CONSTRICTION AND STRIATIONS IN ELECTRIC DISCHARGES

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CONSTRICTION AND STRIATIONS IN ELECTRIC

DISCHARGES

Y. Crispin, E. Wasserstrom

To cite this version:

JOURNAL DE PHYSIQUE CoZZoque C9, suppZ6ment au n o l l , Tome 41, novembre 1980, page

~9-225

-CONSTRICTION AND STRIATIONS IN ELECTRIC DISCHARGES

Y. Crispin and

E.

Wasserstrom.

Department of AeronauticaZ Engineering, Technion

-

IsraPZ I n s t i t u t e of Technology, Haifa, Israel.

Abstract.- The power output of electric lasers is limited due to gas heating and instabilities of the glow discharge. A theory is presented which describes the instability phenomena occurring in the po- sitive column of glow discharges. The theory predicts the neutral stability boundaries of the tran- sition from a stable diffuse glow to a constricted discharge as well as the transition to a striated column (waves of ionization). A comparison between theory and experimental results for neon is pre- sented.

I. I n t r o d u c t i o n

Gases are poor conductor However, when they are

s o f e l e c t r i c i t y . subjected t o a s u f f i c i e n t l y h i g h v o l t a g e they become e l e c t r i c a l l y conducting. For s u i t a b l e c o n d i t i o n s o f pressure and e l e c t r i c c u r r e n t the s e l f sustained discharge i s d i f f u s e and homogeneous

[I].

On t h e other hand, when the c o n d i t i o n s are n o t f a v o r a b l e several modes o f i n s t a b i 1 i t y can appear [ Z ) . This phenomenon i s o f fundamental and p r a c t i c a l importance and has a t t r a c t e d renewed i n t e r e s t , e s p e c i a l l y since glow discharges were found t o be an e f f i c i e n t mechanism f o r l a s e r e x c i t a t i o n

[3-5).

The purpose o f t h i s work i s t o study a n a l y t i c a l l y the two b a s i c k i n d s o f i n s t a b i l i t y o c c u r r i n g i n glow discharges, namely c o n s t r i c t i o n and s t r i a t i o n s

163.

If t h e pressure o r c u r r e n t i n t e n s i t y i s increased the discharge collapses and c o n s t r i c t s i n t o concentrated f i l a m e n t s [7].

On the o t h e r hand, e s p e c i a l l y f o r very low pressures, s t r i a t i o n s o f v a r y i n g l u m i n o s i t y move. along t h e p o s i t i v e column [a-101. There are several mechanisms which can l e a d t o such i n s t a b i l i t i e s such as plasma k i n e t i c processes

[ll], e l e c t r o d e e f f e c t s , gas contamination

[12],

e x t e r n a l c i r c u i t r e s i s t a n c e and thermal e f f e c t s

[13, 141.

I n t h i s work we s h a l l concentrate on the l a t t e r mechanism which can be q u a l i t a t i v e l y described as f o l l o w s :

6 ~ 6f ~ 6vif 1 & n e t 6 ~ ~ ( J E ~ ) ! t b ~ t

Due t o uneven Joule h e a t i n g a l o c a l increase of gas temperature (sT

f

) appears which causes a decrease o f gas d e n s i t y ( ~ N I ) , since the pressure remains constant. Consequently, t h e i o n i z a t i o n

r a t e , vi, being a f u n c t i o n o f the r a t i o o f t h e a x i a l e l e c t r i c f i e l d t o gas d e n s i t y Ex/N, increases. T h i s leads t o an increase o f e l e c t r o n d e n s i t y ( & n e t ), e l e c t r i c c u r r e n t d e n s i t y ( s J

1 )

and Joule h e a t i n g ( 6 ( J E x )

f

), which f i n a l l y ,increases t h e i n i t i a l disturbance o f t h e gas temperature and thus leads t o i n s t a b i l i t y . We adopt the l i n e a r s t a b i l i t y approach which has been e x t e n s i v e l y used i n the study o f hydrodynamic s t a b i 1 i t y

[15].

The p r o p e r t i e s o f steady s t a t e glow discharges have been the subject o f many experimental and t h e o r e t i c a l i n v e s t i g a t i o n s

[16].

On the other hand, the i n s t a b i l i t y phenomenon has been s t u d i e d much l e s s thoroughly. Haas t r e a t e d t h e development o f i n s t a b i l i t i e s i n an i n f i n i t e s p a t i a l l y homogeneous m i x t u r e of r a r e and molecular diatomic gases

[17].

This work was f o l l o w e d by t h a t of Nighan and Wiegand (181 who s t u d i e d t h e e f f e c t s o f plasma k i n e t i c processes, e s p e c i a l l y recombination, on discharge s t a b i l i t y . The thermal and acoustic i n s t a b i l i t i e s developed i n h i g h power e l e c t r i c l a s e r s were s t u d i e d by Jacob and Mani

[19].

Two a c o u s t i c modes and one thermal mode were i d e n t i f i e d . The growth r a t e of t h e disturbances was found t o increase w i t h i n c r e a s i n g power i n p u t . The pressure and c u r r e n t c o n d i t i o n s f o r the onset o f i n s t a b i l i t y were n o t studied. Ecker e t a l .

[13,

141 used l i n e a r s t a b i l i t y t h e o r y t o study the onset o f thermal c o n s t r i c t i o n i n an a x i a l l y homogeneous discharge, t a k i n g i n t o account t h e i n f l u e n c e o f t h e r e s i s t a n c e o f t h e e x t e r n a l c i r c u i t . Their c a l c u l a t i o n s f o r helium a r e a t variance w i t h t h e

JOURNAL DE PHYSIQUE

experiments o f Pfav and Rutscher 7 A d i f f e r e n t approach has been t a k e n by Rogoff [20] who s t u d i e d t h e t i m e development o f gas d e n s i t y and temperature f o r an a x i a l l y u n i f o r m p o s i t i v e column o f a p u l s e d d i s c h a r g e i n hydrogen. I t was found i h a t t h e c u r r e n t d e n s i t y p r o f i l e c o n s t r i c t s smoothly w i t h t i m e d u r i n g t h e growth o f i o n i z a t i o n . A s i m i l a r approach was t a k e n by Oster, Jeager and Phelps [21] i n t h e i r s t u d y o f t h e growth o f thermal c o n s t r i c t i o n s i n h e l i u m discharges. The gas temperature, d e n s i t y and r a d i a l v e l o c i t y as w e l l as charge c o n c e n t r a t i o n were c a l c u l a t e d as a f u n c t i o n o f t i m e f o r a wide range o f pressures. The c o n s t r i c t i o n i s a consequence o f i n c r e a s e d i o n i z a t i o n i n r e g i o n s o f decreased gas d e n s i t y . A d d i t i o n a l c a l c u l a t i o n s o f t h i s t y p e have been c a r r i e d o u t r e c e n t l y b y Oster [22]. There e x i s t s a l a r g e volume o f experiments on t h e c o n s t r i c t i o n o f e l e c t r i c discharges, e.g. [23-271. However, no general t h e o r y e x i s t s which can p r e d i c t t h e c o n d i t i o n s f o r onset o f such i n s t a b i l i t i e s .

The second t y p e o f i n s t a b i l i t y , t h e s t r i a t e d column, has been e x t e n s i v e l y s t u d i e d e x p e r i m e n t a l l y [8-10). Measurements of t h e e l e c t r i c p o t e n t i a l , e l e c t r o n temperature and charge d e n s i t y i n argon d i s c h a r g e s along t h e s t r i a t i o n s as f u n c t i o n o f t i m e were c a r r i e d o u t by Stewart [28] and by G e n t l e

1291.

The response o f t h e p o s i t i v e column t o a r t i f i c i a l l y e x c i t e d d i s t u r b a n c e s was s t u d i e d by Pekarek [30], Pekarek and K r e j c i [31]. Using a m i c r o s c o p i c approach, Pekarek and K r e j c i [31] suggested a one-dimensional mathematical model i n o r d e r t o e x p l a i n t h e mechanism by which s t r i a t i o n s a r e developed f r o m an i n i t i a l d i s t u r b a n c e . Gas temperature v a r i a t i o n s were n o t taken i n t o account. The model o f G e n t l e [29] was a l s o based on a m i c r o s c o p i c approach. He f o u n d t h a t moving s t r i a t i o n s appear i n t h e column f o r charge d e n s i t i e s lower t h a n a c r i t i c a l v a l u e corresponding t o t h e Pupp l i m i t i8-101. Pfau and Rutscher [7] s t u d i e d e x p e r i m e n t a l l y t h e e x i s t e n c e r e g i o n s o f c o n s t r i c t i o n and s t r i a t i o n s . i n t h e plane o f pressure versus c u r r e n t i n t e n s i t y f o r t h e n o b l e gases neon, argon, k r y p t o n and xenon f o r a wide range o f pressures and c u r r e n t i n t e n s i t i e s . However, those s t a b i l i t y r e g i o n s have n o t been p r e d i c t e d t h e o r e t i c a l l y .

To sum up: s e v e r a l t h e o r i e s e x i s t f o r t h e

p r e d i c t i o n o f e i t h e r c o n s t r i c t i o n o r s t r i a t i o n i n s t a b i l i t i e s i n s e l f - s u s t a i n e d e l e c t r i c discharges. None o f t h e s e t h e o r i e s can f u l l y p r e d i c t t h e onset o f these i n s t a b i l i t i e s , i.e., t h e pressure and c u r r e n t f o r which t h e d i s c h a r g e f i r s t becomes unstable. I t i s t h e purpose o f t h i s work t o develop a u n i f i e d macroscopic t h e o r y which d e s c r i l j e s t h e two k i n d s o f i n s t a b i l i t y and can p r e d i c t t h e c o n d i t i o n s f o r t h e i r onset.

11. F o r m u l a t i o n o f t h e Problem

I n o r d e r t o analyze t h e s t a b i l i t y o f e l e c t r i c discharges we f i r s t have t o c o n s t r u c t a time- dependent model o f t h e discharge. T h i s model should be s u f f i c i e n t l y r e a l i s t i c , so t h a t i t c o n t a i n s t h e i n s t a b i l i t y phenomenon. On t h e o t h e r hand, t h e model should be s i m p l e enough t o be amenable t o a n a l y t i c a l treatment. I n o r d e r t o s a t i s f y these c o n d i t i o n s we s h a l l have t o make some s i m p l i f y i n g assumptions, n o t a l l o f which can be r i g o r o u s l y j u s t i f i e d . The v a l i d i t y o f t h e model i s checked b y comparing i t s p r e d i c t i o n s t o experiments.

F i g . 1: Geometry of t h e d i s c h a r g e between two i n s u l a t i n g plane p a r a l l e l w a l l s .

We s h a l l deal w i t h t h e p o s i t i v e column o f an e l e c t r i c d i s c h a r g e which i s m a i n t a i n e d i n a two- dimensional geometry, i.e., between two non-conducting p a r a l l e l i n f i n i t e w a l l s , see F i g .

describe t h e behaviour of t h e discharge are t h e gas-dynamic equations ( i .e. c o n t i n u i t y , momentum, energy and s t a t e ) and Maxwell's electro-rnagnetic equations. I t i s assumed t h a t the n e u t r a l s , e l e c t r o n s and ions behave as i d e a l gases. The i o n gas i s i n thermal e q u i l i b r i u m w i t h the n e u t r a l gas whereas t h e e l e c t r o n temperature i s much higher than the gas temperature. The equations r e l e v a n t t o our model are presented below:

( a ) D r i f t v e l o c i t i e s .

Using the momentum and s t a t e equations o f the e l e c t r o n and ioni gases [33-343, n e g l e c t i n g t h e i n e r t i a l terms and t h e f o r c e s due t o c o l l i s i o n a l momentum t r a n s f e r between e l e c t r o n s and ions as w e l l as the pressure g r a d i e n t s f o r the i o n gas, t h e f o l l o w i n g expressions are obtained f o r t h e d r i f t v e l o c i t i e s o f the charged p a r t i c l e s , see Ref. [6] f o r j u s t i f i c a t i o n :

Here ue, ve, ui, vi are t h e v e l o c i t y components i n the x and y d i r e c t i o n s o f t h e e l e c t r o n s and ions r e s p e c t i v e l y . n i s the charge concentration. Ex i s the a p p l i e d e l e c t r i c f i e l d i n the l o n g i t u d i n a l d i r e c t i o n . e i s t h e e l e c t r o n charge, ue and U . are t h e e l e c t r o n and i o n

1

m o b i l i t i e s r e s p e c t i v e l y , Te i s t h e e l e c t r o n temperature and k i s Boltzmann's constant.

( b ) C o n t i n u i t y equations (conservation o f charge). The c o n t i n u i t y equations f o r t h e ions and f o r t h e e l e c t r o n s are [33, p. 1231:

4

Where t i s time, = u.x+v.y and

1 1

ve

=

u x+vey are t h e v e l o c i t y v e c t o r s o f t h e i o n e

and e l e c t r o n gases r e s p e c t i v e l y . vi and a are t h e i o n i z a t i o n and recombination c o e f f i c i e n t s . x and y are u n i t vectors i n t h e x and y d i r e c t i o n s .

I n t r o d u c i n g the v e l o c i t y components from Eqs.

_-.

(2.1-2.4) one obtains:

Eq. (2.8) i s obtained by s u b t r a c t i n g (2.6) from (2.5) where pi has been neglected w i t h respect t o ue s i n c e u.<<u (see [I, pp. 121-1241).

I e

( c ) Energy equations.

The conservation o f energy o f the n e u t r a l and e l e c t r o n gases leads t o the f o l l o w i n g equations:

aT

a

aT

+L

aT 3

mNNcpX = -(K-) ( 2 ( T - T ) (2.9) ax ax ay ay mN eN 2

where me, rnN, N, T, K and cp are t h e e l e c t r o n and atom masses, gas p a r t i c l e density, gas temperature, heat c o n d u c t i v i t y and s p e c i f i c h e a t a t constant pressure r e s p e c t i v e l y . veN and Vi are t h e c o l l i s i o n frequency between e l e c t r o n s and n e u t r a l s and the i o n i z a t i o n p o t e n t i a l o f t h e gas r e s p e c t i v e l y . The l a s t term i n (2.9) i s due t o t r a n s f e r o f energy from t h e h o t e l e c t r o n s t o t h e c o l d gas. I n Eq. (2.10) t h e increase o f energy o f the f r e e e l e c t r o n s i s due t o t h e ohmic h e a t i n g J E ( t h e f i r s t two terms on t h e r i g h t hand s i d e ) , minus t h e energy which i s t r a n s f e r r e d by c o l l i s i o n s t o t h e n e u t r a l s and t h e energy needed t o i o n i z e t h e gas.

( d ) Maxwell's equation.

Most o f t h e i n f o r m a t i o n i n c l u d e d i n Maxwkll's equations i s n o t needed f o r our purposes. Since magnetic e f f e c t s are neglected, t h e e l e c t r i c f i e l d vector i s i r r o t a t i o n a l , OX? = 0, i.e., i n c a r t e s i a n coordinates

Summary o f t h e Equations

C9-325 JOURNAL DE PHYSIQUE

charge d e n s i t y a t the center i n steady s t a t e c o n d i t i o n s . ExS i s t h e steady s t a t e l o n g i t u d i n a l e l e c t r i c f i e l d . The s u b s c r i p t w

denotes c o n d i t i o n s on the w a l l . S i m i l a r l y Te, N, ue, pi, veN, vi and K are normalized by t h e i r values on t h e w a l l s (which depend on t h e w a l l temperature). Upon n o r m a l i z a t i o n t h e f o l l o w i n g system o f equations i s obtained:

where t h e comma denotes p a r t i a l d i f f e r e n t i a t i o n . Here 6, C, E, L, M, F, G and H are nondimensional parameters defined by

B = viwa/Ui, C = anca/Ui, E = kTew/eaExs, L =

(kw/pwCp)/aUi, M = 1.5HnckTew/pwCpTw, F = G/E

G = 2 ~ ~ w / 3 ~ i w , H = 2 ( m e / m ~ ) a v e ~ w / U i _(2.19) where U i = ujwExs.

Equations (2.13-2.18) c o n t a i n e i g h t nondimen- s i o n a l parameters (2.19). However, most o f these are i n t e r r e l a t e d . I t can be shown t h a t a l l these parameters can be determined by t h e pressure p and t h e c u r r e n t i n t e n s i t y I. Rather than attempting the f u l l s o l u t i o n s o f (2.13-2.18) we s h a l l o b t a i n admissible and r e l a t i v e l y simple steady s t a t e s o l u t i o n s and then i n v e s t i g a t e t h e i r s t a b i T i t y w i t h respect t o small p e r t u r b a t i o n s .

111. Steady State S o l u t i o n s

I t can be shown [6] t h a t t h e system of equations (2.13-2.18) admits steady s t a t e (i .e., t i m e independent s o l u t i o n s ) which are a l s o

independent o f the a x i a l coordinate, i.e. a/ax = 0. I n p a r t i c u l a r , we are i n t e r e s t e d i n t h e influence o f J o u l e h e a t i n g on the discharge

I

c h a r a c t e r i s t i c s since t h i s phenomenon i s important t o the i n s t a b i l i t y mechanism which we s h a l l study i n subsequent sections. For bre;ity, t h e steady s t a t e problem i s n o t t r e a t e d here. However, t h e f o r m u l a t i o n o f t h i s problem, i t s

s o l u t i o n as w e l l as the comparison o f t h e r e s u l t s t o experiments, can be found i n Ref. [6] and w i l l be published elsewhere.

I V . P e r t u r b a t i o n Equations

We f o l l o w the standard method [15] and assume t h a t t h e s o l u t i o n s o f t h e nondimensional system

(2.13-2.18) are o f the form

and s i m i l a r l y f o r N, Te, T, Ex, Ey where Ex, i s constant. For s i m p l i c i t y o f n o t a t i o n , here and i n what f o l l o w s the bar over a l l nondimensional q u a n t i t i e s i s deleted. The s u b s c r i p t s i n (4.1) denotes t h e steady s t a t e s o l u t i o n and t h e s u b s c r i p t 1 denotes t h e small ( i n f i n i t e s i m a l ) amp1 i tude o f the p e r t u r b a t i o n s . c i s t h e r e a l nond~mensional wave number, w =

w _r + i w . i s a complex nondimensional

1

frequency and i =

fl.

The system i s l i n e a r l y s t a b l e i f the p e r t u r b a t i o n s decay i n time, i.e. s t a b i l i t y imp1 i e s

Re(w) = wr

>

0 (4.2) I n order t o a e r i v e the p e r t u r b a t i o n equations, Eqs. (4.1) are s u b s t i t u t e d i n t o Eqs. (2.13-2.18) and the r e s u l t i n g system i s l i n e a r i z e d . For s i m p l i c i t y i t i s assumed here t h a t t h e c o e f f i c i e n t s pi, ue, veN and K are n o t perturbed. On the other hand t h e p e r t u r b a t i o n o f t h e i o n i z a t i o n r a t e vi i s taken i n t o account, since t h i s dependence i s considered important t o t h e s t a b i l i t y mechanism. A f t e r some manipulations t h e f o l l o w i n g l i n e a r homogeneous equations are obtained:

E n ' + ( i c

-

w

-

Bvis + E i s + 2Cns)n +

YS

2

+ ( i c

-

BQ)nsEx + BSnsNs T = 0 (4.3) 2

+ (2ET;s + Eys)nl + (ET;?;

-

c ETeS + Eis+ 2

+

i c ) n + icnSEx + (En;'

-

c En )T _{s e} = 0 (4.4) 2

LT" + (wNS

-

c L)T + MnsTe + MTesn = 0 (4.5) (Tesw + F

-

HTes

-

Vlvis + 2V2ns + icGTeS)n +

where V1 = 2BeVi/ekTe,, V2 = 2CeVi/3kTew

Q = (avi/aEx),

,

S = (avi/aNIs (4.7)

(2.18) has been used. For s i m p l i c i t y t h e s u b s c r i p t 1 f o r t h e p e r t u r b a t i o n s has been o m i t t e d . A f t e r some s t r a i g h t f o r w a r d m a n i p u l a t i o n s Eqs. (4.3-4.6) reduce t o :

where t h e c o e f f i c i e n t s A. t o A4 and Bo t o B3 can e a s i l y be o b t a i n e d from Eqs. (4.3-4.6). The p e r t u r b a t i o n v a r i a b l e s s a t i s f y homogeneous boundary c o n d i t i o n s :

n t ( 0 ) = 0, n ( 1 ) = 0, T 1 ( 0 ) = O,T(l) = 0 (4.10) The s t a b i l i t y problem can now be s t a t e d as f o l l o w s : g i v e n a steady s t a t e system, which i s d e a l t w i t h i n Ref. [ 6 ] and which i s d e f i n e d by n S ( y ) , Ns(y), TeS(y), T s ( y ) , Exs and E V c ( y ) . Choose an a r b i t r a r y wave number c.

J

-

Eqs. (4.8-4.10) d e f i n e an e i g e n v a l u e problem' f o r W. The s t a b i l i t y i s t h e n d e c i d e d by t h e c r i t e r i o n (4.2). I f t h e system i s s t a b l e f o r a l l wave numbers c, i t i s l i n e a r l y s t a b l e ; o t h e r w i s e i t i s u n s t a b l e . V. C o n s t r i c t i o n : A S i m p l i f i e d Model

Two k i n d s o f i n s t a b i l i t y appear i n glow discharges: c o n s t r i c t i o n and s t r i a t i o n s . The b a s i c assumption i n t h i s work i s t h a t t h e mechanism f o r t h e s e - i n s t a b i l i t i e s i s t h e r m a l and t h a t Eqs. (4.8-4.10) can d e s c r i b e b o t h k i n d s of i n s t a b i l i t y . B e f o r e a t t e m p t i n g a f u l l s o l u t i o n o f t h i s c o m p l i c a t e d problem, a s i m p l e r case w i l l be d e a l t w i t h where t h e p e r t u r b a t i o n (4.1) can v a r y o n l y across t h e column b u t i s c o n s t a n t a l o n g it. I n t h i s case t h e wave number c = 0 and t h e p e r t u r b a t i o n s (4.1) reduce t o n ( y , t ) = n S ( y ) + nl(y)exp(-wt) (5.1) P h y s i c a l l y t h i s can be expected t o s i m u l a t e t h e phenomenon o f t r a n s i t i o n f r o m a d i f f u s e glow t o a t r a n s v e r s e l y c o n s t r i c t e d discharge. W i t h c = 0 t h e e i g e n v a l u e problem (4.8-4.10) i s s i m p l i f i e d . The- c o n s t r i c t i o n problem can be s t a t e d as f o l l o w s : g i v e n a steady s t a t e o f a d i f f u s e p o s i t i v e column w i t h g i v e n p r e s s u r e p and c u r r e n t i n t e n s i t y I, t h e complex f r e q u e n c y 61 has t o be c a l c u l a t e d such t h a t (4.8-4 . l o ) has a n o n - t r i v i a l s o l u t i o n . I f wr > 0 t h e steady s t a t e i s s t a b l e and d i f f u s e . I f wr < 0 t h e steady s t a t e i s u n s t a b l e and t h e d i s c h a r g e c o n s t r i c t s . T h i s problem was s o l v e d by t h e G a l e r k i n method [35]

JOURNAL DE PHYSIQUE

assumed constant, so that instead of (4.1) we now

have:

where ns is a known constant which is taken as the

average value of nS(y) and

nl

is

a

constant per-

turbation amplitude.

Physically, the perturba-

tions (6.1) can model the appearance of stria-

tions along the column.

Under this assumption

Eqs. (4.8-4.9) reduce to the algebraic system:

The condition for a nontrivial solution of (6.2)

is:

Developing the determinant ~(w),

a cubic

polynomial for

w

is obtained:

Here,

a l

to a3 and

to

e3

are real and

can be calculated from the coefficients of Eqs.

(4.8-4.9) and i

=\/ri.

We are not interested in

the complete solution of (6.4) but only in its

stability condition:

Re(o)

2

0

for stability.

This can be

decided by

the

Frank-Bilharz

criterion

[36].

NEON

-

I . 2 0 r n A

ap, om Torr

Fig. 3: Stability diagram of the transition from

a longitudinally homogeneous glow to

a

striated column for a current of 20 mA.

ap, crn Torr

Fig. 4: Same as Fig. 3 for a current of 50 mR.

apO rm Torr

Fig.

5:

Same as Fig. 3 for a current of

70

mA.

In Figs.

3 to

5

the curves of neutral

stability for various currents are displayed on a

c

vs.

apo plane for neon discharges.

,The

domain

o f

stability is outside these curves. Two

main conlusions can be drawn from these figures:

(1) For a constant current there is an interval

Pcrl

5

P

5

Pcr2 where the discharge is

stable for a11 wave numbers.

t h e t r a n s v e r s e d i r e c t i o n , so t h a t t h e development

."""L

:

---

E I P PFAU (19681

aPo I -

-

EXP PEKARER cmTorr -

-

THEOR?.

- 1 - D PERTURBATIONS

'

- -

-

THEORI C.O STABLE

H o M O G E l E O U S 100 - -

__-_----

UNSTABLE STRIATED / I

1

UNSTABLE STRIATED

\

o f s t r i a t i o n s a l o n g t h e p o s i t i v e coiumn i s s i m u l a t e d . The n e u t r a l s t a b i l i t y b o u n d a r i e s were c a l c u l a t e d and compared t o t h e e x p e r i m e n t a l r e s u l t s o f Pfau and Rutscher [7] and Pekarek and Novak [37]. Good q u a l i t a t i v e agreement between t h e o r y and experiment was o b t a i n e d .

The b a s i c assumption u n d e r l y i n g t h i s work i s t h a t t h e s t a b i l i t y problem of s e l f - s u s t a i n e a e l e c t r i c a i s c h a r y e s cdn be t r e a t e d DO m a c r o s c o p i c a l l y and t h a t t h e m i c r o s c o p i c F i g . 6: N e u t r a l s t a b i l i t y c u r v e o f t h e t r a n s i t i o n from a l o n g i t u d i n a l l y homogeneous glow t o a s t r i a t e d column i n t h e p l a n e o f reduced p r e s s u r e v e r s u s c u r r e n t i n t e n s i t y . F i g . 6 d e s c r i b e s t h e s t a b i l i t y boundary i n t h e p l a n e o f p r e s s u r e vs. c u r r e n t . For a g i v e n c u r r e n t t h e two c r i t i c a l p r e s s u r e s a t t h e o n s e t o f s t r i a t i o n s d e s c r i b e d above a r e shown i n t h e f i g u r e . I t t u r n s o u t t h a t t h e c u r v e BD a p p e a r i n g i n t h e f i g u r e i s i d e n t i c a l t o t h e n e u t r a l s t a b i l i t y c u r v e which was o b t a i n e d i n t h e p r e v i o u s s e c t i o n f o r c = 0 and which,describes t h e s t a b i l i t y boundary o f t h e t r a n s i t i o n f r o m a d i f f u s e glow t o a c o n s t r i c t e d column. T h e r e f o r e one can conclude t h a t f o r p r e s s u r e s h i g h e r t h a n Pcr2, t h e two b a s i c k i n d s o f i n s t a b i l i t y , namely c o n s t r i c t i o n and s t r i a t i o n s , appear s i m u l t a n e o u s l y . T h i s agrees w i t h t h e e x p e r i m e n t a l r e s u l t s o f Pfau and Rutscher [ 7 ] which a r e a l s o d e s c r i b e d i n t h e f i g u r e . A l s o shown a r e t h e e x p e r i m e n t a l r e s u l t s o f Pekarek and Novak [37] who s t u d i e d t h e appearance o f s t r i a t i o n s i n neon d i s c h a r g e s a t low pressures.

V I I . Summary

A mathematical model which d e s c r i b e s t h e i n s t a b i 1 i t y phenomena o c c u r r i n g i n glow d i s c h a r g e s has been developed. The l i n e a r i z e d e q u a t i o n s l e a d t o a c o m p l i c a t e d e i g e n v a l u e problem. Two s i m p l i f i e d one-dimensional cases were t r e a t e d . I n t h e f i r s t case i t was assumed t h a t t h e d i s t u r b a n c e v a r i e s o n l y i n t h e t r a n s v e r s e d i r e c t i o n b u t n o t a l o n g t h e column, so t h a t t h e d i s c h a r g e c o n s t r i c t i o n i s simulated. I n t h e second case t h e d i s t u r b a n c e was assumed t o o s c i l l a t e a l o n g t h e column b u t was c o n s t a n t i n

processes such as i o n i z a t i o n , r e c o m b i n a t i o n etc., can be modelled by r e l a t i v e l y s i m p l e c o n s t i t u t i v e r e l a t i o n s . I t i s a l s o assumed t h a t ttie major mechanism o f t h e i n s t a b i l i t y i s therrnal, i.e., i s due t o uneven J o u l e h e a t i n g and t h a t t h i s e f f e c t can e x p l a i n b o t h k i n d s of i n s t a b i l i t y : c o n s t r i c t i o n and s t r i a t i o n s .

The macroscopic approach was f o l l o w e d by o t h e r i n v e s t i g a t o r s (13, 14, 19-22]. Ecker e t a l . [13, 141 used a s i m p l e r l i n e a r s t a b i l i t y a n a l y s i s f o r t h e s t u a y of C o n s t r i c t i o n b u t f a i l e d 4 t o i n c l u d e t h e e q u a t i o n of c u r r e n t c o n t i n u i t y

0 - J

= 0. I n s t e a d t h e y i n c l u d e t h e i n f l u e n c e o f t h e e x t e r n a l e l e c t r i c c i r c u i t . T h e i r r e s u l t s f o r c o n s t r i c t i o n i n s t a b i l i t y a r e a t v a r i a n c e w i t h t h e experiments o f Pfau and Rutscher [7]. A d i f f e r e n t macroscopic approach was t a k e n b y R o g o f f [20] and by Oster, Jeager and Phelps [ 2 1 j who s t u d i e d t h e t i m e development o f t h e d i s c h a r g e p r o p e r t i e s . Such an a n a l y s i s cannot p r e d i c t t h e s t a b i l i t y boundaries.

On t h e o t h e r hand t h e i n s t a b i l i t y phenomena were t r e a t e d by o t h e r i n v e s t i g a t o r s u s i n g a m i c r o s c o p i c approach. I n a comprehensive paper, Haas [17] has c o n s i d e r e d t h e s t a b i l i t y o f a p l asma c o n s i s t i n g of n e u t r a l molecules, metastables, n e g a t i v e ions, p o s i t i v e i o n s and e l e c t r o n s . Nighan and Wiegand [18] a p p l i e d Haas' e q u a t i o n s t o processes o c c u r r i n g i n N -C02-He

C9-332 JOURNAL DE PHYSIQUE

C o n s t r i c t i o n o f t h e column and t h e appearance o f s t r i a t i o n s .

/ V I I I . Resume

/

On a developpe un modzle mathgmatique /

d g c r i v a n t l e s ph&m>nes d ' i n s t a b i l i t e dans l e s

/

decharges gazeuses luminescentes. Les g q u a t i o n s l i n e ' a r i s g e s r 6 v 2 l e n t un problzme c o m p l i q u d de v a l e u r s p r o p r e s . On e s t amen:

2

t r a i t e r deux cas uni-dimensionnels s i m p l i f i e $ . Dans l e p r e m i e r cas, on assume que l a p e r t u r b a t i o n v a r i e dans l a d i r e c t i o n t r a n s v e r s a l e seulement, s i m u l a n t a i n s i

/

l a c o n t r a c t i o n de l a decharge. Dans l e s e c ~ ~ ~ d cas, on assume que l a p e r t u r b a t i o n o s c i l l e l e l o n g de l a c o l o n n e p o s i t i v e mais r e s t e c o n s t a n t e dans l a d i r e c t i o n t r a n s v e r s a l e , s i m u l a n t a i n s i l e d&loppement des s t r i e s . Les l i m i t e s de

0 /

s t a b i l i t g n e u t r e s o n t c a l c u l e e s e t comparees aux r g s u l t a t s experimentaux de Pfau e t Rutscher [ 7 ] e t Pekarek e t Novak [37]. On c o n s t a t e a i n s i une bonne compati b i 1 i te/ qua1 i t a t i v e e n t r e n o t r e

/

t h e o r i e e t l ' e x n e r i e n c e .

~ ' h ~ ~ o t h s s e de base de c e t t e r e c h e r c h e e s t l a s u i v a n t e : on assume que l e probl&ne de s t a h i l i t g des d <charges g l e c t r i q u e s p e u t & r e t r a i

te)

p a r une approche macroscopique t a n d i s que des r e l a t i o n s c o n s t i t u t i v e s r e l a t i v e m e n t s i m p l e s peuvent f o u r n i r un modzle aux p r o c e s s i microscopiques t e l s que 1

'

i o n i s a t i o n , l a

/

recombinaison e t c . On assume egalement que l e p r i n c i p a l m&nisme d ' i n s t a b i l it< e s t de t y p e / thermal, a s a v o i r q u ' i l e s t cause p a r un rgchauffement l o c a l de Joule. A i n s i c e t e f f e t / p e u t e x p l i q u e r l e s deux t y p e s d ' i n s t a b i l i t e : l a c o n t r a c t i o n e t l e s s t r i e s . Les a u t e u r s s u i v a n t s o n t s u i v i l ' a p p r o c h e macroscopique: Ecker e t a1 [13, 1 4 1 o n t e t u d i e / l e ph&om&e de l a c o n t r a c t i o n p a r une a n a l y s e ue s t a b i l it: l i n g a i r e p l u s s i m p l e comprenant l l i n f l u e n c e du c i r c u i t g l e c t r i q u e e x t e r n e mais ne / p r e n a n t pas en conlpte ~ l e ) ~ u a t i o n de c o n t i n u i t e du

+

c o u r a n t 6 1 e c t r i q u e , V.J = 0. Leurs r k u l t a t s s u r l l i n s t a b i l i t e / de l a c o n t r a c t i o n p r g s e n t e n t des d i f f g r e n c e s . n o t o i r e s avec l e s r $ s u l t a t s experimentaux de Pfau e t Rutscher [7]. R o g o f f [20] e t Jeager, O s t e r e-t Phelps (211 o n t u t i l i s g une approche macroscopique d i f f g r e n t e en g t u d i a n t l s & o l u t i o n dans l e terhps des p r o p r i g t & de l a

/

dgcharge. Une t e l l e analyse ne p e u t p r e d i r e l e s

l i m i t e s de s t a b i l i t c

D ' a u t r e p a r t , l ' a p p r o c h e m i c r o s c o p i q u e des

# \ 0 /

phenomenes d ' i n s t a b i 1 it: a e t e s u i v i e p a r d ' a u t r e s a u t e u r s . Dans une Gtude t r z s c o m p l ~ t e , Haas [17] analyse l a s t a b i l i t g d ' u n plasma

/ 0

c o n s i s t a n t en molecules n e u t r e s , metastables, i o n s n g g a t i f s e t p o s i t i f s e t g l e c t r o n s . Nighan

0

e t Wiegand [18] o n t a p p l i q u e l e s g q u a t i o n s de Haas aux r g a c t ~ o n s o c c u r a n t dans l e s mglanges

/

N,-CO-He

-

e t o n t c a l c u l e l a l i m i t e de

2 ,

s t a b i l i t e dans un p l a n N/n vs. n. I 1 e s t

2

n o t e r que ces analyses s o n t l o c a l e s e t ne p r e n n e n t pas en compte l e s c o n d i t i o n s des l i m i t e s macroscopiques. Haas remarque q u l i l n ' y a pas de / nioyen de f a i r e une r e l a t i o n e n t r e l ' i n s t a b i l i t e m i c r o s c o p i q u e e t macroscopique ' de l a 8 c o n t r a c t i o n . En c o n c l u s i o n , il e s t demontre dans 0

c e t t e g t u d e que l a t h e o r i e macroscopique pour l e s

0 gaz n o b l e s p e u t d e c r i r e l e s deux t y p e s 0 d ' i n s t a h i l i t e : l a c o n t r a c t i o n de l a colonne e t l ' a p p a r i t i o n des s t r i e s . References

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