ON THE SHAPE OF A LIQUID-METAL FIELD-ION EMITTER

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ON THE SHAPE OF A LIQUID-METAL FIELD-ION EMITTER

R. Forbes

To cite this version:

R. Forbes. ON THE SHAPE OF A LIQUID-METAL FIELD-ION EMITTER. Journal de Physique

Colloques, 1984, 45 (C9), pp.C9-161-C9-166. �10.1051/jphyscol:1984927�. �jpa-00224407�

JOURNAL DE PHYSIQUE

Colloque C9, supplément au n°12, Tome 45, décembre 1984 page C9-161

ON THE SHAPE OF A L I Q U I D - M E T A L F I E L D - 1 ON EMITTER

R.G. F o r b e s

University of Surrey, Department of Electronic and Electrical Engineering, Guildford, Surrey GU2 5XH, U.K.

Résumé - Une source d'ions à métal liquide en fonctionnement a la forme d'une protubérance sur un cône. Les raisons possibles de cette géométrie sont discu- tées et les difficultés associées' à la théorie de la stabilité électromécanique des liquides conducteurs sont présentées.

Abstract - An operating liquid-metal field-ion source has the shape of a cusp on a cone. Possible reasons for this are discussed, and difficulties with the theory of the electromechanical stability of conducting liquids are displayed.

The experimental observations of Zeleny / I / , Taylor / 2 / , and others on electrified fluids showed the presence of cusps and/or jets at the apex of the conical liquid body formed in the presence of a high electric field. Taylor's theory / 2 / success- fully explained the half-angle of the underlying cone, and his theoretically predicted shape has come to be known as the Taylor cone.

The shape of the apex of an operating liquid-metal field-ion source (LMFIS) is of current interest, isomer's theory of LMFIS operation / 3 / assumed this would be a rounded Taylor cone. Electron microscope observations by Clampitt and coworkers /4,5/ had, in fact, already shown that an operating caesium LMFIS had the shape of a cusp on a cone, and it was subsequently shown by Gaubi et al. / 6 / that an operating gold source also had this shape. There are also theoretical reasons for thinking this shape plausible for an operating LMFIS: it has been shown /7-10/

that a rounded Taylor cone could not reasonably support observed emission currents at the fields necessary to sustain field evaporation at its apex.

Thus, although there may be smaller-scale microprotrusions on the rounded cusp apex /8,ll,l2/, it can be assumed that an operating LMFIS has the basic shape of a cusp on a cone. The precise physical reasons for this, however, remain to be agreed. The intentions of this paper are to summarise the possibilities, and look again at some of the relevant theory.

I - BASIC THEORY

(1). First consider the elementary arguments associated with surface tension. In fig.l suppose that an area of surface 6A is created by "pulling sideways", with force T per unit length. The work w done is TySx. If y denotes the surface energy per unit area (in the absence of any field), then clearly:

w = Ty6x = T6A = y6A (1) Surface tension T is equal to the (chemical) surface energy per unit area, y.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984927

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( 2 ) . N e x t consider t h e p r e s s u r e d i f f e r e n c e a c r o s s a curved s u r f a c e i n local equilib- rium. I suarmarise Gibbs' d e r i v a t i o n /13/. Suppose t h a t a s m a l l element ( o f a r e a S ) o f a curved l i q u i d s u r f a c e expands forward t o a new p o s i t i o n , as shown i n f i g . 2 . Let t h e p r i n c i p a l c u r v a t u r e s o f t h e o r i g i n a l s u r f a c e be c l and c l , and suppose t h a t every p o r t i o n o f t h e s u r f a c e moves outwards along its l o c a l normal by d i s t a n c e 6n.

me volume 6v through which the s u r f a c e h a s moved is approximately s6n; and t h e i n c r e a s e 6 s i n a r e a is ( ( c l + c 2 ) s 613).

I f hp denotes t h e mechanical p r e s s u r e d i f f e r e n c e between t h e two s i d e s o f t h e s u r f a c e ( w i t h Ap p o s i t i v e if t h e p r e s s u r e "inside" is t h e g r e a t e r ) then t h e change 6E i n system f r e e energy a s s o c i a t e d with t h e movement of t h e s u r f a c e element S is:

For t h e element t o be i n equilibrium, we must have aE/an = 0 ; hence we g e t t h e elementary formula f o r p r e s s u r e d i f f e r e n c e a c r o s s a s u r f a c e i n equilibrium:

I FIGURES: These schematic diagrams illustrate arguments in the text

1.

r

2.

s+Ss

curvatures Cl ,C2

3. ^Length

T = y

4* I P ~ E ~ F = ~ A

^X

^sina A

T 5P T

r (r si nc(12x Ap

( 3 ) . I n t h e case of a sphere o f r a d i u s R, we o b t a i n Ap = 2y/R. T h i s r e s u l t can a l s o be obtained by t r e a t i n g t h e s u r f a c e t e n s i o n as a f o r c e , and considering the e q u i l i b r i u m o f a hemisphere as shown i n f i g . 3 .

( 4 ) . NOW consider how t o introduce e l e c t r i c a l e f f e c t s , assuming a conducting l i q u i d . The s i m p l e s t approach, followed by Zeleny, Taylor and o t h e r s , is t o note t h a t on every s m a l l a r e a o f s u r f a c e t h e r e is an outwards p r e s s u r e ( 1 / 2 ) e O ~ 2 (where F is t h e l o c a l f i e l d s t r e n g t h ) , and t o assume t h a t t h e new formula f o r t h e equilibrium mechanical p r e s s u r e d i f f e r e n c e is:

( 5 ) . Now consider t h e mechanical s t a b i l i t y o f a Taylor cone, as i l l u s t r a t e d i n f i g . 4 . From e l e c t r o s t a t i c a n a l y s i s , Taylor showed t h a t t h e f i e l d a t t h e cone s u r f a c e would be p r o p o r t i o n a l t o r-112 i f t h e half-angle a had a s p e c i f i c v a l u e c l o s e t o 49.30, and t h e surrounding e l e c t r o d e had t h e shape o f one o f the r e l e v a n t family o f e q u i p o t e n t i a l s . L e t T a y l o r ' s f i e l d d i s t r i b u t i o n be FT = ~ ~ 1 1 2 . Two approaches are now p o s s i b l e , analogous t o the approximations I and ^I1 used by Taylor i n h i s a n a l y s i s o f spheroidal-drop s t a b i l i t y . Approach I, used i n T a y l o r ' s cone a n a l y s i s /2/, i s t o assume t h a t e q . ( 4 ) g i v e s t h e equilibrium p r e s s u r e

d i f f e r e n c e a c r o s s t h e curved cone s u r f a c e . This y i e l d s :

Approach I1 balances t h e f o r c e s shown i n f i g . 4 . W e assume that p r e s s u r e is uniform i n space o u t s i d e t h e cone, and t h a t w i t h i n t h e cone t h e p r e s s u r e is uniform i n any plane normal t o t h e axis. Ap is t h u s t h e mechanical p r e s s u r e d i f f e r e n c e between an i n t e r n a l p o i n t i n t h e plane shown and any e x t e r n a l p o i n t . I n t e g r a t i n g t h e outwards e l e c t r i c a l l y - i n d u c e d f o r c e s g i v e s a r e s u l t a n t f o r c e p a r a l l e l t o t h e cone a x i s o f 2 r ~ r s i n 2 a . Force balance parallel t o t h e a x i s t h e n y i e l d s :

The d i f f e r e n c e between eqns ( 5 ) and ( 6 ) is a new r e s u l t .

T a y l o r ' s r e s u l t is now obtained as follows. P r e s s u r e i s uniform ( i n p r a c t i c e z e r o ) i n space o u t s i d e t h e cone; t h u s t h e cone as a whole w i l l not be i n e q u i l i b r i u m u n l e s s Ap i s independent o f r. W e f o r c e this by t a k i n g t h e parameter T t o be c o n s t a n t ( e q u a l t o y ) , and s e t t i n g t h e expression i n square b r a c k e t s e q u a l t o z e r o , t h u s making Ap z e r o f o r all r. This g i v e s a value f o r B, and hence /2/ a necessary ( i f e q u i l i b r i u m is t o e x i s t ) value f o r the v o l t a g e VT between t h e l i q u i d cone and h i s surrounding e l e c t r o d e .

I1 - POSSIBLE REASONS FOR CUSP FORMATION

We can now d i s c u s s reasons f o r t h e formation o f a cusp on a cone r a t h e r t h a n a e x a c t Taylor cone. Several p o s s i b i l i t i e s e x i s t . To consider t h e first two it is easiest t o assume t h a t t h e exact Taylor cone is a s t a t i c equilibrium configuration a t a c e r t a i n v o l t a g e VT. (Really, t h i s r e q u i r e s t h a t t h e system e l e c t r o d e s have t h e a p p r o p r i a t e shape, b u t t h i s p o i n t is not important h e r e . )

1 ) W e r v o l t i n g . Take a Taylor cone and suppose t h a t t h e a p p l i e d v o l t a g e is increased s l i g h t l y from VT. The f i e l d a t each p o i n t on t h e cone w i l l be increased by a s m a l l amount 6F t h a t v a r i e s from p o i n t t o p o i n t , and a t each p o i n t is such t h a t ~ F / F T = ~V/VT = r) , where r) is a s m a l l c o n s t a n t . The excess p r e s s u r e i n s i d e t h e cone a t d i s t a n c e r from t h e apex is now given by:

C9-164 JOURNAL DE PHYSIQUE

C l e a r l y t h e p r e s s u r e d e c r e a s e s as we approach t h e cone apex. Liquid would tend t o move towards t h e apex. Thus we would expect a cusp t o form and grow as voltage is increased f r o m t h e v a l u e VT. Even i f a Taylor cone is n o t t h e e x a c t e q u i l i b r i u m shape a t any v o l t a g e , one might expect a cusp o f s i z e i n c r e a s i n g w i t h v o l t a g e , due t o dynamic e f f e c t s . Kingham and Swanson have modelled t h i s /10/.

2 ) I n s t a b i l i t y coupled with dynamic overshoot. A l t e r n a t i v e l y , suppose t h a t t h e v o l t a g e is such t h a t a Taylor cone would be i n equilibrium, b u t t h a t the cone h a s t o be formed by the d i s t o r t i o n o f a l i q u i d f i l m i n i t i a l l y adhering t o t h e s u r f a c e o f an underlying needle. As the l i q u i d d i s t o r t s , it w i l l p i c k up k i n e t i c energy.

Thompson and Prewett have modelled this process /14/. Even i f t h e l i q u i d t e n d s i n t o t h e shape o f a cone, i t w i l l overshoot. The q u e s t i o n is whether a cusp, once formed, would continue t o grow,

-

o r whether it would c o l l a p s e back i n t o a Taylor cone. I f it would continue t o grow, then t h e cone is u n s t a b l e with r e s p e c t t o cusp formation, and we should not r e a l l y expect t o observe the e x a c t Taylor cone. The q u e s t i o n f o r t h i s hypothesis is: Is t h e Taylor cone an u n s t a b l e equilibrium 7 3 ) Inadequacy i n surface-tension theory. A more r a d i c a l hypothesis, r a i s e d by C u t l e r and coworkers /15/, i s t h a t T a y l o r ' s approach t o surface-tension theory i s not f u l l y c o r r e c t , and t h a t e q . ( 4 ) does n o t c o r r e c t l y g i v e t h e e q u i l i b r i u m mechan- i c a l p r e s s u r e d i f f e r e n c e a c r o s s a curved s u r f a c e when t h a t s u r f a c e is h i g h l y charged. I n t h i s c a s e t h e Taylor-cone c o n f i g u r a t i o n i s presumed n o t to be an e q u i l i b r i u m shape, and hence would b e expected t o d i s t o r t . This suggestion i s d i s t i n c t from t h e discrepancy between eqns ( 5 ) and ( 6 ), which a l s o might be thought t o i n d i c a t e some inadequacy i n t h e theory.

A l l the p h y s i c a l i n f l u e n c e s mentioned might c o n t r i b u t e t o observed LMFIS shapes, and p o s s i b l y o t h e r f a c t o r s t o o . However, w e now look more c l o s e l y a t t h e t h i r d hypothesis above, because i t h a s i m p l i c a t i o n s f o r all t h e o r i e s o f LME'IS shape.

111

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^{ANO!PIER LOOK} AT TEE SUFFAC5: TENSION OF CaARGED LIQUIDS

The work o f Rayleigh /16,17/ on charged d r o p l e t s is g e n e r a l l y h e l d t o be c o r r e c t i n p r i n c i p l e . From t h i s it is clear t h a t one p o s s i b l e approach is t o i n c l u d e a potential-energy term r e l a t i n g t o t h e e l e c t r i c a l f r e e energy o f t h e system.

I n o u r case t h e l i q u i d e l e c t r o d e is held by a b a t t e r y a t a constant v o l t a g e V r e l a t i v e t o its surroundings. I f a change i n shape o f t h e l i q u i d electrode

i n c r e k s e s its e l e c t r i c a l capacitance, r e l a t i v e t o t h e surroundings, then a charge 6Q flows through t h e b a t t e r y and two energy changes r e s u l t . The p o t e n t i a l energy s t o r e d i n t h e c a p a c i t o r c o n s t i t u t e d by t h e l i q u i d e l e c t r o d e and its surroundings i n c r e a s e s by (1/2)V6Q; and t h e p o t e n t i a l energy o f t h e b a t t e r y decreases by VSQ.

The n e t change i n t h e e l e c t r i c a l f r e e energy o f t h e system is -(1/2)V6Q.

Now c o n s i d e r a p l a n a r arrangement as i l l u s t r a t e d i n f i g . 5 , with t h e l i q u i d elec- trode c o n s t i t u t i n g t h e upper e l e c t r o d e , and being of f i n i t e s i z e . By analogy w i t h t h e argument used earlier, consider a new a r e a o f surface c r e a t e d i n t h e middle s e c t i o n o f t h e l i q u i d e l e c t r o d e by " p u l l i n g sideways" with a f o r c e T p e r u n i t l e n g t h . There are now two energy terms a s s o c i a t e d w i t h t h e a r e a 6A o f new s u r f a c e c r e a t e d : a n i n c r e a s e i n energy y6A as before, associated with t h e breaking o f bonds between atoms as t h e surface is formed; and t h e e l e c t r i c a l term discussed above. I f we may assume t h a t t h e i n c r e a s e i n capacitance is €06A/b, where d is t h e e l e c t r o d e s e p a r a t i o n , then - i f P is t h e f i e l d i n t h e c e n t r e o f t h e region between t h e e l e c t r o d e s

-

simple a l g e b r a l e a d s t o t h e r e s u l t :

It is clear from f i g . 5 t h a t t h e r e is an downwards pressure o f ( 1 / 2 ) e O ~ 2 on t h e charged l i q u i d e l e c t r o d e , due t o the "pull" o f the lower e l e c t r o d e ' s charge. So it seems t h a t , i n t h i s quasi-planar c o n f i g u r a t i o n , two e l e c t r i c a l e f f e c t s exist: the outwards p r e s s u r e on t h e s u r f a c e ,

and

a r e d u c t i o n i n t h e s u r f a c e t e n s i o n T.

A t t h i s s t a g e it c l e a r l y becomes necessary t o ask whether Taylor was c o r r e c t i n assuming t h a t e q . ( 4 ) would g i v e t h e equilibrium p r e s s u r e d i f f e r e n c e a c r o s s a charged s u r f a c e , o r whether it w a s c o r r e c t i n e q . ( 6 ) t o set T = y. The main a l t e r n a t i v e is t o t a k e T as a non-constant quantity, which would be c o n s i s t e n t with t h e d i s c u s s i o n leading t o eq.(8) and with t h e arguments of C u t l e r and coworkers.

Currently, I do not know what approach is c o r r e c t . Discussion continues below, b u t f i r s t look at p o s s i b l e consequences of t a k i n g T as non-constant i n t h e context of e q . ( 6 ) . Suppose T has t h e form:

where t i s a correction, presumed p o s i t i v e , t h a t depends on f i e l d , voltage and p o s s i b l y o t h e r f a c t o r s . We cannot reasonably assume t h a t t h a s the same form f o r a cone as f o r t h e Eig.5 geometry, b u t i t seem reasonable t o assume t h a t i t decreases i n s i z e with decrease i n l o c a l f i e l d . I n t h i s case t would become negligible i n comparison with y a t s u f f i c i e n t l y l a r g e d i s t a n c e r, and we should expect t h e d i s t a n t p a r t s of t h e l i q u i d body t o have t h e Taylor configuration. On t h e o t h e r hand, l a r g e t might be p r e d i c t e d near t h e apex o f a Taylor cone, making T s m a l l o r negative t h e r e . The Taylor cone could not then be i n s t a t i c equilibrium near the apex: dynamic e f f e c t s , and divergence from c o n i c a l geometry, might be expected.

For argument, suppose t h a t T f o r a Taylor cone w e r e given by e q . ( 8 ) . It is r e a d i l y shown t h a t t h e d i s t a n c e r a t which T would become z e r o i s given by:

Voltages i n t h e range 5-10 kV lead, f o r t y p i c a l y-values (-1 N/m), t o r-values o f o r d e r 100-500 m. But t h e r e s u l t s o f Gaubi e t al. /6/ show t h a t i n t h e l i m i t o f low LMFIS emission c u r r e n t , c o n i c a l geometry is w e l l e s t a b l i s h e d a t d i s t a n c e s o f a t most a few p n from t h e geometrical t i p apex. The experiments would seem t o confirm that, even i f eq. ( 9 ) be q u a l i t a t i v e l y c o r r e c t , eq. ( 8 ) is not a p p l i c a b l e t o a cone.

I t is, of course, conceivable t h a t i n t h e above arguments t h e concept of surface t e n s i o n is being used i n v a l i d l y , and t h a t t h i s is responsible f o r t h e discrepancy between eqns ( 5 ) and ( 6 )

.

(Though t h i s assumption would a l s o raise doubts about t h e v a l i d i t y of T a y l o r ' s approximation I1 i n h i s spheroidal drop a n a l y s i s . ) On the o t h e r hand, it s e e m s clear t h a t T a y l o r ' s argument is incomplete and t h a t t h e v a l i d i t y o f eq. ( 4 ) ought t o be proved. Let us attempt t o do t h i s by incorporating a Raleigh-type electrical term i n t o t h e Cibbs-type argument used earlier.

IV

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AN EZECTRICAL VERSION OF THE GIBES APPROACH

I f q i s t h e s u r f a c e charge a s s o c i a t e d with element of a r e a S, and 6 q i s t h e i n c r e a s e i n s u r f a c e charge when t h e element moves forward by a d i s t a n c e 6n. then equation ( 2 ) i s replaced by:

Proceding as before would give a term involving aq/an. A l t e r n a t i v e l y , w e may write q=us, where U is t h e s u r f a c e charge d e n s i t y on t h e l i q u i d e l e c t r o d e . P u t t i n g 6q=u6s+s6u, and r e q u i r i n g t h a t aE/an be zero, w e o b t a i n a f t e r some manipulation:

The f i r s t r h s term c l e a r l y involves t h e "reduced s u r f a c e tension" t h a t appears i n e q . ( 8 ) , b u t t h e r e i s now an e x t r a term. Evaluation o f eq.(12) can be done i n the two simple cases, giving:

C9-166 JOURNAL DE PHYSIQUE

p a r a l l e l p l a t e s : &p =

-

^(1/2

^{OF^}

^{( 1 3 )}

sphere : AP = 2y/R

-

(1/2 )€OF' ( 1 4 )

These r e s u l t s are t h e same as assumed i n elementary theory. B u t i n more general cases t h e second term i n eq.(12) i s very d i f f i c u l t t o e v a l u a t e , because t h e q u a n t i t y aF/an seems not t o be well-defined. aF/an seems t o depend, not only on what i s happening t o t h e l o c a l small element o f s u r f a c e , b u t a l s o on how t h e rest of t h e s u r f a c e behaves.

I t is important t o note t h a t both terms i n e q . ( l 2 ) h a v e combined t o g i v e t h e F2 term i n e q . ( 1 4 ) . So it i s not g e n e r a l l y p o s s i b l e t o i n t e r p r e t t h e second term i n eq.(12) as a simple "hydrostatic pressure" term. This i s c o n s i s t e n t with t h e supposition i n t h e previous s e c t i o n , t h a t T i n e q . ( 6 ) is not given by e q . ( 9 ) . B u t n e i t h e r have I been a b l e t o prove t h a t t h e two e l e c t r i c a l t e r m i n e q . ( l l ) , taken together, i n g e n e r a l produce t h e F2 term t h a t appears i n eq. ( 4 ) . The implication is t h a t , as of now, I am unable t o prove Taylor c o r r e c t .

V

-

CONCLUSIONS

From t h i s work I draw two main conclusions:

(1) There is no d i f f i c u l t y i n accounting f o r t h e observed cusp-on-a-cone shape o f an o p e r a t i n g -1s.

( 2 ) There s e e m t o be unresolved o b s c u r i t i e s i n t h e t h e o r y o f t h e s u r f a c e t e n s i o n of charged l i q u i d s , and derived phenomena. I n p a r t i c u l a r , t h e discrepancy between eqns ( 5 ) and ( 6 ) remains t o be f u l l y explained, and I am c u r r e n t l y unable t o prove t h a t t h e basic equation ( 4 ) , assumed a - p r i o r i by Taylor i n h i s cone a n a l y s i s , is t h e correct equilibrium condition f o r a charged l i q u i d s u r f a c e . Worse, I am unable t o prove t h a t any well-defined equilibrium condition i n f a c t e x i s t s f o r shapes o t h e r than t h e sphere o r p a r a l l e l - p l a t e configuration. Conceivably t h e r e is some flaw i n t h e thermodynamic arguments i n s e c t i o n IV; b u t perhaps we should at least contemplate t h e hypothesis t h a t charged-liquid shapes s u b s t a n t i a l l y d i f f e r e n t from t h e t w o simple cases may not be able t o exist i n static equilibrium.

I warmly acknowledge many s t i m u l a t i n g d i s c u s s i o n s with D r Graeme L R Mair.

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