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STABILIZING EFFECT OF EMITTED CHARGES ON THE CONE-LIKE SHAPES OF ELECTRIFIED
MENISCI
G. Joffre, M. Cloupeau
To cite this version:
G. Joffre, M. Cloupeau. STABILIZING EFFECT OF EMITTED CHARGES ON THE CONE-LIKE SHAPES OF ELECTRIFIED MENISCI. Journal de Physique Colloques, 1986, 47 (C7), pp.C7-359- C7-364. �10.1051/jphyscol:1986761�. �jpa-00225956�
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Colloque C7, supplément au n o 11, Tome 47, Novembre 1986
STABILIZING EFFECT OF EMITTED CHARGES ON THE CONE-LIKE SHAPES OF ELECTRIFIED MENISCI
G.H. JOFFRE and M. CLOUPEAU
Laboratoire d'Aérothermique du C.N.R.S., 4ter. Route des Gardes, F-92190 Meudon, France
Résumé : Lorsqu'un ménisque e s t formé à l ' e x t r é m i t é d ' u n c a p i l l a i r e e t p o r t é à un p o t e n t i e l é l e c t r i q u e suffisamment élevé, l e ménisque p e u t p r é s e n t e r l a forme d'un cône. Suivant l e s c o n d i t i o n s expérimentales, on observe à Ta p o i n t e du cône, l ' é m i s s i o n de charges p a r e f f e t de champ, l a p r o d u c t i o n de g o u t t e l e t t e s ou une dé- charge corona. La forme conique e s t e s s e n t i e l l e c a r e l l e permet l ' e x i s t e n c e des deux premiers phénomènes. Dans l e présent t r a v a i l , l e s p r o f i l s des ménisques é l e c t r i s é s o n t é t é c a l c u l é s en t e n a n t compte de l ' e f f e t r é g u l a t e u r que l e d é p a r t des charges exerce s u r l e champ é l e c t r i q u e au somnet du cône. Les c a l c u l s montrent notamment que l ' é q u i l i b r e d'un ménisque conique e s t p o s s i b l e pour t o u t e une gamme de v a l e u r s du p o t e n t i e l é l e c t r i q u e e t de l a p r e s s i o n hydrostatique.
A b s t r a c t : When a meniscus i s f o m e d a t t h e free end o f a c a p i l l a r y tube and i s sub- j e c t e d t o t h e e f f e c t o f a s u f f i c i e n t l y i n t e n s e e l e c t r i c f i e l d , t h e meniscus may d i s - p l a y t h e shape o f a cone. Depending upon t h e experiment c o n d i t i o n s , a f i e l d e f f e c t charge emission, the p r o d u c t i o n o f d r o p l e t s o r corona discharge may be observed a t t h e apex o f t h e cone. The cone shape i s e s s e n t i a l t o generate t h e f i r s t two pheno- mena. I n t h i s study, t h e p r o f i l e s o f e l e c t r i f i e d menisci were c a l c u l a t e d t a k i n g account o f t h e r e g u l a t i n g e f f e c t t h a t charge emission may e x e r t on the e l e c t r i c f i e l d a t t h e apex o f t h e cone. The c a l c u l a t i o n s p a r t i c u l a r l y show t h a t c o n i c a l me- niscus e q u i l i b r i u m i s p o s s i b l e over a whole range o f e l e c t r i c p o t e n t i a l and hydros- t a t i c pressure values.
1. INTRODUCTION
I n 1914, Zeleny ( 1 ) s t u d i e d corona discharges produced from l i q u i d menisci, u s i n g an assembly i n which a c a p i l l a r y tube ( f i g . 1 ) was f e d w i t h l i q u i d under a var5able pressure, close t o zero. A v o l t a g e o f a few thousand v o l t s was then a p p l i e d between the meniscus formed a t t h e f r e e end o f t h e c a p i l l a r y tube and a p l a t e . Zeleny obser- ved t h a t d r o p l e t s c o u l d be e m i t t e d under t h e a c t i o n o f t h e e l e c t r i c a l f i e l d . Among t h e o p e r a t i n g modes which he described ( Z ) , t h e most i n t e r e s t i n g i s c e r t a i n l y t h a t i n which t h e meniscus takes t h e shape o f a cone, w i t h i t s apex extended by a v e r y f i n e f i l a m e n t , t h e breakage o f which r e s u l t s i n charged m i c r o d r o p l e t s .
It would seem t h a t these phenornena were n o t the s u b j e c t o f any f u r t h e r study u n t i l they were rediscovered ( 3 ) i n t h e e a r l y f i f t i e s . However, t h e y have since been t h e s u b j e c t o f many works ( 4 ) . I n p a r t i c u l a r , several authors (5,6) have attempted t o use t h i s process f o r e l e c t r i c a l p r o p u l s i o n i n space. Thus, i n 1961,,as p a r t o f a study concerning t h e p r o d u c t i o n o f charged d r o p l e t s i n a vacuum, u s i n g a l i q u i d me- t a l , Krohn ( 7 ) observed t h a t metal i o n s were a l s o e m i t t e d by t h e cone-shaped menis- cus. Various authors have attempted t o o b t a i n t h i s phenomenon w i t h o u t t h e r e being simultaneous emission o f d r o p l e t s . The i o n sources thus created were s t u e i e d f o r various a p p l i c a t i o n s , such as i o n i c p r o p u l s i o n , i o n i m p l a n t a t i o n , i o n i c microscopy, m i c r o l i t h o g r a p h y , etc.. . (8,9,10,11).
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986761
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F i g . 1 - Experiment set-up and d e f i n i t i o n o f a few symbols
I n t h e case o f b o t h i o n emission and t h a t o f a f i n e spray, t h e cone-like shape o f the meniscus i s e s s e n t i a l as, i n t h e former case, i t a l l o w s i o n s t o be e x t r a c t e d by f i e l d e f f e c t and, i n t h e second case, i t a l l o w s a l i q u i d f i l a m e n t t o be created, t h e diameter o f which i s much s m a l l e r than t h a t o f t h e c a p i l l a r y tube.
Further, i n a l 1 t h e o r e t i c a l s t u d i e s concerning c a l c u l a t i o n o f the shape o f c o n e - l i k e menisci, no author has considered s p e c i f i c phenomena o c c u r r i n g a t t h e apex o f t h e cone and, notably, t h e presence o f t h e space charge (12).
The f i r s t work c a r r i e d o u t t o t h e o r e t i c a l l y j u s t i f y t h e existence o f cone-like menis- c i was t h a t o f T a y l o r (13). T h i s a u t h o r considered a s p e c i f i c geometrical c o n f i g u r a - t i o n formed by a s t r i c t l y c o n i c a l meniscus ( i .e, w i t h s t r a i g h t generating l i n e s ) , supported by a s e m i - i n f i n i t e cone t r u n k . He demonstrated t h a t , a t zero h y d r o s t a t i c pressure, c a p i l l a r y and e l e c t r o s t a t i c pressures can be balanced a t a l 1 p o i n t s , f o r a c r i t i c a l value o f e l e c t r i c a l p o t e n t i a l , p r o v i d i n g t h e h a l f - a n g l e o f t h e cone apex be 49.3 degrees .
This mode1 corresponds t o a v e r y p a r t i c u ' i a r case and i s s t i l l t h e s u b j e c t o f some discussion (14,15). However, more s o p h i s t i c a t e d s t u d i e s have since been c a r r i e d o u t . This i s n o t a b l y t h e case o f t h e works o f Basaran and Scriven (16) who, i n 1981, obtained t r a n s i e n t forms o f cone-shaped menisci using a f i n i t e element method.
However, t h e above works do n o t cover c e r t a i n experiments f a c t s concerning s t a b l e cone-shaped menisci :
- f i rs t l y , these cone-shaped menisci can e x i s t through a whole area o f e l e c t r i c a l p o t e n t i a l and h y d r o s t a t i c pressure values,
- secondly, t h e y have never been observed w i t h o u t a simultaneous charge emission due, depending upon circumstances, t o a corona disc.harge, t h e p r o d u c t i o n o f charged d r o p l e t s o r i o n emission. I n a l 1 these cases, a space charge i s c r e a t e d which e x e r t s a r e g u l a t i n g e f f e c t on the e l e c t r i c a l f i e l d a t t h e apex o f t h e cone and which seems indispensable f o r e q u i l i b r i u m t o be maintained.
Therefore, i n t h i s work, t h i s e f f e c t has been modelled and used i n a method (17), which a l lows e l e c t r i f i e d meni s c i shapes t o be c a l c u l a t e d w i t h no s i m p l i f i c a t i o n assumptions. For reasons discussed l a t e r i n t h e t e x t , t h i s m o d e l l i n g has been applied t o t h e case i n which a corona discharge occurs a t the apex o f t h e cone. I n fact, the c a l c u l a t i o n s performed a c t u a l l y r e s u l t i n t h e observed cone-1 i k e shapes.
2. THEORY 2.1 Equations
The problem t r e a t e d i s t h a t o f a meniscus formed a t t h e end o f a v e r t i c a l c a p i l l a r y tube ( f i g . 1 ) s e t t o e l e c t r i c a l p o t e n t i a l Uo i n r e l a t i o n t o a s e m i - i n f i n i t e horizontal p l a t e , l o c a t e d a t a d i s t a n c e D from t h e e x i t s e c t i o n o f t h e c a p i l l a r y tube, o f r a d i u s R. The shape o f the meniscus i s axisymmetrical and i s represented by t h e f u n c t i o n y ( r ) , where i s the r a d i a l coordinate normalised i n r e l a t i o n t o R. The meniscus i s formed by a l i q u i d o f d e n s i t y p i and w i t h surface t e n s i o n A.
The balance equation f o r t h e surface, considered as conducting, i s w r i t t e n i n a non- dimensional form (18,19) :
I n t h e f i r s t term, which represents h y d r o s t a t i c pressure, B i s equal t o p~ gR2/A, where g i s the a c c e l e r a t i o n due t o g r a v i t y . KH i s equal t o H/R, where H i s t h e h e i g h t (always counted p o s i t i v e l y upwards from the c a p i l l a r y tube e x i t s e c t i o n ) o f a column o f t h e same l i q u i d as t h a t forming the meniscus. The c o n s t a n t E i s equal t o 1 f o r a pendent meniscus and -1 f o r a s e s s i l e meniscus.
I n t h e second term, which represents c a p i l l a r y pressure, R1 and R2 are t h e main r a d i i -
o f c u r v a t u r e normalised i n r e l a t i o n t o R. I n c y l i n d r i c a l s y m e t r y , they a r e c a l c u l a - t e d as a f u n c t i o n o f t h e f i r s t d e r i v a t i v e & and second d e r i v a t i v e % o f 7, i n r e l a t i o n t o 7 .
I n t h e t h i r d tenn, which represents e l e c t r o s t a t i c pressure, Es i.s t h e f i e l d a t t h e -
s u r f a c e normalised i n r e l a t i o n t o Uo/D. KD i s equal t o D/R, EO i s t h e d i e l e c t r i c constant o f the surrounding medium.
To s o l v e equation ( l ) , Ës must be known a t each p o i n t on t h e surface, which means, i n p r i n c i p l e , t h a t the d i s t r i b u t i o n s o f f i e l d Ë and charges must be c a l c u l a t e d f o r a l 1 t h e space between t h e surface o f t h e l i q u i d and t h e p l a t e .
F o r t h i s , Poisson's equation and the c u r r e n t conservation e q u a t i o n must be s i m u l t a - neous solved. Considering t h e e l e c t r i c a l m o b i i i t y o f the i o n s t o be a constant K, t h i s i s w r i t t e n as f o l l o w s :
-+ -
d i v E = - T C $
peoD2
where 4 = - , peo being a c h a r a c t e r i s t i c value o f t h e d e n s i t y o f the charges O O
e m i t t e d a t the meniscus surface.
f
The equation system (1) (2) and ( 3 ) , i n which t h e unknowns a r e 7 , E and p, must t h e r e f o r e be solved.
2.2 Boundary c o n d i t i o n s
The boundary conditions. used f o r t h e meniscus p r o f i l e are : - z ( 1 ) = O and YF(0) = O
The c o n d i t i o n s adopted f o r p o t e n t i a l Ü a r e represented i n f i g u r e 2.
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Fig. 2 Fig. 3
Block diagram of meshing and boundary conditions Example of cone-shape p r o f i l e obtained
by cal culation Finally, as demonstrated by F e l i c i ( 2 0 ) , a condition fixing the value of emitted charges density p , must be imposed on the meniscus surface. This density will be expressed using a$ equation model l i n g charge emission.
2.3 Charge emission law
In the case of ion emission, i t i s very d i f f i c u l t t o define a s u i t a b l e charge emis- sion equation. In addition, extremely f i n e meshing would be required t o represent the t i p of the cone apex, which i s extremely pointed.
In the case of spraying, complex hydrodynamic phenomena should a l s o be considered.
This i s the reason f o r which i t was preferred t o model t h e regulating e f f e c t of charges in the case of corona e f f e c t . In order t o cover c e r t a i n c h a r a c t e r i s t i c pro- p e r t i e s of the corona discharge, the following assumptions were nade :
- charges are only emitted l o c a l l y f o r a f i e l d a t the surface, Es, above a c e r t a i n c r i t i c a l value, Ec, which i s an increasing function of the surface curvature 5,
- the density of the emitted charges, p , increases with the difference between the surface f i e l d , E s , and the c r i t i c a l f i e f d , Ec.
The equation adopted i s written a s follows :
I n t h i s expression, the constant G w i l l be between 9 and 14 (21,22).
RESULTS
3. -
Equations ( l ) , (2) and (3) a r e transformed i n t o d i s c r e t e format using f i n i t e d i f f e - rence formulae i n v a r i a b l e p i t c h meshing ( f i g . 2 ) .
Equation ( 2 ) has been handled using a p o i n t by p o i n t o v e r - r e l a x a t i o n method and equation ( 3 ) u s i n g the c h a r a c t e r i s t i c s method (19). Equation (1) was solved u s i n g Newton's method.
The c a l c u l a t i o n s g i v e t h e d i s t r i b u t i o n o f p o t e n t i a l s and charges i n space, t r a n s - f e r r e d c u r r e n t , 1, and d i s t r i b u t i o n o f surface charges on t h e meniscus.
This method gives r e s u l t s b a s i c a l l y s i m i l a r t o experiment observations concerning both meniscus c o n e - l i k e shapes and t h e values o f the p o t e n t i a l a p p l i e d and t h e c u r r e n t f o r which these menisci a r e obtained.
As an example, f i g u r e 3 shows the p r o f i l e c a l c u l a t e d f o r one o f t h e c a p i l l a r y geo- m e t r i e s adopted. The r a d i u s o f curvature a t the apex o f the cone i s , i n t h i s case, l i m i t e d by t h e p i t c h o f t h e meshing used. Except around t h e apex, increased meshing d e n s i t y has v i r t u a l l y no e f f e c t on t h e shape o f the cone, t h e generating l i n e o f which i s p r a c t i c a l l y s t r a i g h t , i n t h i s p a r t i c u l a r case.
Such cone-shape menisci a r e obtained over a s i g n i f i c a n t range o f h y d r o s t a t i c pressu- r e and e l e c t r i c a l p o t e n t i a l values.
Concerning t h e p o t e n t i a l , i t should be p o i n t e d o u t t h a t the e x t e n t o f t h i s range v a r i e s considerably w i t h t h e shape o f t h e c a p i l l a r y tube used.
Generally, t h e o r e t i c a l r e s u l t s demonstrate the main types o f phenomena observed i n experiments ( 1 8 ) .
4. CONCLUSION
The r e s u l t s obtained u s i n g t h e method developed t o determine the shape o f e l e c t r i f i e d menisci e m i t t i n g charges j u s t i f y the existence o f s t a b l e cone-shape menisci observed d u r i n g experiments.
The emission equation used models t h e case o f a corona discharge. The theory gives a c o r r e c t q u a l i t a t i v e d e s c r i p t i o n o f t h e main phenomena observed, notably, cone- shape menisci may be obtained over a s i g n i f i c a n t range o f p o t e n t i a l and h y d r o s t a t i c pressure values.
Whichever process i s used f o r charge emission, t h e cone-shape menisci obtained are o f s i m i l a r form. A meniscus shape c a l c u l a t i o n more s u i t a b l e f o r t h e i o n source case c o u l d be achieved, u s i n g a charge emission equation c o r k s p o n d i n g more a c c u r a t e l y t o t h e f i e l d emission phenomenon, i n t h e method described above, and by considerincj the f a c t t h a t i o n speed i s then a f u n c t i o n o f l o c a l e l e c t r i c a l p o t e n t i a l .
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