Transient cataphoresis in a discharge tube with end volumes

262

Physica 114C (1982) 262-268 North-Holland Publishing Company

TRANSIENT CATAPHORESIS IN A DISCHARGE TUBE WITH END VOLUMES C. S A N C T O R U M , J. O N G E N A * a n d W. W I E M E Laboratorium voor Natuurkunde, Rijksuniversiteit Gent, Rozier 44, B-P000 Gent, Belgium Received 2 February 1982

A theoretical and experimental study has been made of time-dependent cataphoresis in a binary gas mixture when a dc discharge is initiated in a tube with end volumes behind the electrodes. The theoretical models as proposed by Shair et al. and Metze and Chanin are compared. Transient response curves of the impurity density, computed in accordance with the two models, are presented and /discussed. The gas separation in the positive column of a dc discharge is also studied experimentally for a mixture of 0.025% argon in neon. The argon density is monitored at two locations within the positive column of the discharge using a mass spectrometer. The experimental data are compared with calculations based on the model formulated by Metze and Chanin. A good agreement between theory and experiment is obtained for the end-volume configuration under consideration.

1. Introduction W h e n a dc discharge passes t h r o u g h a gas m i x t u r e consisting of a m a i n gas a n d an admixture gas that is easily ionized with respect to the m a i n gas, a s e p a r a t i o n of the gas c o m p o n e n t s usually occurs in the axial direction of the discharge tube. T h e ions of the m i n o r i t y gas are d r i v e n t o w a r d s the c a t h o d e u n d e r the influence of the axial electric field, w h e r e they are n e u tralized. A s the m i n o r i t y gas a c c u m u l a t e s in the c a t h o d e region it will diffuse b a c k w a r d towards the a n o d e . S o m e time after the ignition of the discharge, the ion m i g r a t i o n is b a l a n c e d by the back-diffusion a n d a steady-state segregation is established. This segregation phenomenon, caused by the selective t r a n s p o r t of the m i n o r i t y gas towards the c a t h o d e , has b e e n t e r m e d c a t a p h o r e s i s [ l]. In this paper, a theoretical a n d e x p e r i m e n t a l i n v e s t i g a t i o n of the v a r i a t i o n in the c o n c e n t r a t i o n of the readily ionized c o n s t i t u e n t following the i n i t i a t i o n of the discharge is p r e s e n t e d . T h e t i m e - d e p e n d e n c y of c a t a p h o r e s i s in a b i n a r y gas m i x t u r e is c o n s i d e r e d at different places in a * Aspirant N.F.W.O. 0378-4363/82/0000-0000/$02.75 © 1982 N o r t h - H o l l a n d

discharge t u b e with e n d v o l u m e s b e h i n d the electrodes.

2. Theory When mixture

in a h o m o g e n e o u s l y d i s t r i b u t e d gas a discharge is switched on, the

c a t a p h o r e t i c segregation increases asymptotically to a final d i s t r i b u t i o n d e p e n d i n g o n the composition of the mixture, the discharge t u b e c o n f i g u r a t i o n a n d the o p e r a t i n g c o n d i t i o n s of the discharge. F o r a small partial p r e s s u r e of the m i n o r i t y gas, the axial m i n o r i t y d e n s i t y dist r i b u t i o n at steady-state segregation decreases e x p o n e n t i a l l y [2] from c a t h o d e (z = 0) to a n o d e : n ( z ) = n(O) e x p ( - a z ) ,

(1)

w h e r e a = tx+Eqb/D is the slope of the e x p o n e n tial d i s t r i b u t i o n of the m i n o r i t y density. /x + r e p r e s e n t s the m o b i l i t y of the m i n o r i t y ions in the m a i n gas, E is the m a g n i t u d e of the axial electric field, 4, the i o n i z a t i o n degree of the m i n o r i t y gas a n d D the diffusion coefficient of the m i n o r i t y gas in the m a i n gas. T h e m i n o r i t y d e n s i t y in the e n d b u l b s is, at steady-state, equal

C. $anctorurn et al. / Transient cataphoresis

to the minority density at the respective electrodes. When diffusion is coupled with the action of an external force, the rate of change of the admixture density with time (at a position z) is given [3] by the equation

o2

OOtn (z, z) = D ~

n(z, t) + Iz +E~

n (z, t) .

(2)

The time-dependence of the axial segregation can be obtained by solving eq. (2). Shair et al. [4, 5] w e r e the first to propose a theoretical model for time-dependent gas separation in a discharge tube with end volumes. An outline of the discharge tube configuration considered is shown in fig. la. Note that 8 and e are, respectively, the ratios of the volumes of the cathode and anode end bulbs to the volume of the discharge region. They transformed eq. (2) into a non-dimensional equation:

3 3 2 3 o-g O(n' ~) = --on~ O(n, ~) + o~ -~n O(n' ~') ,

(3)

where 0 = n(z, t)/n* and a = Iz+EchL/D = aL, when n ( z , t ) / n * , > n + ( z , t ) / n *. n + and n* are, respectively, the density of the ions of the minority gas and the minority density uniformly cathode -~ end o volume ~

discharge region

Ci

'

A V

;

L

I

•

Z

I

I

•

~[]

o

1

C I

0

I

A I

dispersed throughout the tube prior to the discharge. L here represents the length of the discharge region of the tube. ~- = t D / L 2 and r / = z / L are, respectively, the dimensionless time and distance coordinates. Eq. (3) was solved for the discharge region using boundary conditions believed to be appropriate when accounting for the influence of the bulbs behind the electrodes. The solution proposed by Shair et al., obtained using Laplace transform calculus, was completed by Sanctorum et al. [6]. In the theoretical model mentioned above, when developing the boundary conditions, the assumption was made that the end bulbs were "well-mixed". The admixture density throughout each end bulb was uniform and equal to the value of the admixture density at the nearest end of the discharge region at all times. In a discharge tube with end volumes this assumption does not hold, as both in the cathode and anode end volumes density gradients will a p p e a r as soon as the discharge is initiated and the cataphoretic segregation starts. Recently, Metze and Chanin [7] examined the problem of time-dependent cataphoresis for a cylindrical discharge tube with uniform crosssectional area over its entire length, including the end volumes (see fig. lb). The diffusion equation (3) was rewritten as follows:

a 0(77, r) a~

anode ~ end c° volume

I

Zc

ZA

L

"qc

"qA

1

•

Z

Fig. 1. Diagrams of the discharge tube configurations considered respectively by Shair et al. (a) and Metze and Chanin (b) to formulate a model for transient cataphoresis when end volumes are present.

263

=

~02 o(n, ",9 +

1,~(n)o(,7,-,-)],

(4)

and it now describes the temporal behaviour of the admixture density in the whole tube. The quantity ~ is no longer a constant but a discontinuous function of r/; i.e., equal to a constant in the discharge region and zero in the end volumes. L now represents the overall length of the discharge tube including the end volumes, instead of the length of the discharge region. Metze and Chanin replaced the physically incorrect boundary conditions at the anode and the cathode, as postulated by Shair et al, by conditions stating that there are no gas losses at the

C. Sanctorum et al. / Transient cataphoresis

264

ends of the tube:

On 007, ~')J~=0 = 0

O(n, 'T)[~=I = 0 ,

and

for all r .

(5)

They numerically solved diffusion equation (4), satisfying the boundary conditions (5) and the initial condition: 0(r/, ~- 0) =

1,

=

for all r/ ,

(6)

and obtained computed time-dependent admixture density profiles for the discharge tube geometry under consideration. As we found it interesting to compare the solutions obtained using the two models mentioned above, we independently solved eq. (4) and conditions (5) and (6) using an implicit finite-

difference method [8]. In this way it became possible to study the minority density distribution along the tube at any time, and the change of the minority density with time at any point in the tube. This can be performed for any cylindrical end-volume configuration and for any discharge in any mixture. Figs. 2-4 show an example of computed transient cataphoresis. The full lines are the results of numerical calculations (model: Metze and Chanin), while the dashed curves are obtained using the complete analytical solution as presented in ref. [6] (model: Shair et al.). The values of the quantities describing tube configuration and discharge conditions, used in the computations, are given in table I for both models. Corresponding transient response curves of the impurity density at the electrodes and in the end volumes, computed in accordance with the two

4

J

I

MODEL:

I

Metze and Chanin

!

"c

5

v

C

I

CATHODE

FROM CATHODE I

CATHODE END OF THE TUBE

CATHODE AND IN CATHODE END

m

~

"'-Scm

¢,/

.

.

.

.

.

.

.

20cm FROM CATHODE

.

.

.

FROM CATHODE

t

T ANODE AND IN ANODE Met ze and

MODEL:

....

60

Chonin

~

Shair et at.

120

T I M E (s)

END VOLUME

'~...,.~

180

0

60

120

TIME Is)

180

Figs. 2 a n d 3. C a l c u l a t e d m i n o r i t y d e n s i t y as a f u n c t i o n of t i m e at different l o c a t i o n s in a d i s c h a r g e t u b e w i t h e q u a l size e n d b u l b s (e = 8 = 0.333). T h e full lines r e f e r t o n u m e r i c a l c a l c u l a t i o n s ( a = 16.66), w h i l e t h e d a s h e d c u r v e s are o b t a i n e d using t h e c o m p l e t e a n a l y t i c a l s o l u t i o n of ref. [6], ( a = 10).

C. Sanctorum et al. / Transient cataphoresis

265

3. E x p e r i m e n t

I I

/T,.E

A,,/

j

JX7

;,, 7",,

_L/ /-%-,-,---/

,

-Os

#/" ~ I~ '

I

J I

I I

C

I

l I ....

I I

I I

A

MOOEL: Metze and Ohanin Sho,= ~ o,

DISTANCE ALONG THE TUBE

Fig. 4. Calculated axial minority density profile at different times after the ignition of a discharge in a tube with equal size end bulbs (e = 8 = 0.333). The full lines refer to numerical calculations (a = 16.66), while the dashed curves are obtained using the complete analytical solution of ref. [6]

(a = ]o).

models, show, as expected, little agreement. In the discharge region of the tube however, at a sufficient distance from the cathode and some time after the ignition of the discharge, the deviations are less pronounced and the general behaviour of the corresponding curves is similar.

Table I Values of the parameters used for the computation of the curves of figs. 2 to 4 del e = 8 L (cm) a (cm -1) a D(cm2/s)

Shair et al.

Metze and Chanin

0.333 100 0.1 10 20

0.333 166.6 0.1 16.66 20

Transient cataphoresis was studied experimentally in a neon-argon mixture containing 0.025% argon. The pyrex cataphoresis tube has an overall length of 166 cm and a uniform inner diameter of 2.8cm. The electrode distance is 98 cm and the length of both end volumes behind the electrodes equals 34cm. The cathode and anode electrode assemblies are identical and consist of open-ended gold cylinders with an outside diameter of 1.7cm and length 1.5 cm. The electrodes are shielded to avoid difficulties due to cathode sputtering effects [9]. The discharge tube is mounted on a standard U H V pumping and gas-handling system. After an outgassing procedure at 350°C for a period of 24 h the background pressure in the system is less than 5 × 10-1° Torr. The premixed research-grade mixture is supplied in a metal vessel by the L'Air Liquide company. The gas pressure is measured with a capacitance manometer and the discharge is maintained using a current-regulated dc power supply. The experimental data are obtained by mass analysis [1] of gas continuously sampled through a molecular leak mounted in the wall of the discharge tube at a suitable distance from the electrodes. As the discharge tube is symmetrical, a reversal of the polarity of the electrodes results in a second sampling position in the discharge region. In order to study the time-dependence of the density of the impurity gas, the ratio of the mass spectrometrical peak heights of 4°Ar and 22Ne is monitored [10] as a function of time. A block diagram of the fully automatic measuring system is shown in fig. 5. The procedure of data acquisition is controlled by a desktop computer. The computer is interfaced with the measuring apparatus using a selfbuilt "synchronizer". This instrument provides all timing pulses and makes data transfer, via an IEEE-bus, to the computer possible. A sampling instruction, generated by the computer program,

266

C. Sanctorum et al. / Transient cataphoresis I

DC I

I

I

POWER SUPPLY

1

~.. 1.5

DISCHARGE I Ium MASS TUBE ~ECTROMETERDETEC

I

I

z=52cm

c~

1

SPECTROMETERI ELECTRONICS I

0.025% Ar in Ne p =4.40 Torr T=320 K

LECTROMETER

1.G EXPERIMENT -

[°++ I

-- -- THEORY

6=0.t,2

E:=0.37

COMPUTER

0(.=7.0 0.5

L=166cm

Fig. 5. Block diagram of the measuring system. D = 63 cmZls

is fed to the mass scan p r o g r a m m e r via the synchronizer. The p r o g r a m m e r drives the quadrupole mass spectrometer in scanning the masses +OAr and Z~Ne in sequence. The output current of the ion detector is amplified by the electrometer and the analog signal available at the output is digitized. The information is transmitted to the computer via the synchronizer for further processing using a suitable program.

4. Experimental results and comparison with theory Figs. 6 and 7 show typical examples of the transient response of the density of the argon impurity after a discharge is initiated in a mixture of 0.025% argon in neon. D a t a concerning the enhancement of the argon impurity in the cathode region of the tube, and the depletion of the argon in the anode region are given for the pressures p = 4.40 and 9.0Torr. These data are obtained using the experimental technique and apparatus described in section 3 of this paper. A 100 m A discharge current was chosen in both cases to m a k e sure that the axial density of the argon decays

z=l14cm

I 60

I 120

I 180

TIMEIsI

2z.O

Fig. 6. Comparison between the experimental and theoretical time-dependent variation of the argon density in the cathode and anode regions of the discharge; p = 4 . 4 0 T o r r , I = 100 mA.

exponentially over the length of the positive column [11]. The average gas t e m p e r a t u r e was determined from the measured pressure increase in the cataphoresis tube after the ignition of the discharge. The t e m p e r a t u r e amounts to 320 and 330 K, respectively. Gas samples are taken at axial distances z = 5 2 and l l 4 c m from the cathode end of the tube; that is at 18cm from the cathode and from the anode, respectively. C o m p u t e d transient response curves of the normalized argon density, corresponding with the experimental profiles, are also shown in figs. 6 and 7. These are obtained using numerical calculations in accordance with the theoretical model formulated by Metze and Chanin (see section 2). T o m a k e a comparison between the experi-

C. Sanctorum et al. / Transient cataphoresis I

!

I

N-1,5 c<

f-

z=52cm 0.025%Ar in Ne p =g.o Torr T=330K

1.0 ....

EXPERIMENT THEORY

~=0.42 X~ 0.5

E=0.37

O.=9,13 ~

L=166crn ~

D=33cm21s

z=11&crn I

60

I

120

I

180

T(ME(s)240

Fig. 7. Comparison between the experimental and theoretical time-dependent variation of the argon density in the cathode and anode regions of the discharge; p = 9 . 0 T o r r , I = 100 mA.

mental data and theoretical predictions, the values of the quantities ~5, e, t~ and D must be known. A value of 0.35 for both ~$ and e can be calculated from the dimensional data of the discharge tube in section 3. The a-values are obtained from the slopes of the experimental steady-state distributions. The D-values are taken from the literature [12]. Calculations using the above p a r a m e t e r s yield poor agreement between theory and experiment for the sampling site in the cathode region. An examination of the problem led to the conclusion that the observed deviations were due to a mischoice of the end-volume configuration used for the calculations. In formulating the theoretical model, the assumption was made that the positive column extended from anode to cathode.

267

Visual inspection of the discharge, however, shows that the positive column extends from very close to the anode to about 5 to 6 cm from the cathode. The space between the cathode and positive columns is occupied by the negative zones. A p a r t from the cathode dark space and the transition area from the Faraday dark space to the positive column, the electric field in the cathode zone is very small, if not zero. This means that the quantity a can be set equal to zero in that part of the discharge, and as a consequence the cathode end volume extends beyond the cathode. If the distance from the cathode to the head of the positive column is 5 cm, the length of the positive column reduces to 9 3 c m and the length of the cathode end volume becomes 39cm. This yields an endvolume configuration where 8 = 0.42 and e = 0.37, and the cathode sampling position is now situated at 1 3 c m from the end of the positive column. Figs. 6 and 7 show that for the above-mentioned end-volume configuration, the agreement between experiment and theory is good. Experimental and computed curves show similar temporal behaviour. In the anode region, the c o m p u t e d density decays somewhat faster to an equilibrium than is experimentally observed. In the cathode region, the calculated density increases exponentially to a steady-state condition, while the experimentally observed density rises to a peak slightly higher than the calculations predict, and then slowly decays to the steadystate value. A possible explanation for the observed deviations, is that the electrode assemblies somewhat restrict the flow of the impurity particles in the tube. When in a discharge the impurity gas accumulates at the cathode, the particles will diffuse into the cathode end volume and also will back-diffuse towards the anode. An obstruction of the diffusion into the end volume caused by the cathode, results in a temporal rise of the impurity density above the value that would be attained when no obstacle is present. As the

268

C. Sanctorum et al. / Transient cataphoresis

impurity gradient near the cathode grows above the expected value, the drift of the impurity gas from the anode region is slowed down. The model used for the calculations does not include the possibility to take into account the restricting influence of the electrodes on the flow of the impurity particles. In this m a n n e r it is not possible to eliminate the remaining discrepancies between theory and experiment. Although the above-presented comparison between experimental data and theory is very satisfactory, an attempt will be made in the near future to build a discharge tube with electrode assemblies that do not restrict the gas flow. Care will also be taken to transform the experimental set-up and conditions so that data can be taken under conditions where the positive column ends very close to the cathode. Results will be reported in a future publication.

Acknowledgements T h e quadrupole mass spectrometer and the desktop c o m p u t e r were m a d e available through

grants from the National Fonds voor Wetenschappelijk O n d e r z o e k (N.F.W.O.), Brussels. J. O n g e n a wishes to thank the N.F.W.O. more particularly for the award of a research mandate.

References [1] L.M. Chanin, Gaseous Electronics, M.N. Hirsch and H.J. Oskam, Eds., Vol. 1, Part 2.4 (Academic Press, New York, 1978) p. 133. [2] J. Freudenthal, Physica 36 (1967) 354. [3] H.J. Oskam, Physica 40 (1969) 594. [4] F.H. Shair and D.S. Cohen, Chem. Eng. Sci. 24 (1969) 39. [5] F.H. Shair and D.S. Remer, J. Appl. Phys. 39 (1968) 5762. [6] C. Sanctorum, L. Bonte and L. Jacques, Physica 104C (1981) 457. [7] A. Metze and L.M. Chanin, J. Appl. Phys. 51 (1980) 2483. [8] W.F. Ames, Numerical Methods for Partial Differential Equations (Academic Press, New York, 1977) p. 49. [9] J.P. Gaur and L.M. Chanin, Rev. Sci. Instr. 39 (1968) 1948. [10] J. Freudenthal, Physica 36 (1967) 365. [11] C. Sanctorum, Physica 83C (1976) 367. [12] W. Hogervorst, Physica 51 (1971) 59.