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\bigskip
\centerline {\bf Current-Voltage Characteristic of a Partially}
\centerline {\bf Ionized Plasma in Cylindrical Geometry}
\bigskip
\bigskip
\centerline {\bf by}
\centerline {Joel L. Lebowitz and Alexander Rokhlenko}
\centerline {Department of Mathematics}
\centerline {Hill Center, Busch Campus}
\centerline {Rutgers University}
\centerline {New Brunswick, NJ 08903}
\bigskip
\bigskip\bf
\centerline {ABSTRACT}
\medskip\rm
The properties of a partially ionized plasma in a long cylindrical
tube subject to a uniform axial electric field are investigated. The
plasma is maintained by an external ionizing source balanced by bulk
and surface recombinations. Collisions between neutrals, whose density
greatly exceeds the density of charged particles, and of neutrals with
ions are sufficiently effective for their velocity distribution to be
close to a Maxwellian with the same uniform temperature, independent
of the external field. The behavior of the plasma is described by a
collisional two-fluid scheme with charge neutrality in the interior of
the tube. Approximate nonlinear equations for the hydrodynamical
moments are obtained from a Boltzmann equation in which
electron-neutral, electron-ion and electron-electron collisions are
all important. It is found that under certain circumstances the
current, and the temperature of the electrons undergo a drastic
change, with hysteresis, as the electric field is varied.
\bigskip
\bigskip\noindent
PACS codes: 52.25.Fi;\ 52.65.Kj;\ 52.20.Fs;\ 05.20.Dd
\bigskip\bf
I. Introduction
\medskip\rm
Instabilities are ubiquitous in strongly ionized plasmas. They
dominate the behavior of such systems and their study forms the core
of the subject. The origin of the instabilities lies in the nature of
the plasma interactions: on the one hand they are long range and thus
can produce strong cooperative effects and on the other hand they
become 'weaker' locally at high energies (or temperatures) as
manifested by the decrease of the Coulomb cross section with energy
rise$^1$. The situation is different in weakly ionized cold plasmas,
systems which have attracted much attention recently$^{2,3}$. In such
systems collective phenomena play a smaller role and instabilities are
less common. Nevertheless there are cooperative phenomena in these
systems too, which, as we have shown earlier$^{4,5}$ for homogeneous
idealized systems and will show here for more realistic laboratory
situations, can lead to dramatic abrupt changes in the state of the
plasma, when such systems are driven by external fields.
There is a large literature on the behavior of plasma in a cylindrical
geometry (see for example Refs.6,7,8,9). The reason for considering
this system again here is that we are interested in a regime in which
the electrons are colder than in the self-sustained discharge plasmas
generally investigated. The plasma has a sufficiently high degree of
ionization to make the electron- electron and electron-ion collisions
important. The condition of low electron temperature permits us to
neglect inelastic collisions but requires that we include volume
recombination effects.
The phenomenon in which we are most interested is a rapid, essentially
abrupt, change in the electron temperature and current as the external
electric field crosses a certain critical value which depends on the
neutral--ion temperature, degree of ionization, etc. The phenomenon is
related to the well known runaway effect in fully ionized plasmas
caused by the decrease of the electron-ion collision cross section as
the external field increases the electron energy$^{10,11,12}$. In the
partially ionized gas the presence of the neutrals prevents such a
runaway. In fact, if one neglects collisions between the electrons
(e-e), as is done in the swarm approximation$^{13}$, valid when the
degree of ionization is sufficiently small, then the stationary
distribution (in the absence of inelastic collisions) will be of the
Druyvesteyn form$^{14}$ for which the current and electron mean energy
are smooth functions of the external field. The situation is similar
when the size of the plasma is smaller than the energy relaxation
length of the electrons$^{9}$. There are, however, other regimes, even
in weakly ionized plasmas, where the effect of the e-e collisions is
sufficient to keep the electron distribution close to a
Maxwellian$^{10,4}$. Such conditions can lead to cooperative abrupt
changes in the temperature and drift velocity of electrons as the
external field is varied. The origin of the phenomenon is made clearer
if one starts with a kinetic approach rather than with a macroscopic
description.
While it is possible, even likely, that effects related to those
discussed here have already been observed indirectly in the behavior
of discharges it would be useful for both theoretical and practical
reasons to have experiments in which the parameters can be controlled,
so as to study the phenomena in a quantitative way. We expect that the
transition will be seen as hysteresis in the current--voltage (I-V)
characteristic of the plasma when the external electric field is
slowly varying in time$^{4,5}$.
We shall consider an experimental arrangement consisting of a weakly
ionized gas in a tube of radius $R$ subjected to a constant external
axial electric field ${\bf E}.$ The plasma is assumed for simplicity
to be produced through uniform ionization inside the tube, by some
external source, at a constant rate $a$. It is balanced by two kinds
of recombination processes: a bulk one and a surface process on the
tube wall. The main bulk recombination for the regime we are
interested in, are$^{10}$ three body processes involving two electrons
plus an ion,
and dissociative recombinations in which a metastable atom-ion complex
recombines with an electron. The rates depend on the temperatures and
densities of the electrons, ions and neutrals in a rather complicated
way; see Ref.15.
{}For the sake of simplicity we lump the two processes together and
assume an effective rate of bulk recombination proportional to
$T^{-3/2}$, where $T$ is the electron temperature. We ignore the
dependence of this rate on the neutral and ion temperature and on
the pressure which we keep more or less constant.
The recombination at the wall is also treated phenomenologically.
In particular we assume that the energy is absorbed by the wall which
is kept at a fixed temperature, see section 4.
\medskip
We have in mind here a situation in which the great majority of
neutral atoms are some kind of noble gas to which may, or may not, be
added a small amount of a more easily ionized second species, though
we realize that in the latter case the analysis would be more
complicated$^{16}$. This will be reflected mainly in the rates of
ionization $a$ and recombination $\gamma$ since we shall always
consider a regime in which the density of electrons $n(r,t)$ is much
lower than that of the neutrals, $N$, but big enough, due to the great
disparity between the electron mass, $m$, and the ion--neutral mass,
$M$, for binary electron-electron collisions to dominate the energy
exchanges in electron-ion and electron-neutral (e-i and e-n)
collisions. This requires$^4$ that $$ {\sigma m\over 2\pi Me^4}(kT)^2
<<{n\over N}<<1.\eqno(1a)$$ Here $k$ is the Boltzmann constant, and
$\sigma$ is the total electron-neutral particle collision cross
section, which is taken to be a constant in our work. Putting in
appropriate values for the parameters on the left side of (1a) gives
(see Ref.4),$$ 1>>n/N >> 8\cdot 10^{-7},\ 4\cdot 10^{-8},\ 1.5\cdot
10^{-8}\eqno(1b)$$ for He, Ne, Ar plasmas respectively, when $kT$ is
approximately 1 eV and it decreases as $T^2$ for colder electrons.
The upper bound relates to the fact that we ignore any collective self
induced electrostatic or magnetic interactions.
\medskip
The ions in our model are assumed to have a uniform temperature,
$T_i$, the same as the neutrals, while their density is $n(r,t)$,
i.e. the plasma is treated as locally quasi-neutral. We assume axial
symmetry and longitudinal homogeneity so the spatial dependence is
only in the radial variable $r\leq R$. The different mobilities of
ions and electrons are compensated by an internally generated radial
ambipolar electric field ${\bf F}(r,t)$. We are thinking of a
quiescent, long positive column which fills the tube$^6.$
\medskip
The reason for considering external rather than field induced
ionization is that in our previous works$^{4,5}$, in which we
considered a spatially homogeneous case with an a priori fixed plasma
density, we found that as we varied the external field $E$, there was
a transition in the electron distribution between regimes of weak and
strong coupling to the ions. (Ref.5 presents a rigorous proof of such
a transition for a greatly simplified model system). This mechanism of
a kinetic transition, which we investigate here in a more realistic
physical situation, requires a low electron temperature and relatively
weak electric field, which seems hard to achieve when the ionization
is produced by the field, see Sec.4.
\medskip
The basic idea in Ref.4 and here is to consider situations in which
the collisions between electrons are strong enough to force their
velocity distribution $f(r,{\bf v}, t)$ to stay close to a Maxwellian
$M_f$ with temperature $T$ and drift velocity ${\bf w}$. The values of
$T$ and ${\bf w}$, which are simply related to the first two velocity
moments of $f$, are then determined by self consistent
``hydrodynamic'' equations, i.e. we evaluate the integrals entering
the time evolution of $T$ and ${\bf w}$ with the help of replacing $f$
by this Maxwellian. {}For the spatially homogeneous case this yields
ordinary differential equations in time for $T$ and ${\bf w}$ which
can be reduced to a couple of transcendental equations for stationary
values $T(E)$ and ${\bf w}(E)$ yielding, in some cases, S-shaped
curves as functions of $E={\bf |E|}$. In the region where $w$
decreases with $E$ the system is unstable. The current and the
electron temperature can therefore be expected to jump upwards
(downwards) when the field increases (decreases). This will occur at
different values of the field intensity, see Fig.3: in the lower
(upper) part of the loop the electrons will be supercooled
(superheated) creating in this way a hysteresis loop. The origin of
this behavior lies in a changeover from e-i to e-n coupling as the
dominant factor when the electron energy is increased by the field.
In the situation analyzed in this paper we obtain nonlinear partial
differential equations for $n(r,t),\ T(r,t)$, ${\bf w}(r,t)$ and
$F(r,t).$ Their stationary solutions are the main concern of this
work. This is presented in section 3 following the mathematical
formulation of the problem in section 2. A discussion of physical
situations where the transition predicted by the two-fluid model used
in this paper might be observed is presented in section 4. This is a
much more restrictive domain than that given by (1), it requires
essentially $n/N \sim 10^{-4}$ while $N \sim 10^{15} cm^{-3}$ for
light noble gases when $T_i$ is close to room temperature. At the
transition the electron temperature jumps within a range of several
$T_i$, staying significantly lower than the usual temperatures in
self-sustaining gas discharges. This determines the experimental set
up we study here. The nature of the various approximations made is
discussed in section 4 and in the Appendix.
\bigskip\bf
II. Mathematical Description
\medskip\rm
The behavior of the system is described by kinetic equations for
the electron and ion distribution functions $f(r,{\bf v}, t)$ and
$f_i(r,{\bf v}, t)$ normalized to the same density $n(r,t)$. Under the
above assumptions the Boltzmann equation for electrons has the
form$^4$ $$ {\partial f(r,{\bf v}, t)\over \partial t} - {e\over
m}\left [{\bf E} +{{\bf r}\over r}F(r,t)\right ]\cdot {\bf \nabla_v}
f(r,{\bf v}, t)+{\bf v\cdot \nabla_r} f(r,{\bf v}, t)=$$
$$a\psi ({\bf v})- \gamma n f(r,{\bf v,}t)+
{m\over Mv^2}{\partial \over \partial v}
\left \{ (bn+v^4 /l)\left [{\bar f}(r,v,t)+{kT_i\over mv}{\partial
{\bar f}(r,v,t)\over \partial v}\right ]\right \} +$$
$${bn\over v^3}\hat L f(r,{\bf v},t)+
{v\over l}[{\bar f}(r,v,t)- f(r,{\bf v}, t)] +Q[f].\eqno(2)$$
In (2) the first two terms on the right side represent the
ionization and recombination respectively:
$\psi ({\bf v})$ is the normalized distribution function of
newly born electrons with mean kinetic energy $mv^2_0 /2$,$$
\int \psi ({\bf v})d^3 v=1,\ \ \int {\bf v}\psi ({\bf v})d^3 v
=0,\ \ \int v^2\psi ({\bf v})d^3 v = v_0^2.$$ The third term with the
prefactor $m/M$, represents the diffusion in the speed of the
electrons due to collisions with ions and neutrals. The effectiveness
of the e-i collisions is proportional to $n$ and is strongly peaked at
small speeds. The constant $b$ is given$^{4,11}$ by $b=4\pi e^4 L/m^2$
($L\sim 10$ is the Coulomb logarithm), while $l=1/N\sigma$ is the mean
free path of electrons in e-n collisions. The next two terms in the
right hand side of (3) represent respectively the effects of e-i and
e-n collisions on the angular parts of electron velocities with$$
\hat L = \Sigma_{\mu,\nu}\ {\partial \over \partial c_{\mu}}
(c^2 \delta _{\mu,\nu} - c_{\mu} c_{\nu} ){\partial \over
\partial c_{\nu}},\qquad (\mu ,\nu =1,2,3) $$
where ${\bf c = v-W}$ and ${\bf W}$ is the ion drift velocity.
We neglect the difference between $|{\bf c}|$ and $|{\bf v}|$,
in particular in the energy exchange term in (2), since
$|{\bf W}|<r'$ and $r'$ marks the boundary of the
plasma sheath. On the other hand the thickness of the sheath is$^6$
of the order of a few Debye lengths $\lambda _D = \sqrt{kT_i/4\pi ne^2
}\approx 10^{-4}cm$ when $T_i=300 K, \ n\approx 10^{12} cm^{-3}$.
Assuming that $R>>\lambda _D$ we can neglect the difference between
$r'$ and $R$ in Eq.(16) and instead use (16) as the boundary condition
for (12-14), as is proposed by Persson$^7$ and other authors (see
Refs.6,8).
Our task now is to solve (12)--(14) subject to the conditions $$
{dn\over dr}(0)=0,\ \ {du\over dr}(0)=0,\ \ w_r(0)=0\eqno(17)$$ at
$r=0$ and to $w_r(R)$ given by (16) at $r=R$. The functions $n(r),\
u(r)$ are finite on the tube axis, $r=0$, Eqs.(12), (14), (17)
therefore imply $$ {dw_r\over dr}(0)={1\over
2n(0)}[a-cn^2(0)/u^3(0)]\eqno(18)$$ and $A(0)=0$. The last
relationship is very important and we rewrite it as$$
\sqrt{\pi \over 6}\left ({eE\over m}\right )^2 =
{gn(0)+4u^4(0)/9l\over u^3 (0)\ n(0)}\Biggl\{
2\sqrt{6\over \pi}{m\over M}
\left [1-{v^2_i \over u^2(0)}\right ]n(0){bn(0)+8u^4(0)/9l\over
u(0)}$$
$$- a\left [ v_0^2 -{5\over 3}u^2(0)\right ] -
{2\over 3}c{n^2(0)\over u(0)}\Biggr\}.\eqno(19)$$
\medskip
Solving the set (12-14) with conditions (16-19) we determine the
density and temperature profiles as well as the average temperature
$\bar T(E)$ and the total longitudinal current $I(E)$: $$
\bar T(E)= {2m\over 3kR^2} \int^R_0 u^2 (r)rdr,\qquad
I(E)= 2\pi e\int^R_0 rn(r)w_z (r)\ dr.\eqno(20)$$
\medskip
When we studied in Ref.4 the spatially homogeneous problem without
ionization and recombination, we had $c=0,\ a =0,\ n=const,\ u=
const$, and (19) served as the equation for the determination of $u$,
or the electron temperature, as a function of $E$. We have a similar
situation here when $l/R <<1$. In this case one can disregard the
recombination on the tube walls and we have again a spatially
homogeneous problem $n=n(0),\ u=u(0)$. The only difference is that we
have now, in virtue of (12),$$ n=\sqrt {a u^3 \over c}.$$ Defining $$
\phi ={9\over 8}\sqrt{\pi\over 6}{lv_i^{-3/2}\over
\sqrt{ac}}\left ({eE\over m}\right )^2 ,\ \omega = {9 \over 4}
blv^{-5/2} _i \sqrt {a\over c},\ x={u\over v_i} ,\ \theta =
{v_0^2\over v_i^2},\ \mu = {c\over b},\
\epsilon^2={m\over M} \eqno(21)$$
(19) can be rewritten in the dimensionless form $$
\phi = {\omega +x^{5/2}\over x^3}\left [ \sqrt{6\over \pi}{
\epsilon ^2\over \omega \mu }(x^2 -1)(\omega +2x^{5/2})+
(x^2 -\Theta)/2 \right ].\eqno(22)$$
\medskip
It is obvious from (22) that $\phi $ is not monotone in $x$ and
therefore $T(E)$ is not single-valued if the parameter $\omega$ is big
enough. In a Helium plasma where $M/m \approx 7000,$ we take $\
c=4\cdot 10^{13}cm^6 sec^{-4},\ \theta =10$ and find that $\omega $
should be larger than about 113 in order to have a transition
corresponding to an S-shaped $T(E)$ curve; see Fig.1, where $\omega
=250$. For smaller $c$ the critical value of $\omega$ is lower.
In terms of the ion temperature and the degree of ionization
$\omega$ can be represented as$$
\omega =\pi L{e^4\over \sigma (kT_i)^4}{n\over N}.$$
This is equal approximately $2\cdot 10^6 n/N$ for Argon with the
ambient temperature of the background gas. We see that the transition
requires $n/N \sim 10^{-4}$ or higher. Real values for the electric
field and current density $j$ can be found with the help of following
relations:$$ El=1.28\cdot 10^{-4}\left (T_i\over 300\right
)\sqrt{\omega \phi},
\ \ jl=2.6\cdot 10^{-3}\left (T_i\over 300\right )^{5/2}{x^3\sqrt
{\omega^3\phi}\over \omega +x^{5/2}}.$$ Here $l$ is in $cm$, $E$ and
$j$ are in $volts/cm$ and $mA/cm^2$ respectively.
\bigskip\noindent
\centerline{\bf The T = const Approximation}
\medskip
As a first step in solving the nonuniform case (12-14) let us assume
that the electron temperature is constant, $u(r)=u(0)$. It seems
reasonable to study such a simplified problem both as a guide for the
more general case which we shall consider later and as an
approximation which often yields reliable results for the positive
plasma column$^6$.
\medskip
The energy balance equation (14) cannot hold now for all $r\leq R,$
but at the point $r=0$ it reduces to (19), and our task is to solve
(12,13) with the conditions (16--18). {}For fixed E and $n=n(0)$ Eq.(19)
determines $u$ and therefore the electron temperature in the tube. We
then integrate (12,13), find the profiles $n(r),\ w_r(r)$, compute
$w_z(r)$ with the help of (11), and using (20) find the total current
$I(E)$, which gives the current-voltage characteristic of the
plasma. We can neglect the difference between $b$ and $g$, because in
noble gases $g/b-1\sim \mu \sim 10^{-4}$.
\medskip
To solve Eq.(12,13) numerically we pick a value of $u$ and find a
suitable $n(0)$ which allows to satisfy (16). Eq. (19) is used for
the calculation of $E$ for each choice of $u$ and $n(0)$. In this way
we obtain the functions $n(r),\ w_z(r)$ and substitute them into
(20). {}Fig.2 shows the electron temperature and the electric current
thus computed versus $E^2$ in relative units when $l/R =0.2.$ Curves
$T(E)$ are single-valued for $\omega\leq 66$ with the same $\Theta ,c$
as before in a Helium plasma.
\medskip
\bigskip\noindent
\centerline {\bf T$\not \equiv $ const}
\medskip
We solved numerically Eqs. (12)--(14) with boundary conditions
(16--19). Technically we choose for each $u(0)$ a trial value of
$n(0)$, compute E with the help of Eq.(19), and solve the differential
equations (12--14) for this triple. The resulting $w_r(r)$ will
generally not satisfy the boundary condition (16) at $r=R$ and we then
iterate with a different $n(0)$. This search can be easily optimized,
the procedure usually converges very fast and yields the profiles of
$n(r)$ and $u(r)$ for $r\leq R$. {}From these we can obtain $F(r),\
w_z(r),\ w_r(r)$ for a given external field E. In addition we also
find the total current and the mean electron temperature over the tube
cross section (20). The parts of the $T(E)$ and $I(E)$ curves with
negative derivatives are unstable and physically
non-accessible$^{4,5}$, which also shows up in the computation being
extremely unstable there when we try to find the solutions
numerically. Thus for $E=E'$ in {}Fig.3 and any trial temperature on the
tube axis inside the gap region (0.046-0.065 V), $T(r)$ immediately
jumps to a point corresponding to $T'$ after a few steps of
integration. This difficulty does not occur when we keep $T=const$.
\medskip
A qualitative description of the results is as follows:
\medskip\noindent
a) $n(r)$ always decreases monotonically. {}For some range of parameters
a kinetic transition takes place, $n(0)$ is then not a single-valued
function of the external field or the mean electron temperature. The
decrease of the volume recombination rate $c$ enhances the transition.
\medskip\noindent
b) The I-V characteristic for the total current I(E) and the mean
electron temperature show transition like behavior for $\omega >
60$. This is close to that obtained for the model $T=const$.
\medskip\noindent
c) $T(r)$ is a smooth monotonically decreasing curve when $E$ is
large; for very small $E$ the electron temperature passes a maximum at
some $r7.6$ (see Ref.4).
The solution of (A6,A7) for $x$ can be found from the relation$$
E^2 ={12\over \pi}{x^2-1\over x^6}(q+4x^4/9 )
(q+8x^4/9 ).\eqno(A8)$$
Eq.(A8) is close to Eq.(33), which was obtained in a rough
approximation without referring to the kinetic equation.
\medskip
It is natural to expand $f({\bf v},t)$ for the spatially homogeneous
case in a series in $\epsilon$
$$f({\bf v},t)=\sum _{j=0} f_j ({\bf v},t)\epsilon ^j .\eqno(A9)$$
Substituting (A9) into (A2) gives a
set of coupled equations for $f_j$ where higher
components can be expressed through lower ones.
We do not use this method in the present work, instead we have solved
Eq.(A1) using the method of moments with $\epsilon =1.4\times 10^{-4},
\ \delta =0.2, \ \lambda \approx 0.1,$ and $\Theta =10.$ In order to
have a closed set of differential equations we have taken$$
f_1=\left ({3{\bf v\cdot w}\over x^2}
\right ){\bar f}$$
which comes from the shifted Maxwellian. We expect the effect of our
additional approximation to be small.
\vfill
\eject
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\bigskip
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\bigskip
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\bigskip
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\bigskip
A.V.Rokhlenko and J.L.Lebowitz, Phys.Fluids B {\bf 5},
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\bigskip
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\vfill
\eject
\centerline {\bf FIGURE CAPTIONS.}
\bigskip
\centerline {Fig.1}
\medskip
The dependence of $x$ (representing the electrons thermal speed) and
the current density on $\phi$ (the electric field squared) for the
spatially homogeneous case $(R=\infty )$ when $\omega =250.$ The
dimensionless units are defined in Eq.(21) in the text.
\bigskip
\centerline {Fig.2}
\medskip
The electron temperature $T$ and total current $I\cdot R$ (in
$mA\cdot cm$) vs the electric field $E\cdot R$ (in volts) when
$T$ is assumed constant in the tube cross section,
$T_i=300^0$ and $\omega =90.$
\bigskip
\centerline {Fig.3}
\medskip
Plots of the mean electron temperature and total current when
$T\not \equiv const$ and $\omega =90.$ (The same units as in Fig.2).
The hysteresis loop is indicated by arrows. At values of the field
between the end points of the loop, like $E'$, the computation
leads to values on the lower or upper branches of the loop,
determined by how close the starting point is to one of them.
\bigskip
\centerline {Fig.4}
\medskip
The radial profiles of the mean electron speed and ambipolar
electric field $F(r)\cdot R$ (in volts) when $\omega =90$ for two
regimes: the solid lines correspond to a small field
$(E\cdot R\approx 0.04 V,\ {\bar T}/T_i \approx 2.5),$ the dotted
lines to a larger field $(E\cdot R\approx 0.07 V,\ {\bar T}/T_i
\approx 8).$
\end