AN ELECTROHYDRODYNAMIC ANALYSIS OF THE EQUILIBRIUM SHAPE AND STABILITY OF STRESSED CONDUCTING FLUIDS : APPLICATION TO LMIS

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AN ELECTROHYDRODYNAMIC ANALYSIS OF THE EQUILIBRIUM SHAPE AND STABILITY OF STRESSED CONDUCTING FLUIDS : APPLICATION

TO LMIS

M. Chung, P. Cutler, T. Feuchtwang, E. Kazes, N. Miskovsky

To cite this version:

M. Chung, P. Cutler, T. Feuchtwang, E. Kazes, N. Miskovsky. AN ELECTROHYDRODYNAMIC

ANALYSIS OF THE EQUILIBRIUM SHAPE AND STABILITY OF STRESSED CONDUCTING

FLUIDS : APPLICATION TO LMIS. Journal de Physique Colloques, 1986, 47 (C7), pp.C7-351-C7-

358. �10.1051/jphyscol:1986760�. �jpa-00225955�

AN ELECTROHYDRODYNAMIC ANALYSIS OF THE EQUILIBRIUM SHAPE AND

STABILITY OF STRESSED CONDUCTING FLUIDS : APPLICATION TO LMIS

M. CHUNG, P.H. CUTLER, T.E. FEUCHTWANG, E. KAZES a n d

N.M. MISKOVSKY*

Department o f P h y s i c s , The P e m s y l v a n i a S t a t e U n i v e r s i t y , U n i v e r s i t y Park, P A 16802, U.S.A.

' ~ e p a r t m e n t o f P h y s i c s , Altoona Campus, The P e m s y l v a n i a S t a t e U n i v e r s i t y , A l t o o n a , PA 16603, U.S.A.

~ é s u m é

-

Une a n a l y s e e l e c t r o h y d r o s t a t i q u e du modele du cone d e Taylor u t i l a n t a l a f o i s l e s c r i t e r e s d e Taylor e t d e Zeleny f a i t r e s s o r t i r p l u s i e u r s contra- d i c t i o n s . On p e u t montrer qu'un t r a i t m e n t dynamique d e l a geometrie d ' e q u i - l i b r e e t d e l a s t a b i l i t y p e u t en résoudre l e s c o n t r a d i c t i o n s apparentes.

E t precisement, e n u t i l i s a n t l ' e q u a t i o n electrodynamique l i n e a r i s e e t d e s c o r r e c t i o n s a u 1 s t o r d r e , on montre qu'au s e u i l d ' u n s t a b i l i t e l e cone prend une forme, " c u s p i d a l e " . La t e n s i o n c r i t i q u e d e s e v i l d ' i n s t a b i l i t e e s t obtenue pour l e gallium a p a r t i r de l a r e l a t i o n d e d i s p e r s i o n . M v a l e u r c a l c u l e e d e 5.8 kV correspond b i e n aux mesures experimentales d e - 4 - 7 kV. Enfin, l e c a r a c t e r e t r e s l o c a l i s e d e l l i n s t a b i l i t e e s t en accord avec l e s o b s e r v a t i o n s experimentales obtenues en TEM p a r Sudraud, e t . a l .

A b s t r a c t

-

^An e l e c t r o h y d r o s t a t i c a n a l y s i s of t h e Taylor cone model, using both t h e Taylor and Zeleny s t a b i l i t y c r i t e r i a h a s r e v e a l e d s e v e r a l i n c o n s i s t e n c i e s i n t h e model. It i s shown t h a t a dynamical t r e a t m e n t of t h e e q u i l i b r i u m shape and s t a b i l i t y c a n r e s o l v e t h e s e a p p a r e n t c o n t r a d i c t i o n s i n t h e Taylor model.

S p e c i f i c a l l y , u s i n g t h e l i n e a r i z e d electrohydrodynamic e q u a t i o n s w i t h correc- t i o n s up t o f i r s t - o r d e r , i t is shown t h a t , a t t h e o n s e t of i n s t a b i l i t y , t h e cone deforms i n t o a c u s p i d a l shape. From t h e d i s p e r s i o n r e l a t i o n s , t h e c r i t i - c a l v o l t a g e f o r t h e o n s e t of i n s t a b i l i t y i s o b t a i n e d f o r l i q u i d gallium. The c a l c u l a t e d value of 5.8 kV compares w e l l w i t h experimental v a l u e s of -4-7 kV.

F i n a l l y , t h e i n s t a b i l i t y i s p r e d i c t e d t o be h i g h l y l o c a l i z e d , which a g r e e s w i t h t h e experimental o b s e r v a t i o n s i n t h e T m images o r Sudraud, e t . a l .

1. INTRODUCTION

For t h e p a s t s e v e r a l y e a r s , we have been i n v e s t i g a t i n g i n a systematic and rigor- ous way some of t h e b a s i c p h y s i c s of conducting f l u i d s i n s t r o n g e l e c t r i c f i e l d s t 5 f . The u l t i m a t e o b j e c t i v e of t h i s s t u d y i s t o h e l p e x p l a i n t h e mechanism f o r i o n (and n e u t r a l ) emission from a f i e l d emission l i q u i d metal i o n source (LMIS).

I n t h i s s t u d y t h r e e s p e c i f i c q u e s t i o n s a r e addressed:

1. What i s t h e e v o l u t i o n of t h e dynamical e q u i l i b r i u m shape of t h e l i q u i d m e t a l e l e c t r o d e w i t h a p p l i e d f i e l d ?

2. When and how d o e s i n s t a b i l i t y o c c u r (i.e., breakdown o r d i s i n t e g r a t i o n of t h e l i q u i d e l e c t r o d e , r e s u l t i n g i n emission)?

3. what i s t h e s p e c i f i c mechanism f o r i o n formation?

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986760

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The answer t o t h e f i r s t , and a l s o t o a l a r g e e x t e n t t h e second q u e s t i o n , r e q u i r e s a macroscopic t h e o r y u s i n g f l u i d mechanics and e l e c t r o s t a t i c s . The p h y s i c a l model(s) and a n a l y s i s r e q u i r e d t o answer t h e l a s t q u e s t i o n a r e microscopic. and use quantum physics. I t i s obvious t h a t a r e l a t i o n s h i p e x i s t s between t h e quantum mechanism f o r i o n formation and t h e 'macroscopic' f l u i d shape. Although this w i l l n o t b e d i s c u s s e d h e r e , some i n i t i a l r e s u l t s r e l a t i n g t o t h e f l u i d shape and i o n formation w i l l be pub- l i s h e d elsewhere. I n t h e c u r r e n t paper, we s h a l l be concerned with t h e macroscopic p a r t of t h e problem, s t a t e d i n q u e s t i o n s one and two. S p e c i f i c a l l y , we have used a f u l l electrohydrodynamic (EHD) t h e o r y t o c a l c u l a t e t h e dynamical shape of a t h r e e - dimensional e l e c t r i f i e d f l u i d a t t h e o n s e t of i n s t a b i l i t y . The r e s u l t s a r e a p p l i e d t o a G a LMIS and b o t h q u a l i t a t i v e and q u a n t i t a t i v e agreement a r e found w i t h experimental o b s e r v a t i o n s . The dynamical a n a l y s i s a l s o r e s o l v e s c e r t a i n c o n t r a d i c t i o n s and incon- s i s t e n c i e s i n t h e Taylor cone model.

In s e c t i o n II, we d e s c r i b e b r i e f l y t h e macroscopic h y d r o s t a t i c and hydrodynamic concepts of i n s t a b i l i t y . The r e s u l t s of t h e e l e c t r o h y d r o s t a t i c a n a l y s i s a r e summa- r i z e d i n S e c t i o n III. The electrohydrodynamic c a l c u l a t i o n s a r e o u t l i n e d i n S e c t i o n I V , and r e s u l t s and c o n c l u s i o n s p r e s e n t e d i n S e c t i o n V.

II. MACROSCOPIC DESCRIPTION OF INSTABILITY

An e x a c t t h e o r y f o r t h e o n s e t of i n s t a b i l i t y , o r breakdown of a f l u i d s u r f a c e , would t r e a t t h e problem dynamically, u s i n g t h e complete s e t of EHD e q u a t i o n s . I n p r i n c i p l e , t h e s e t of e q u a t i o n s would be s o l v e d , f o r a p p r o p r i a t e boundary c o n d i t i o n s , from z e r o f i e l d t o t h e c r i t i c a l v o l t a g e when i n s t a b i l i t y i n t h e f l u i d occurs. For convenience, we d e f i n e two l i m i t i n g regimes f o r an i n s t a b i l i t y :

1. The e l e c t r o h y d r o s t a t i c l i m i t : This i s c h a r a c t e r i z e d by low f i e l d s and s m a l l f l u i d v e l o c i t i e s . I n t h i s regime, a q u a s i - s t a t i c e q u i l i b r i u m shape i s assumed t o e x i s t . ~y q u a s i - s t a t i c , we mean t h e f l u i d s u r f a c e deforms i n response t o t h e e l e c t r i c f i e l d , b u t t h e f l u i d i s s t a t i c . I n this l i m i t , t h e e q u i l i b r i u m s u r f a c e of t h e f l u i d s a t i s f i e s t h e Laplace-Young s t r e s s balance c o n d i t i o n (LY):

where t h e n o t a t i o n i s s t a n d a r d , and Ap=constant#O, i n g e n e r a l . P h y s i c a l l y , an i n s t a - b i l i t y o c c u r s i f t h e outward d i r e c t e d s t r e s s e s a r e e q u a l t o o r g r e a t e r t h a n t h e in- ward s t r e s s e s . Equation 1 can be solved a n a l y t i c a l l y i n some c a s e s [ 2 ] , and numerical- l y i n o t h e r s [ l , 2 ] f o r q u a s i - s t a t i c e q u i l i b r i u m shapes. By applying a c r i t e r i o n f o r t h e s t a b i l i t y of t h e s o l u t i o n ( s e e t h e following d i s c u s s i o n ) , one can s o l v e f o r Vc, t h e c r i t i c a l o r t h r e s h o l d v o l t a g e f o r breakdown i n t h e s t a t i c ( o r q u a s i - s t a t i c ) model, This i s an i m p o r t a n t parameter, s i n c e it can b e compared w i t h e x p e r i m e n t s [ î l . I t i s e s s e n t i a l t o know whether t h e shape determined from t h e s o l u t i o n of ~ q u a t i o n 1 i s s t a b l e . * We have used two c r i t e r i a to analyze t h e s t a b i l i t y of h y d r o s t a t i c equi- l i b r i u m shapes. Given i n r e v e r s e c h r o n o l o g i c a l o r d e r i n which t h e y f i r s t appeared, t h e y a r e t h e ~ a y l o r [ S ] and Zeleny[6] s t a b i l i t y c r i t e r i a . We b r i e f l y c o n s i d e r each of them :

( i ) Taylor s t a b i l i t y c r i t e r i o n : In t h e Taylor model[51, i t i s assumed t h a t f o r e q u i l i b r i u m a c r o s s t h e f l u i d i n t e r f a c e , t h e e l e c t r i c a l s t r e s s i s balanced by t h e mechanical o r s u r f a c e t e n s i o n s t r e s s , s o t h a t Eq. 1 becomes:

E~ 1 1

S = .-

-

^{T (-}

+ -

^{) = O} w i t h Ap = 0.

8ïi R 1 Rz

Although Eq. 2 has been used i m p l i c i t l y by T a y l 0 ~ [ 5 1 and o t h e r s a s a s t a - b i l i t y c r i t e r i o n i n t h e a n a l y s i s of LMIS[7], i t i s , i n f a c t , o n l y a l i m i t - As i s w e l l known, "...There must be an e x a c t s o l u t i o n of t h e e q u a t i o n s of f l u i d dynamics f o r any problem w i t h g i v e n s t e a d y e x t e r n a l c o n d i t i o n s ; . . . y e t n o t every s o l u t i o n of t h e e q u a t i o n of motion, even i f i t i s e x a c t can a c t u a l l y occur i n Nature. The flows that occur i n Nature must n o t o n l y obey t h e e q u a t i o n s of f l u i d dynamics, b u t a l s o be s t a b l e . " [ 4 1

c o u n t e r - e l e c t r o d e , a r e s u l t recognized by s e v e r a l o t h e r groups[8,9].

(ii) Zeleny s t a b i l i t y c r i t e r i o n : Formally, t h i s c r i t e r i o n s t a t e s t h a t t h e f i r s t d e r i v a t i v e of t h e s t r e s s e s ( S ) w i t h r e s p e c t t o a c o o r d i n a t e , Say f3, c h a r a c t e r i z i n g t h e f l u i d s u r f a c e i s z e r o a t a p o i n t where t h e i n s t a b i l i t y may a r i s e . This i s j u s t t h e g e n e r a l c o n d i t i o n f o r mechanical s t a b i l i t y based on energy c o n s i d e r a t i o n s . It i s e q u i v a l e n t t o t h e mathematical s t a t e m e n t t h a t a 2 ~ / a ~ 2 = 0 , where U i s t h e f r e e energy of t h e system, and 3 i s t h e s u r f a c e c o o r d i n a t e [ 1 O]

.

Somewhat i n e x p l i c a b l y , t h e Zeleny c r i t e r i o n f o r s t a b i l i t y of f l u i d i n t e r f a c e s h a s g e n e r a l l y been ignored o r overlooked i n t h e a n a l y s i s of s t r e s s e d conducting f l u i d s and LMIS.

2. The electrodydrodynamic l i m i t : I n t h i s dynamic l i m i t , i t i s assumed t h e r e a r e p r e s s u r e g r a d i e n t s which g i v e r i s e t o f l u i d flow. Hence, t h e p r e s s u r e i n t h e LY s t r e s s balance c o n d i t i o n i s t r e a t e d a s a time-dependent q u a n t i t y . Fluid flow i s now included i n t h e d e s c r i p t i o n of t h e deformation of t h e f l u i d s u r f a c e i n response t o t h e a p p l i e d e l e c t r i c f i e l d . I n t h e EHù l i m i t , t h e i n s t a b i l i t y i s determined by solv- i n g boundary c o n d i t i o n s f o r t h e v e l o c i t y and e l e c t r o s t a t i c p o t e n t i a l s , and t h e f l u i d s u r f a c e deformation. Then, from t h e t h e - d e p e n d e n t LY s t r e s s b a l a n c e c o n d i t i o n , one o b t a i n s t h e d i s p e r s i o n r e l a t i o n

w h e r e w i s t h e a n g u l a r frequency of t h e p e r t u r b e d eigenmode and k is t h e wavevector.

A hydrodynamic i n s t a b i l i t y o c c u r s when w2 <0[4,111. p h y s i c a l l y , t h e amplitude of t h e eigenmode u i n c r e a s e s w i t h o u t l i m i t l e a d i n g t o an i n s t a b i l i t y , because t h e f l u i d can- n o t f o l l o w t h e o s c i l l a t i o n of t h a t mode.

It i s i m p o r t a n t t o recognize t h e d i f f e r e n t p h y s i c s , and c r i t e r i a , n e c e s s a r y t o d e s c r i b e t h e o n s e t of f l u i d i n s t a b i l i t y i n t h e EHS and EHù c a s e s .

Chung, e t . a 1 [ 2 ] have g i v e n a d e t a i l e d a n a l y s i s of an EHS i n t e r p r e t a t i o n of t h e Taylor cone model. We b r i e f l y summarize some of t h e i r conclusions.

Their r e s u l t s of t h e a n a l y s i s of t h e Taylor model can be s t a t e d a s follows:

A c o n i c a l i n t e r f a c e between two f l u i d s can e x i s t i n e q u i l i b r i u m i n t h e presence of an e l e c t r i c f i e l d , o n l y i f i ) t h e cone h a s a semi-vertex a n g l e of 49.3O(Fcx); ii) Ap=0 a c r o s s t h e i n t e r f a c e , and iii) t h e c o u n t e r - e l e c t r o d e has a shape given by r=r, [R,(cos 8 ) I - ~ . [ 5 , 7 ] F u r t h e m o r e , using E q . 2 a s a s t a b i l i t y c r i t e r i o n , t h e c r i t i c a l v o l t a g e f o r o n s e t of i n s t a b i l i t y i s found t o be15,71

vCT

= [ 4 s i n 0 ~ / ~ - + cos^) 1 / 2 I T r O ~ c o t ~ where t h e n o t a t i o n i s d e f i n e d i n Refs. 5 o r 7.

Three d i f f i c u l t i e s , o r c o n t r a d i c t i o n s , a r e a p p a r e n t i n t h e Taylor model.

1. The Taylor form of t h e s t r e s s b a l a n c e c o n d i t i o n , E q . 2, i s o b t a i n e d assum- i n g Ap=0 i n t h e g e n e r a l LY c o n d i t i o n . What i s t h e v a l i d i t y of assuming Ap=O? This q u e s t i o n has been addressed and d i s c u s s e d i n Ref. 7 , 8, and 9 ,

and w i l l n o t b e t r e a t e d f u r t h e r h e r e o t h e r t h a n t o observe t h a t i n Our dynamic t r e a t m e n t of t h e " c o n i c a l " model, we f i n d t h a t ApfO a f t e r defor- mation ( s e e S e c t i o n I V ) .

2. The cone i s t r e a t e d a s a h y d r o s t a t i c o r q u a s i - s t a t i c e q u i l i b r i u m shape.

However, t h e Taylor s t r e s s c o n d i t i o n i s regarded a s simultaneously a c r i t e r i o n f o r s t a b i l i t y , y i e l d i n g

vcT.

I f t r u e ( i . e . , i f Eq. 4 i s v a l i d ) , t h e n t h e Taylor s t a b i l i t y c r i t e r i o n p r e d i c t s an o v e r a l l

-

o r g l o b a l

-

breakdown simultaneously a c r o s s t h e e n t i r e s u r f a c e of t h e cone. This has never been observed e x p e r i m e n t a l l y i n

3

s t u d y of s t r e s s e d conducting f l u i d s o r LMIS[8,9,12,131.

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3. The cone h a s a s i n g u l a r i t y a t t h e apex, where t h e f i e l d beconies i n f i n i t e . This i s obviously incompatible w i t h a s t a t i c e q u i l i b r i u m shape. The s i g - h i f i c a n c e of s i n g u l a r i t i e s on s t a b i l i t y and breakdown was f i r s t d i s c u s s e d by Chung, e t . a1.[141 i n t h e a n a l y s i s of an i d e a l c u s p i d a l model a s a s t a t i c e q u i l i b r i u m shape.

The r e s u l t s of Our c a l c u l a t i o n s s u g g e s t t h a t t h e dynamical t r e a t m e n t of an e q u i l i b r i u m shape and s t a b i l i t y can r e s o l v e t h e a p p a r e n t c o n t r a d i c t i o n s and/or in- c o n s i s t e n c i e s i n t h e Taylor model. It i s shown t h a t a t t h e o n s e t of i n s t a b i l i t y , t h e ' s t a t i c ' cone deforms i n t o a c u s p i d a l form, a h i g h l y l o c a l i z e d i n s t a b i l i t y i s p r e d i c t e d and l a s t l y t h a t ApfO. It i s a l s o concluded t h a t A p = O i s a n e c e s s a r y b u t n o t s u f f i c i e n t c o n d i t i o n f o r t h e o n s e t of a dynamic i n s t a b i l i t y . This i s i n aqree- ment with Gabovich.[9]

To o b t a i n t h e s e r e s u l t s , we have assumed t h a t t h e most g e n e r a l form of the LY s t r e s s c o n d i t i o n i s

where s u b s c r i p t I o ' r e f e r s t o t h e s t a t i c l i m i t , and ,C2 i s t h e v e l o c i t y p o t e n t i a l . Since t h e f l u i d v e l o c i t y

;=-?a,

^{f l o w} i s no" i n t r o d u c e d i n t o t h e problem. Before d i s c u s s i n g t h e EHD c a l c u l a t i o n s , we b r i e f l y review t h e EHS a n a l y s i s of t h e s t a b i l i t y of t h e Taylor cone; Chung, e t . a1.121 have g i v e n a cornpiete t r e a t m e n t of t h e s t a - b i l i t y of b o t h simple and a r b i t r a r y shaped f l u i d s u r f a c e s i n t h e EHS l i m i t .

III. AN ELECTROHYDROSTATIC ANALYSIS OF THE STABILITY OF SIMPLE COORDINATE SURFACES:

THE TAYLOR CONE.

The o b j e c t i v e of t h i s a n a l y s i s i s to determine t h e s t a b i l i t y of some known s t a t - i c ( o r q u a s i - s t a t i c ) e q u i l i b r i u m shapes. It i s n e c e s s a r y t o know S, which r e q u i r e s a knowledge of t h e c u r v a t u r e and the e l e c t r o s t a t i c f i e l d a p p r o p r i a t e f o r t h a t shape when an e x t e r n a l v o l t a g e is a p p l i e d . I n g e n e r a l , f o r a r b i t r a r y shapes, t h e curva- t u r e and f i e l d s must be obtained n u m e r i c a l l y [ l , 8 1 r and t h e a n a l y s i s f o r t h e o n s e t of i n s t a b i l i t y done by a s p e c i a l procedure of numerical s i m u l a t i o n of t h e f l u i d s u r f a c e a s a f u n c t i o n of a p p l i e d v o l t a g e . This i s d e s c r i b e d i n t h e papers by Chung, e t . a l .

11 i21.

C e r t a i n simple c o o r d i n a t e s u r f a c e s ( o r shapes) a r e e s p e c i a l l y u s e f u l because t h e r e a r e experimental images[5,13] t o s u g g e s t t h a t each of t h e s e shapes h a s been ob- served p r i o r t o o r d u r i n g t h e o n s e t of i n s t a b i l i t y . Furthermore, f o r each of t h e s e c o o r d i n a t e s u r f a c e s , t h e cone, t h e c u s p and t h e hyperboloid of r e v o l u t i o n , t h e curva- t u r e and t h e e l e c t r o s t a t i c f i e l d can be o b t a i n e d a n a l y t i c a l l y , s o t h a t S can be e v a l u a t e d . Therefore, t h e s t a b i l i t y a n a l y s i s can a l s o be done a n a l y t i c a l l y , a l b e i t sometimes approximately.

We h e r e c o n s i d e r o n l y t h e s t a b i l i t y a n a l y s i s of t h e cone usinq t h e Taylor and Zeleny c r i t e r i a . The d e t a i l s a r e q i v e n i n Ref. 2 f o r t h e cone and t h e o t h e r equi- l i b r i u m shapes.

When t h e Taylor c r i t e r i o n ( i . e . ,

m.

2 o r S=0 w i t h Ap=O) i s a p p l i e d , Fiq. 4 re- s u l t s . I n a d d i t i o n , t h e following c o n c l u s i o n s emerge:

i ) The Taylor cone i s a n i s o b a r i c s u r f a c e , with, i n f a c t , PO.

i i ) Other cones w i t h semi-vertex a n g l e n o t e q u a l t o 49.3O a r e

net

i s o b a r i c s u r f a c e s . That i s , Ap v a r i e s i n v a l u e from p o i n t t o p o i n t on t h e s u r f a c e . I n t h e n e x t c o n c l u s i o n , we a n t i c i p a t e r e s u l t s from t h e complete EHS a n a l y s i s of s t a b i l i t y .

iii) The Taylor cone i s unique i n t h a t i t i s t h e o n l y shape of a simple coordi- n a t e system which i s allowed h y d r o s t a t i c a l l y .

f o r ro=lmm and

=

17.1 kV f o r ro=2mm. The measured vcexP

=

4-7 kV[131.

When a p p l i e d t o t h e Taylor cone, t h e Zeleny s t a b i l i t y c r i t e r i o n t a k e s t h e form ( a ~ / a 0 ) & 0 ~ = 0 w h e r e S i s s t i l l g i v e n by Eq. 2 a p p l i e d t o t h e Taylor cone. T h i s can b e s o l v e d f o r t h e c r i t i c a l v o l t a g e

where V o i s t h e n o n - i n t e g r a l index of t h e Legendre f u n c t i o n and r is t h e p o s i t i o n co- o r d i n a t e w i t h o r i g i n a t t h e apex of t h e cone. A n a n a l y s i s of E q . 6 l e a d s t o t h e following conclusions:

i ) (&S/a0),,, <O a s *O, s o t h e cone i s u n s t a b l e n e a r t h e apex. A t r = O ( i . e . , a p e x ) , t h e 0 c o n e spontaneously d i s i n t e g r a t e s .

ii) v C Z = v c Z ( r ) , and t h e r e f o r e p r e d i c t s l o c a l breakdown.

When a s i m i l a r a n a l y s i s i s done f o r t h e c u s p i d a l model a s an e q u i l i b r i u m shape, i t i s found t h a t [ 2 1

i ) The cusp i s n o t an i s o b a r i c s u r f a c e , and t h e r e f o r e cannot be a s t a t i c e q u i l i b r i u m shape.

i i ) The cusp is spontaneously u n s t a b l e n e a r and a t t h e apex. The reason is t h a t t h e high f i e l d s t r e s s e s , due t o t h e s i n g u l a r shape, exceed t h e mechanical s t r e s s e s .

S e v e r a l i m p o r t a n t c o n c l u s i o n s emerge from t h e complete EHS a n a l y s i s . A. Taylor cone model:

1. The parameter vcT, o b t a i n e d from t h e Taylor s t a b i l i t y c o n d i t i o n i s t h e v o l t - age n e c e s s a r y t o form a s t a t i c f l u i d cone w i t h t h e Taylor angle. I t i s n o t t h e c r i t i c a l o r t h r e s h o l d v o l t a g e f o r t h e o n s e t of i n s t a b i l i t y .

2. I n t h e EHS l i m i t t h e e x i s t e n c e of a s i n g u l a r i t y a t t h e apex (whether mathe- m a t i c a l l y , o r " p h y s i c a l l y " induced) i s e s s e n t i a l f o r t h e development of an i n s t a b i l i t y t h e r e .

3. The q u a n t i t y ,

vcZ,

c a l c u l a t e d from t h e use of Zeleny s t a b i l i t y c r i t e r i o n , s ' = O , i s t o b e i n t e r p r e t e d a s t h e a d d i t i o n a l v o l t a g e , o v e r and above vcT f o r t h e f l u i d t o d i s i n t e g r a t e i n t h e EHS model. F u r t h e m o r e , Vc"O a s rtO and no a d d i t i o n a l v o l t a g e beyond

vCT

i s n e c e s s a r y f o r i n s t a b i l i t i e s , t o a r i s e a t t h e apex.

B. Cuspidal shape:

1. ~t o r n e a r o n s e t of i n s t a b i l i t y where c u s p i d a l shapes a r e observed experi- m e n t a l l y i l 3 1 , a n EHD t r e a t m e n t i s n e c e s s a r y t o d e s c r i b e a c c u r a t e l y t h e shape and c r i t i c a l v o l t a g e f o r t h e o n s e t of i n s t a b i l i t y .

C. It i s c o n j e c t u r e d t h a t none of t h e simple c o o r d i n a t e s u r f a c e s , e x c e p t t h e Taylor cone, a r e allowed s t a t i c e q u i l i b r i u m shapes. However, according t o t h e Zeleny c r i t e r i o n , t h e Taylor cone i s spontaneously u n s t a b l e .

D. F i n a l l y , a r b i t r a r y shaped f l u i d s u r f a c e s w i t h a x i a l symmetry e x h i b i t h y d r o s t a t i c e q u i l i b r i u m , b u t o n l y f o r A&O.

C7-356 JOURNAL DE PHYSIQUE

I V . ELECTROHYDRODYNAMIC ANALYSIS OF EQUILIBRIUM SHAPE AND STABILITY

It i s w e l l e s t a b l i s h e d t h a t t h e Taylor cone model has been, and i s u s e f u l f o r t h e s t u d y of some q u a l i t a t i v e and q u a n t i t a t i v e f e a t u r e s of LMIS. However, i t i s a s t a t i c model, and a s h a s been demonstrated, i t h a s i n h e r e n t i n c o n s i s t e n c i e s (e.g., spontaneous d i s i n t e g r a t i o n a t t h e apex). I t i s a p p a r e n t t h a t a dynamical t h e o r y i s n e c e s s a r y f o r a c o r r e c t and a c c u r a t e d e s c r i p t i o n of t h e b a s i c p h y s i c s involved.

S p e c i f i c a l l y , i n o r d e r t o e x p l a i n t h e development and l o c a l i z a t i o n of i n s t a b i l i t i e s , a s w e l l a s t h e dynamical shapes observed e x p e r i m e n t a l l y , a n EHD t r e a t m e n t i s r e q u i r e d . We have t h e r e f o r e used an EHD t h e o r y , v a l i d t o f i r s t - o r d e r i n t h e l i n e a r approxima- t i o n , t o c a l c u l a t e t h e dynamical shape of a three-dimensional s t r e s s e d f l u i d s u r f a c e . I n this a n a l y s i s t h e zero-order ( u n d i s t o r t e d ) s u r f a c e corresponds t o one of t h e coor- d i n a t e s u r f a c e s of a s e p a r a b l e c o o r d i n a t e system (e.g., cone, cusp, etc...). I n t h e c a l c u l a t i o n r e p o r t e d below, t h e l i n e a r i z e d EHD e q u a t i o n s a r e a p p l i e d t o t h e e x a c t Taylor cone model because i t i s a u s e f u l ( i . e . , t r a c t a b l e ) zero-order approximation.

A s w i l l be shown, u s i n g t h i s dynamical t h e o r y , t h e cone deforms i n t o approximately t h e c u s p i d a l shape ur.der t h e p e r t u r b i n g a c t i o n of t h e e x t e r n a l e l e c t r i c f i e l d .

Since t h e c a l c u l a t i o n i s v e r y l e n g t h y and t e d i o u s , although s t r a i g h t f o r w a r d , o n l y t h e forma1 d e t a i l s a r e p r e s e n t e d here. The mathematical a n a l y s i s and c a l c u l a - t i o n a l d e t a i l s w i l l be published elsewhere.

The forma1 s e t of EHD e q u a t i o n s to b e solved a r e :

1. The Laplace Equations f o r t h e e l e c t r i c p o t e n t i a l @, and t h e v e l o c i t y poten- t i a l Q:

v2@

= O Outside t h e conducting f l u i d A'Q = O I n s i d e t h e conducting f l u i d

+ +

where E=-?@ and u=-$Q a r e t h e e l e c t r i c f i e l d and v e l o c i t y of t h e f i e l d , r e s p e c t i v e l y . 2. B e r n o u l l i ' s Equation

13 a0 + %P (VQ)' + P

+

Pgz = Po = c o n s t a n t ( 8 )

This i s t h e e q u a t i o n of motion f o r an incompressible and i r r o t a t i o n a l f l u i d of den- s i t y p .

3. Boundary c o n d i t i o n s on t h e p o t e n t i a l s .

@=vo

on t h e f r e e f l u i d s u r f a c e . ( 9 a )

@=O on t h e r i g i d c o u n t e r - e l e c t r o d e . (9b)

?i*$$d =O, t h e normal component of t h e f l u i d v e l o c i t y i s z e r o on t h e ( 9 c ) a x i s of symmetry.

4. L e t Z ( o , t ) b e a ( s h a p e ) f u n c t i o n which d e s c r i b e s t h e deformation of t h e f l u i d s u r f a c e , and 6=B0 be t h e c o o r d i n a t e d e s c r i b i n g t h e u n d i s t o r t e d f l u i d s u r f a c e . W e d e f i n e a f u n c t i o n F 6

-

Z ( r 2 , t ) - B o , where

B

d e f i n e s t h e deformed s u r f a c e . Then

d e s c r i b e s t h e i n s t a n t a n e o u s shape of t h e f l u i d s u r f a c e . The f u n c t i o n F must s a t i s f y [ 151

5. The time-dependent LY s t r e s s b a l a n c e c o n d i t i o n ,

e v a l u a t e d on t h e (deformed) f r e e s u r f a c e of t h e f l u i d , 6=6,

+

Z ( a , t ) . The "exact"

d i r e c t s o l u t i o n of t h e s e e q u a t i o n s i s impossible. ~ i n e a r i z a t i o n of t h e EHü Eqs.

makes some problems t r a c t a b l e , by which we mean t h a t we can s o l v e t h e e q u a t i o n s f o r approximate s o l u t i o n s t h a t d e s c r i b e t h e p e r t u r b e d motion on t h e f l u i d s u r f a c e i n terms of simple harmonic o s c i l l a t i o n s .

I n s t a b i l i t y i n t h e f l u i d i s a m a n i f e s t a t i o n of developing n o n - l i n e a r i t y i n t h e s u r f a c e waves. Therefore, t o f i n d t h e i n s t a b i l i t y , it is n e c e s s a r y t o c o n s i d e r non- p e r i o d i c motion d e s c r i b e d by t h e higher-order c o r r e c t i o n s t o t h e l i n e a r i z e d EHü eqs.

The procedure f o r o b t a i n i n g and s o l v i n g t h e l i n e a r i z e d EHù e q s . t o f i r s t - o r d e r i s a g a i n s t r a i g h t f o r w a r d , b u t v e r y t e d i o u s . A d e t a i l e d mathematical t r e a t m e n t i s given by Chung[l6] and w i l l be published elsewhere. We h e r e merely n o t e t h a t t h e procedure f i r s t e n t a i l s expanding t h e deformation and p o t e n t i a l s a s sums of 'even' and 'odd' c o n t r i b u t i o n s . Assuming t h e deformation Z t o be s m a l l , t h e p o t e n t i a l s a r e expanded i n a Taylor s e r i e s a b o u t t h e u n d i s t o r t e d s u r f a c e , and s u b s t i t u t e d i n t o t h e s e t of t h e EHD e q u a t i o n s and boundary c o n d i t i o n s . By e q u a t i n g terms of e q u a l o r d e r i n t h e r e s u l t i n g s e t of e q u a t i o n s , a new s e t of e q u a t i o n s i s o b t a i n e d f o r each of t h e s u c c e s s i v e l y higher-order c o r r e c t i o n s t o t h e p o t e n t i a l s and deformations, i n terms of lower-order c o n t r i b u t i o n s . Therefore to s o l v e t h e e q u a t i o n s t o f i r s t - o r d e r , we need t h e zeroth-order s o l u t i o n s f o r Q> and Cl on t h e undeformed s u r f a c e . !&en t h e undeform- ed s u r f a c e s correspond t o simple c o o r d i n a t e s u r f a c e s (e.g., cone, c u s p s , e t c . . . ) , t h e s o l u t i o n s f o r O o ( R o ) a r e known, and t h e f i r s t - o r d e r e q u a t i o n s can b e solved f o r t h e p o t e n t i a l s O l ( C l l ) and deformation 21.

I t can be shown[l6] t h a t t h e f i r s t - o r d e r s t r e s s c o n d i t i o n y i e l d s the d i s p e r s i o n r e l a t i o n f o r w, t h e frequency ( o r energy) a s s o c i a t e d w i t h a p e r t u r b e d s u r f a c e wave.

From t h e c o n d i t i o n u2=0, t h e c r i t i c a l v o l t a g e f o r t h e o n s e t of an EHD i n s t a b i l i t y i s o b t a i n e d

.

V. RESULTS AND CONCLUSIONS

The above mathematical procedure was f i r s t a p p l i e d t o 2-dimensional hydrodynamic f l u i d geometries c o n s i s t i n g of a p l a n a r s u r f a c e w i t h a normal e l e c t r i c f i e l d and a c y l i n d r i c a l f l u i d w i t h a r a d i a l f i e l d [ l 6 ]

,

r e s p e c t i v e l y . For l i q u i d Ga, Vc -" 50 kV and 35 kv f o r t h e 2-dimensional geometries.

We used t h e Taylor cone model i n t h e EHD c a l c u l a t i o n s f o r t h r e e reasons: 1. I t i s a simple c o o r d i n a t e s u r f a c e and t h e r e f o r e t h e zeroth-order s o l u t i o n s f o r t h e poten- t i a l s a r e known. 2. It i s an allowed EHS e q u i l i b r i u m shape, and hence a good zeroth- o r d e r model f o r a dynamical t r e a t m e n t . 3. 1t i s a ' r e a l ' 3-dimensional model, and, t o Our knowledge, t h e f i r s t - o r d e r c o r r e c t i o n s have n o t been a p p l i e d t o any t h r e e - dimensional geometry.

Consider a s m a l l deformation c ( r , t ) o f t h e l i q u i d e l e c t r o d e a b o u t t h e s t a t i c Taylor cone. We d e s c r i b e t h e deformation by O = O o + < ( r , t ) , 00=130.70 f o r t h i s model, Eqs. 9a and 11 a r e a p p l i e d a t @=Oo+<, Eq. 9b a t r=ro [P_i,(cos 0 0 ) ] - 2 and Eq. 9c a t 0=s. ( 1 71. The t e c h n i c a l d e t a i l s and s o l u t i o n s a r e d i s c i s s e d elsewhere. Here, we merely n o t e t h a t a r a t h e r complicated d i s p e r s i o n r e l a t i o n y i e l d s t h e following e x p r e s s i o n f o r t h e c r i t i c a l v o l t a g e :

which can be compared w i t h experiment and t h e o t h e r t h e o r e t i c a l e x p r e s s i o n s , vcT and

vCZ

(Eqs. 4 and 5 ) .

Vc i s r a t h e r i n s e n q i t i v e t o t h e r e s t r i c t e d v a l u e s of t h e two parameters s and a , where r t a < r o and OtstO.Z+n, ~ t t l . For l i q u i d G a , r =2mm, and ( r / a ) = 0 . 5 , Vc=5.5 kV.

It can be shown[l6] t h a t i n t h e l i m i t ( r / a ) + l , t h e c r i t i c a l v o l t a g e has t h e c o n s t a n t v a l u e Vc=5.8 kv, which is i n v e r y good'agreement with vcexP 1: 4-7 kV. For 0 t h e r

C7-358 JOURNAL

DE

PHYSIQUE

l i q u i d m e t a l s ,

vc

s c a l e s a s JT.

Since Vc

-

rs/2, t h e i n s t a b i l i t y i s h i g h l y l o c a l i z e d , which a g r e e s with experi- mental o b s e r v a t i o n s [ 1 3 1

.

F i n a l l y , t h e c a l c u l a t e d deformation of t h e cone shows t h a t i t deforms i n t o a c u s p i d a l shape. This c u s p i d a l shape a g r e e s w e l l w i t h r e c e n t o b s e r v a t i o n s of Ben Assayag, e t . a l . [ 1 3 ] , and p r e d i c t i o n s of Chung, e t . a l . [141 and of Kingham and Swanson [ 1 8 1

.

I n summary, t h e r e s u l t s of t h e p r e s e n t electrohydrodynamic c a l c u l a t i o n s have demonstrated t h a t an a p r i o r i t r e a t m e n t of t h e e q u i l i b r i u m shape and s t a b i l i t y of a s t r e s s e d conducting f l u i d i s f e a s i b l e . I n a s p e c i f i c a p p l i c a t i o n , we have p r e d i c t e d t h e dynamic shape and c r i t i c a l v o l t a g e of an o p e r a t i n g LMIS a t t h e o n s e t of i n s t a b i l - i t y t h a t i s i n good agreement w i t h e x p r i m e n t .

ACKNOWLEDGEMENTS

One of t h e a u t h o r s (P.H.C.) would l i k e t o e x p r e s s h i s s i n c e r e thanks to D r S . H.

Launois, and P. Sudraud of t h e CNRS L a b o r a t o i r e d e M i c r o s t r u c t u r e s e t d e Microelec- t r o n i q u e , Bagneux, France, f o r t h e i r generous h o s p i t a l i t y d u r i n g t h e Summer, 1986, when t h i s paper was w r i t t e n . He would a l s o l i k e t o acluiowledge u s e f u l c o n v e r s a t i o n with D r . G. Ben Assayag and a s s i s t a n c e from M. J. Gierak.

REFERENCES

/1/ P. H. C u t l e r , M. Chung, T. E. Feuchtwang and E. KaZeS, Jour. d e physique Colloque C2,

g

(1986) c2-87.

/2/ M. Chung, P. H. C u t l e r and N. M. Miskovsky, submitted to J. Vac. Sci. and Technol

.

^{( 1986)}

.

/3/ See, f o r example, A. Wagner, Appl. Phys.

3,

(1982) 440.

/4/ L. D. Landau and E. M. ~ i f s c h i t z , F l u i d Mechanics

e erga am on

P r e s s , New York, 19591, Chapter III.

/5/

sir.

G. I. Taylor, Proc. ROY. soc., London

e,

(1964) 383.

/6/ J. Zeleny, Proc. Camb. p h i l . Soc.

E,

(1915) 71.

/ 7 / M. Chung, P. H. C u t l e r , T. E. Feuchtwang, N. M. Miskovsky and N. S u j a t h a ,

Scanning E l e c t r o n Microscopy I V , (1984) 1547.

/8/ G. H. J o f f r e , J o u r . d e Mecanique t h e o r i q u e e t a p p l i q u e s

3,

(1984) 545.

(The a u t h o r s wish t o c o r r e c t a m i s i n t e r p r e t a t i o n of Ref. 8 p r e s e n t e d i n Ref. 1 above; s p e c i f i c a l l y , J o f f r e does i n c l u d e h y d r o s t a t i c p r e s s u r e i n h i s a n a l y s i s . This i s shown e x p l i c i t l y i n G. H. J o f f r e and M. Cloupeau, t o be published i n t h e proceedings of t h e 1986 IFES B e r l i n , m l y 7-11.)

/9/ M. D. Gabovich, Sov. Phys. Usp.

26

( 5 ) (1983) 447.

/IO/ H. G o l d s t e i n , C l a s s i c a l Mechanics (Addison-wesley Pub. Co., Reading, MA, 1950).

/ I l / P. G. Drazin and W. H. Reid, Hydrodynamic S t a b i l i t y (Cambridge U n i v e r s i t y P r e s s , Cambridge, 1981 ) Chap. 1.

/12/ N. M. Miskovsky, P. H. C u t l e r and E. KaZeS, J. Vac. Sci. Technol. (1983) 202.

/13/ G. Ben Assayag, P. Sudraud and B. J o u f f r e y , ultramicroscopy (1985) 1.

/14/ M. Chung, P. H. C u t l e r , T. E. Feuchtwang, E. KazeS and N. M. Miskovsky, Jour.

d e physique Colloque Cg, 45, (1984) Cg-153.

/15/ A.

L.- ett ter

^and

3.

D. walecka, T h e o r e t i c a ï Mechanics of P a r t i c i e s and c o n t i n u a ( ~ c ~ r a w - H i l l , New York, 1980). Chap. 10.

/16/ M. Chung, Ph.D. T h e s i s i n p h y s i c s , August, 1986, The ~ e n n s y l v a n i a S t a t e univer- s i t y , and t o b e published.

/17/ M. Chung, P. H. C u t l e r , T. E. Feuchtwang and N. M. Miskovsky, Jour. d e physique Colloque Cg,

g

(19841, Cg-1 53 ( s e e Fig. 7 a ) .

/18/ D. R. Kingham and L. W. Swanson, M p l . Phys.

A39

(1984) 123.