameters. Some final remarks are made in Section 4. 
2. T h e plasma-contactor model 
Figure 1 sketches the two kinds of potential response we find for the plasma 
contactor. The contactor is a sphere of radius R, biased to a positive potential Vp 
1 
relative to the undisturbed ambient plasma. It emits ions (which are accelerated 
outwards) and electrons (which remain confined around the anode), the flow of this 
emitted plasma being characterized by the ion curent / , . The undisturbed density 
and temperature of the ambient plasma are N^o and Too, respectively. The current-
collection problem consists basically in determining the ambient-electron current Je 
to the anode as a function of Vp, I„ R and the thermodynamic state of the species; 
alternatively, we assume Ie given and determine R. The solution we worked out 
corresponds to a steady state without magnetic effects; we also assumed that the 
thermal energy of all species is much less than eVp, and that collisions! and ionization 
processes are negligible. The problem then reduces to solving Poisson's equation, 
Hi^)-*"--"-**--* >. ( i ) 
where r is the radial distance to the center of the anode, JV,, (Nia) is the density 
of ambient electrons (ions), and Nec, JV, are the correaponding densities of emitted 
species; ions are positive and singly-charged. 
Sheath 
. „ Presheath 
\ 
' _*-
\ \ 
1 
t 
h / / 
* 
* 
[Ambient plasma: 
nC0 .Tcoi 
/ / 
i i 
1 . 
\ \ 
\ \ 
% 
Core 
(Ano<te) 
electrons 
Double 
** * v / 
/\ 
""*• V •Sk * 
V 
\ 
K1]" ' 
*A'X.\ 
-Layer 
, \ ^ Preshealh 
V \ 
\ » I » 
. 1 1 
. 1 ' 
' 1 1 
• 1 1 
1 1 
x ^ 
(a) (b) 
Figure 1.- Potential profile regions for the (a) no-core and (b) core modes. The plasma 
is quasineutral at presheath and core. 
Kinetic theory was used to study the ambient plasma. For eVp >• Too, the 
density of the repelled species (ions) is accurately given by the Boltzmann equilibrium 
2 
law, [3]. 
Nu^Ifa>exp(-~\. (2) 
Far from the contactor the attracted species (electrons) is assumed to be monoen-
ergetic with a uniform distribution of angular momentum J; a relation between the 
undisturbed energy £ « , and the electron temperature Too must be provided. A com-
parison of this model with the exact maxwellian formulation, for a passive probe 
[4], has shown very good agreement when R is much larger than the ambient Debye 
lenght A/j, the case of interest here; note, furthermore, that a maxwellian population 
would require numerical solution while an asymptotic analysis is possible for the mo-
noenergetic model. Calling Jg the particular angular momentum, such that electrons 
with J < JB are absorbed by the anode while those with J > JB are turned back 
without reaching it, one has [2] 
JVe = (JVc/2) ( V l + eV/Ex ± ^ 1 + eV/E* - b%/rA , (3) 
with b2B = Jg/2meE<x, given by 
Ie = jrl^eN„y/2E00/mt. (4) 
For the emitted species we made an analysis similar to the ambient species. Al-
though the exact distribution of confined electrons is difficult to compute, a plausible 
assumption, supported by experimental data, is that at least for a steady state, it 
also corresponds to a Boltzmann equilibrium 
2V« = JVoexp[e(V-V,) /T 0 J , (5) 
where the constant Na is later determined by imposing that the total plasma is 
quasineutral around the contactor; T0 is the (known) electron temperature, small 
compared with eVp. For the emitted ions and assuming E{ <C cVp, where Ei is their 
kinetic energy when leaving the contactor, we may neglect the angular momentum 
distribution (as in the ABR model for collection of zero-temperature ions by a sphere 
[5]), because transverse velocities decrease like 1/r as the ions move outwards; the 
expansion is then radial and we have 
N - -Ji~ / m-/2 rt» 
1Wi
 i^eVe(Vp-V) + Ei- W 
3 
Equations (l)-(6) constitute a complete set. To integrate it the easiest way 
is to start from r = +00 towards r — R, considering Rt instead of Ie, as the un-
known parameter. Far away from the anode there is a region, the presheath, where 
quasineutrality holds and the ambient species are dominant. Eq. (1) then becomes 
Ne ~ Nia, yielding 
2r/bB = expCeV/TciKl + eV/E^)^2 expfcV/T^) - l}'1'2 (7) 
for the potential profile V(r). This solution asymptotically matches with a non-
neutral sheath around the point, r = r i , where dV/dr —^  —00 [3]. Note that 7*1 
depends on the ratio Eco/To,; a choice -Eoo/^co = 3/2 will be later justified. We 
then obtain eV(i"i)/Too ~ 0.75 and ri/bs ~ 0.84; eliminating now &B in (4) yields a 
relation between Ie and T\, 
Ie ~ 1.54 x 4*r? x e iV 0 0 v / ^W2 i r m e - (8) 
For the sheath equations the following dimensionless variables and parameter 
are denned: 
F = (32«S7r 2 /m e ) 1 / S V/J?/ 3 , r = n / r , /1 = Urn]'2 j hrnxJ\ (9) 
From a consistent use of asymptotic arguments, Eq.(l) may be simplified to 
,#F 1 fi 
T
 dr2~ 4F JF-^F (10} 
with boundary conditions 
T = 1+ : F = 0, d f / t J r = 0. (11) 
These equations express that inside the sheath (i) only the densities JVj and JVe of the 
accelerated species are important, (ii) the electric field is much higher than outside 
it, and (iii) the presheath potential and the thermal energies of the particles may 
be neglected. Integrating (10) from the sheath outer boundary with conditions ( H ) 
shows that dF/dr starts increasing, reaches a maximum when F = Fp/(1 + fi2) and 
decreases afterwards. Two kinds of solutions are found depending on the values of /1 
and Fp: 
— Core mode.- For p, > jis(Fp), where ft,s(Fp) is later determined, dF/dr 
becomes zero at certain r = r2 and F = F^, with Fz smaller than Fp = F(VP); in 
dimensional form this gives 
r2 = rj/TatFp./i), V2 = VpFt(F„n)/Fp. (12) 
4 
Asymptotic analysis again shows that the confined electron density Nec becomes of 
dominant order in Eq.(l) when dF/dr tends to zero; around r = r>> there is then a 
transition from Eq.(lO) to the quasineutral equation 
Nec -Ni+Ne~0. (13) 
As the details of this transition are relevant only locally, we simply take r = r% as 
the inner boundary of the non-neutral region, which has now all the characteristics 
of a double layer. In the region r < r2, a quasineutral core is formed; the potential 
V(r) is obtained from (13), and setting V = Vi and V ~ Vp> the constant N0 in (5) 
and the probe radius R, 
%=exPL^)Xfi+e^~1/2»-Jw^mmE± (14) 
are finally obtained; when Vp — Vi <C Vp we have Ne -C Nec in (13) and the last factor 
in (14) is approximately one. We must note that Eq.(13) gives a univalued solution 
for V(r) only if the Bohm-like condition E{ > T0 /2 is satisfied, i.e. ions cannot leave 
the anode subsonically; the double layer inner boundary r = TI is then, contrary to 
point 1, a regular supersonic point. From solutions (12) and (14) it is now clear that 
the core mode disappears when V2 —* Vp, thai is when dF/dr becomes zero just at 
F = Fp; the limiting relation fi = [ia(Fp) comes out from Fp = Fi(ft, Fp). 
- No-c ore mode.- When fi < Ms(-^p)i Eqs.(lO) and (11) give dF/dT > 0 
at F = Fp. Thereafter, the non-neutral region is no longer a double layer and no 
quasineutral core is formed. The emitted electron population is now confined to 
a thin boundary layer around the contactor where it locally modifies the potential 
profile. Its effect on the global current-voltage response is however insignificant, the 
contactor operating now exactly as a pure ion-emitter. Calling r = rp the point 
where .F = Fp in the integration of (10) and (11), the probe radius R is 
R^rJr^F,). (15) 
Notice that the limiting case fi — fJ-s(Fp) corresponds to an ion-emitter operating at 
space-charge-limited conditions [6]. 
3 . Current-voltage diagrams 
Equations (8), (9) and (15) yield the dimensionless C-V characteristic for the 
no-core mode (or for an ion emitter) in the form je(ji, Xp) where 
U,i = (Ie , i /*J?eJV0 < >)(me , i /2£:0 0)1 /2 , XP = (eV./E^iXo/R)^3, 
5 
or, measuring R in meters, Vp in volts, Ie<x in amperes, Noa in 1011 m 3 and T ^ in 
0.1 eV, 
i , l f m 86.6 x (m^/m^hjR^N^TU2, 
Xp a 3.2 X W~2 x VyN^T^/R4'3. (16) 
In the following we shall use a convenient approximation of je(jt, xp), 
U * 2 . 1 # ' T + 0.25j,(je/xP)1J\ (17) 
valid for xp 3> 1, corresponding to the thick sheath limit, ri/R > 1 [1]. The 
transition to the core mode is obtained by adding fi = fj.t,(Fp) to Eqs.(8), (9) and 
(15), showing that , for given xp, there is an emission current value, j , = j„(xP), 
such that if j , < j i 3 the contactor does not develop a core; the following aproximate 
expressions, 
j „ a 2 .8 j e 8 ( ln i e 8 ) 1 / 2 , x P ^ 0-48J2/3 W e , , (18) 
of Jia(Xp) a n a Jea(Xp) = Je(iisi Xp) are valid for x > 1. For zero-emission, the sheath 
is negatively biased by the attracted electrons. With j , increasing the total charge 
decreases, vanishing just as the sheath becomes a double layer; the increase of j e over 
this entire process is only moderate. Since we have j , cc xP' , roughly, the higher x„, 
the larger the ion current required to electrically balance the sheath. 
In the coTe mode we have j t > j,*{xP) and Eqs.(8), (9), (12) and (14) give the 
dimensionless C-V response in the form je(j„ Xf^E^R4^3/T0XJJ ,E,/T0). Figure 
2 shows j e and j , versus xP for i) the zero-emission limit, ii) the transition to the 
core mode, and iii) several core-mode cases, each one corresponding to a fixed core 
potential drop ratio, (Vp — Vz)/Vp. Points B to F show different values of j , and Xp 
that lead to the same collected current: j e = 2 X 10 s . Two important conclusions 
turn out: i) the minimum value of j , ( ~ 3.9 X 103) corresponds to relatively low values 
of xp', and ii) in the core-mode, solutions are weakly dependent on xp. Here again 
j'e and j , must balance the electric charge at the double-layer: this roughly yields 
Ji ~ Je(ln j e ) 1 / 2 for a thick layer, and j , ~ j e for a thin layer. In points B to F the core 
plus double layer thickness is constant: T\jR ~ 0.84j'</ ~ 37.6, according to Eq.(8), 
but the thickness ratio at the double layer decreases with xp: we have T\jr-i = r±/R 
at B and r\/r2 ?i 1-6 at E; for xP smaller (point F), j , and T\fri increase due to 
a decrease of the last factor in (14). Moreover, for j e given, orbital-motion-limit 
(OML) effects set a lower-bound to xp: ~L + XP{R/\D)4^ > je', for instance, to collect 
6 
j ' e = 2 X 103 with RJXD — 25 requires that x.v > 27.3. Plasma contactors and ion 
emitters operating in an OML regime are discussed in [1]. 
J e ' J i 
Figure 2.- Dimensionless C-V curves: solid (dashed) lines represent the collected (emit-
ted) current versus the potential at the contactor for several no-core and core cases. 
The (a)-line corresponds to the zero-emission limit, j ; s= 0, and the (b)-lines to the 
transition from the no-core to the core mode, JJ = jis(Xi>) atl<i Vl/Vp — 1- Each pair 
of lines representing a core-mode case has a constant value of the potential drop at the 
core: 1 - V3/Vp is equal to .03 in (c), .06 in (d), .16 in (e), and .36 in (f). Points B to 
F represent five solutions of the C-V equation 2 X 103 = j„ (j'j, xP, 2,0.5). In all curves 
we took EoaR^I3/TQX^ = 2 and Ei/To — 0.5. Dimensionless-variables xpihi and Ji 
are defined in Eq.(16). 
4. Final r e m a r k s 
Regarding the kinetic treatment of the ambient electrons, a remarkable point 
is that Eqs,(3)-(4) are equivalent to the following fluid-like equations: 
me 
2 
(19) 
(20) 
W W . ' 
Therefore, under the monoenergetic model, the electrons behave as a fluid with radial 
and centripetal motion; re being the radial velocity and Jg/y2 the angular momen-
tum; note that for Eq.(20) exactly represent the momentum equation of a polytropic 
7 
fluid with specific heat ratio equal to 3 (as in the case of electron-plasma waveg in 
a collisionless plasma) we must set -EQO — 3Too/2. This fluid equivalence is relevant 
in two different directions. Firstly, compared with the kinetic equations, it might 
lead to a much simpler (still consistent) analysis of the stability of the steady-state 
solution. And secondly, it shows that ad hoc radial fluid models do not recover an 
essential feature of the particle motion: the reflection of particles with high angular 
momentum. This has a double consequence; i) radial models overestimate the current 
collected (for instance, the widely used, isothermal, radial model, gives in Eq.(§) a 
factor of 2.5 instead of 1.54, i.e. about a 60% overestimate on Ie); and ii) radial 
models cannot properly recover the OML regime [1]. 
Refe rences 
[1] E. Ahedo, M. Martinez-Sanchez, and J.R. Sanmartih, submitted to Phys. Fluids B. 
[2] I. Bernstein and I. Rabinowitz, Phys. Fluids 2, 112(1959). 
[3] S. Lam, Phys. Fluids 8, 73(1965). 
\4] 3. Laframboise, Report 100, UTIAS, Toronto (1966). 
[5] P.M. Chung, L. Talbot, and K.J. Touryan, 'Electric probes in stationary and flowing plasmas', 
Springer-Verlag, New York (1975). 
[6] R. Wei and P. Wilbur, J. App. Phys. 60, 2280(1986). 
8 


Current-voltage response of a spherical plasma contactor

A theoretical model for a contactor, collecting electrons from an ambient, unmagnetized plasma and emitting a current Iiis discussed. The relation between Ii and the potential bias of the contactor is found to be crucial for the formation of a quasineutral core around the anode and, consequently, for the current colleted. Approximate analytical laws and charts for the current-voltage response are provided

Current-voltage response of a spherical plasma contactor

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