chemical physics of discharges - Argonne National Laboratory

13
field per electron is and per collision ttrereforc
11. 2. iliffusion of Ciiargeci Species.
2 e2L 2 ,
in u
e
rnGT *
In the discliarges of interest iiere, tile concentration of charged species is
greater ellan and a large fractional separation of elrctrong alid positive ions
becomes impossible, because it would set up a very large opposiug field. The currents
of electrons and ions reaching the wall must tiicn be equal, i = - i) Vn - nu E'
i+ = - D+Vn + nu E' where n = n
+
IJ the diffusion coefficient, LI the mobility, and E' the field due to tile (small) space
charge. The subscripts e and + refer to clectror. and positive ion. Equating i and
i, and eliminating C' one obtains
r
Dew+
I = -
+ U+we
?+ + Ue
= n+ is the electron density, 011 the densitv gradient,
Vn = 1) Vn
which serves as the definition of the anbipolar diffusion coefficient, Da. SubstitukTe
ting P - for i)
e c
and the equivalaut expression for 1)+ otic obtains
v+k
which approximately equals - ('1
c e + T ) or D+(1
+
T
+ ")
T+
because the electron mobility
pee' is much larger than the ionic mobility P+. In active glow discharges T /T+ is
typically 20 to 100, whereas in the afterglow the electrons thermalize rapidly ana
Te/T+ = 1.
Thus, D 2 20 to 100 D+ in the active discharge, and 2 D+ in the afterglow.
The disappearance of charged species bv ambipolar diffusion in the absence of
6n
a source term is described by the diffusion equation = uavZn where n is a function
of r, 8, z, and t. The well-known solution of this equation for the case of an infinite
m
cylinder is n(r,t) = Z
-kit
AiJo(% I) e where al is the i th root of Jo, the Bessel
i=1 0
a.
2 Da
function of zero order and k = ($) . The diffusion length, A, therefore
i
Da =
r 0
equals . The first few zeroes of Jo are a1 = 2.405, a2 = 5.520. a3 = 8.654,
'i
a,, = 11.792 which shows that, as diffusion proceeds, tile time decay will be increasingly
governed by kl = (-) Da, the first (lowest) diffusion mode, because the next three
r0
higher modes are damped out more rapidly by factors of 5.3, 12.9, and 24. After a
short transient, the diffusion-controlled electron decay or the diffusion controlled
loss under steady-state conditions with a spatially well distributed source term can
therefore be closely approximated by a first-order rate constant, k = 5.78 Da/ro2.
This analysis applies when there are only positive ions and electrons present.
h'hen negative ions are also present, their principal effect is to accelerate the
ambipolar diffusion of the electrons, (Da)e, which now becomes approximately
T
Te
(Da)e = (1 + A) D+ (1 + -) + X - 1) which for active discharges (Te/T+ >> 1)
*+ T+
can be further approximated by (1 + 2 A) Te/T+, where
ratio of negative ions and electrons.
= n-/ne. the concentration
It can be seen that for A >> 1 electrons