IN DEFENCE OF THE TAYLOR CONE MODEL : APPLICATION TO LIQUID METAL ION SOURCES

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IN DEFENCE OF THE TAYLOR CONE MODEL : APPLICATION TO LIQUID METAL ION SOURCES

D. Kingham, A. Bell

To cite this version:

D. Kingham, A. Bell. IN DEFENCE OF THE TAYLOR CONE MODEL : APPLICATION TO LIQUID METAL ION SOURCES. Journal de Physique Colloques, 1984, 45 (C9), pp.C9-139-C9-144.

�10.1051/jphyscol:1984924�. �jpa-00224404�

JOURNAL DE PHYSIQUE

Colloque C9, supplément au n°12, Tome 45, décembre 1984 page C9-139

IN DEFENCE OF THE TAYLOR CONE MODEL : APPLICATION TO LIQUID METAL ION SOURCES

D.R. Kingham* and A.E. Bell

+

Cavendish Laboratory, Madingley Road, Cambridge CBS OHE, U.K.

^Oregon Graduate Center, 19600 W Walker Road, Beaverton, OR 97006, U.S.A.

Résumé - Sujatha et al ont récemment suggéré que l'hypothèse du cône de Taylor était incorrecte. Nous avons examiné leurs arguments et trouvé que la situation physique qu'ils considèrent est fondamentalement différente de celles dans la- quelle s'est placé Taylor et de celle qui intervient dans une source ionique à métal liquide. Nous présentons des résultats expérimentaux pour étayer l'hypo- thèse du cône de Taylor et attribuons les écarts observés par rapport à des effets dynamiques qui n'ont pas plus été pris en considération par Taylor que par Sujatha et al.

Abstract - Sujatha et al. have recently suggested that the Taylor cone hypothesis is incorrect. We examine their arguments and suggest that the physical situation that they consider is essentially different from that considered by Taylor and from that occuring in a real liquid metal ion source

(LMIS). We present experimental support for the Taylor cone hypothesis and we attribute observed deviations from the ideal Taylor cone shape in LMIS to dynamic effects which were not considered either by Taylor or by Sujatha et al.

In a recent publication Sujatha et al./l/ have strongly suggested that there is no justification for the Taylor cone hypothesis 111 which has been used as a standard model for the shape of liquid metal ion sources (LMIS) /3,4/. Sujatha et al. also claim to have derived equations for the equilibrium configuration of a conducting fluid in an electric field, which cannot be satisfied by a cone of any angle. This work was presented at the 30th IFES where it attracted much interest. In this paper we examine the basis of their arguments which, we suggest, are incorrect in some respects.

1. CRITICISM OF THE TAYLOR CONE MODEL

Sujatha et al.'s major criticism of the Taylor cone model is that Taylor erroneously omitted the pressure difference term in the Laplace formula. It is certainly true, as Sujatha et al. point out, that Taylor stresses the significance of the pressure difference term when considering the stability of ellipsoidal droplets and that he omits this term when considering a conical shape. Our interpretation o.f Taylor's work, however, is that this omission was both deliberate and correct! Taylor's equation of equilibrium has the form

T ( l/

ri

+ l/r

2

) - p = \ e

^Q

E

2

(1)

where T is the surface tension, r and r are the principal radii of curvature of the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984924

C9-l40 JOURNAL DE PHYSIQUE

surface, p is the excess pressure inside the surface and the term &

E

E is the

2

stress due to the electric field. This equation can certainly be satisfied by an isolated, charged, spherical drop with a constant excess internal pressure. For a conical shape, however, the situation is different and the cone must be either infinite in extent or else some awkward termination of the cone must be arranged.

Taylor treated the infinite case where the surface tension and electric stress terms must tend to zero at large distances from the cone apex as the surface radius of curvature becomes very large. Thus one of the boundary conditions on the problem is zero pressure difference at infinity. In the static case there can be no pressure differences within the fluid so the excess pressure must be zero throughout the cone and the equilibrium condition reduces to

where r2 is infinite for a cone. This condition is satisfied by the Taylor cone shape (i.e. a cone of half-angle 49.3") with a counter electrode of the idealised form r

=

ro/(P, (cos ell2 at a particular value of potential difference between liquid and

4

counter electrode

Sujatha et ales second criticism of Taylor's analysis is that he uses only an approximate solution to the electrostatic cone problem. However, Taylor's potential, V

=

A + ^{B$ P,} (cos B), is the exact solution to the problem he considered, of

4

infinite cone and idealised, infinite counter-electrode. Of course, it is difficult to perform experiments on infinite systems, but Taylor's apparatus was ingeniously designed to simulate an infinite system as far as possible by eliminating any finite edge effects and he certainly succeeded in observing cones of half angle very close to the predicted 49.3O.

2. SUJATHA ET AL.'S VARIATIONAL FORMULATION

Sujatha et al. attempt to improve on Taylor's analysis by deriving variational equations for the static equilibrium condition for a fixed, finite, volume of fluid under purely electrostatic stress. They claim that the energy integral to be minimised is

1'

=

Is T ds - Q o f S u d s + X I d v (3)

where

U

is the surface charge density, V is the constant potential difference maintained by a battery and ds and dv are surface and volume elements of the fluid.

The first term, fs Tds, represents the surface free energy, although they seem

somehow to neglect the base surface of the cone. The last term represents the

supposed c o n s t r a i n t on volume. The r e a s o n f o r i t s i n c l u s i o n i s n o t c l e a r b e c a u s e i n n e i t h e r t h e e x p e r i m e n t s of T a y l o r n o r i n normal o p e r a t i o n of LMI s o u r c e s i s t h e r e a f i x e d c o n s t r a i n t on f l u i d volume. T a y l o r ' s a p p a r a t u s was of macroscopic dimensions s o t h a t he had t o a d j u s t t h e l e v e l of f l u i d i n h i s r e s e r v o i r t o o f f s e t g r a v i t a t i o n a l p r e s s u r e d i f f e r e n c e s , b u t t h i s does n o t amount t o a volume c o n s t r a i n t . I n normal LMI s o u r c e s t h e f l u i d i s a g a i n s u p p l i e d from a r e s e r v o i r a t t h e ambient p r e s s u r e and t h e s m a l l s i z e of t h e s o u r c e means t h a t g r a v i t y can be n e g l e c t e d . The volume c o n s t r a i n t , which i s c o r r e c t f o r a n i s o l a t e d d r o p i s i n a p p r o p r i a t e f o r a f l u i d i n c o n t a c t w i t h a r e s e r v o i r . By i n c l u d i n g t h i s volume c o n s t r a i n t i t i s a p p a r e n t t h a t S u j a t h a e t a l . a r e c o n s i d e r i n g a n e s s e n t i a l l y d i f f e r e n t problem t o t h a t c o n s i d e r e d by T a y l o r .

S u j a t h a e t a 1 c l a i m t o v e r i f y t h e v a l i d i t y of t h e i r v a r i a t i o n a l e q u a t i o n s by a p p l y i n g them t o t h e c a s e of a charged d r o p l e t . Not s u r p r i s i n g l y t h e y f i n d t h e e q u i l i b r i u m s h a p e i s a s p h e r e , b u t t h i ~ t r i v i a l c a s e i s h a r d l y a p r o p e r v e r i f i c a t i o n of t h e i r eq. 1 2 when a l l b u t two of t h e t e r m s a r e z e r o . Furthermore t h e i r eq. 9 was a p p a r e n t l y d e r i v e d assuming a c o n s t a n t p o t e n t i a l d i f f e r e n c e whereas t h e i s o l a t e d d r o p i s n o t connected t o any b a t t e r y , b u t h a s c o n s t a n t c h a r g e i n s t e a d .

3. APPLICATION TO, AND EXPERIMENTAL OBSERVATIONS OF, LIQUID METAL I O N SOURCES The t r e a t m e n t s of b o t h S u j a t h a e t a l . and T a y l o r o n l y c l a i m t o b e v a l i d f o r t h e s t a t i c e q u i l i b r i u m of a c o n d u c t i n g f l u i d i n a n e l e c t r i c f i e l d and c a r e s h o u l d be t a k e n when making comparison w i t h o p e r a t i n g LMI s o u r c e s . S u j a t h a e t a l . s u g g e s t t h a t o b s e r v a t i o n s of o p e r a t i n g s o u r c e s / 5 , 6 / i n d i c a t e " t h a t t h e e q u i l i b r i u m s h a p e b e f o r e t h e o n s e t of i n s t a b i l i t i e s , i . e . c u r r e n t f l o w , i s n o t w e l l r e p r e s e n t e d , o r even g e n e r a l l y d e p i c t e d by a T a y l o r cone". However, i t seems u n l i k e l y t h a t any t r u e l y s t a t i c e q u i l i b r i u m s h a p e c a n e x i s t and Thompson and P r e w e t t

171

s u g g e s t t h a t t h e o n s e t of i o n e m i s s i o n o c c u r s b e f o r e t h e i o n e m i t t i n g f e a t u r e h a s f u l l y formed. The T a y l o r cone, o r any o t h e r s t a t i c model s h a p e , c a n o n l y b e m e a n i n g f u l l y compared w i t h a n o p e r a t i n g LMIS i n t h e l i m i t of low c u r r e n t , o r a t l a r g e d i s t a n c e s from t h e i o n e m i t t i n g r e g i o n . O b s e r v a t i o n s of f r o z e n - i n LMIS c o n e s

/a/,

and t h e y a r e i n d e e d cones a p a r t from some rounding n e a r t h e apex, do show a h a l f - a n g l e remarkably c l o s e t o T a y l o r ' s p r e d i c t e d 49.3O. These o b s e r v a t i o n s may b e c r i t i c i s e d b e c a u s e t h e l i q u i d s h a p e cannot b e a c c u r a t e l y m a i n t a i n e d d u r i n g t h e f r e e z i n g p r o c e s s and t h i s l e d Sudraud and co-workers / 5 , 6 / t o u s e i n - s i t u e l e c t r o n microscopy f o r LMIS o b s e r v a t i o n . I n r e f . 5 t h e y show a s e r i e s of p i c t u r e s of a g o l d LMIS from b e f o r e o n s e t t o a c u r r e n t i n e x c e s s of 100 uA. The o p e r a t i n g LMIS a p p e a r s b a s i c a l l y c o n i c a l w i t h a p r o t r u s i o n a t t h e apex which i n c r e a s e s i n s i z e w i t h i n c r e a s i n g c u r r e n t . A t low c u r r e n t s t h e s i d e s of t h e cone a p p e a r s l i g h t l y convex, b u t a s t h e c u r r e n t i n c r e a s e s , t h e p r o t r u s i o n grows, t h e s i d e s become concave and t h e shape i s c u s p - l i k e . S u j a t h a e t a l . c l a i m t h a t " t h e s e images s u g g e s t a c u s p a s t h e most p r o b a b l e c o n f i g u r a t i o n of t h e l i q u i d s u r f a c e " , b u t a c o n v i n c i n g c u s p s h a p e i s o n l y shown a t h i g h c u r r e n t s , f a r from t h e e q u i l i b r i u m s i t u a t i o n t h a t t h e y c l a i m t o b e c o n s i d e r i n g .

We have observed an LMIS of Au on a n e e d l e c o n s i s t i n g of a 49' cone on t h e end of a l mm w i r e w i t h t h e end of t h e cone t r u n c a t e d i n a c i r c l e o f d i a m e t e r 0.25 mm. T h i s s h a p e of n e e d l e was chosen t o g i v e a p p r o x i m a t e l y t h e r i g h t boundary c o n d i t i o n s f o r t h e T a y l o r cone h y p o t h e s i s and t h e s i z e was chosen l a r g e enough f o r o p t i c a l o b s e r v a t i o n of t h e l i q u i d shape. The b a r e n e e d l e i s shown i n f i g . 1 and t h e o p e r a t i n g LMIS i n f i g . 2. The LMIS s h a p e i s v e r y c l o s e i n d e e d t o a T a y l o r cone, though d e t a i l s of t h e e m i t t i n g a r e a a r e n o t r e s o l v e d .

Kingham and Swanson / 9 / have developed a s i m p l e f l u i d dynamic t r e a t m e n t of LMIS i n a n a t t e m p t t o go beyond s t a t i c e q u i l i b r i u m models. T h e i r work f a v o u r s a model of LMIS s h a p e c o n s i s t i n g of a j e t - l i k e p r o t r u s i o n on t h e end of a T a y l o r cone, w i t h t h e p r o t r u s i o n i n c r e a s i n g i n s i z e a s t h e c u r r e n t i n c r e a s e s . T h i s i s n i c e l y i n agreement w i t h Sudraud's o b s e r v a t i o n s / 5 , 6 / . The j e t - l i k e p r o t r u s i o n model i s c e r t a i n l y c u s p - l i k e a t h i g h c u r r e n t s , b u t d i f f e r s s i g n i f i c a n t l y from S u j a t h a e t a l . ' s c o n c l u s i o n s b e c a u s e i t r e d u c e s t o t h e T a y l o r cone model i n t h e s t a t i c l i m i t . We n o t e t h a t t h e j e t - l i k e p r o t r u s i o n model overcomes t h e problem, mentioned by S u j a t h a e t a l , t h a t t h e Tayfor cone s h a p e cannot s u s t a i n t h e i o n c u r r e n t s observed i n LMIS. T h i s problem had p r e v i o u s l y been a d d r e s s e d by Kang and Swanson /10/ who s u g g e s t e d a

C9-142 JOURNAL DE PHYSIQUE

F i g .

1 -

A tungsten n e e d l e made from

a 1

m diameter wire ground t o a cone o f h a l f a n g l e 49' w i t h a 0 . 2 5 mm f l a t on t h e end.

F i g . 2 - An operating Au l i q u i d metal i o n source supported on t h e needle shown i n

f i g .

1. The s c a l e i s t h e same i n both f i g u r e s .

C9-144

JOURNAL DE PHYSIQUE

cylindrical protrusion shape as a possible solution.

It is important to note that, in contrast to Taylor, Sujatha et al. do at least attempt to use a method which would allow the stability of the fluid shape to be considered. Taylor simply showed that electric and surface tension forces could be made to balance all over a conical shape of half-angle 49.3", but he did not show that this configuration was theoretically in stable equilibrium. Instead he turned to experiment in order to demonstrate that the shape was stable, apart from some jetting at the apex.

4. CONCLUSION

In conclusion we find that the Taylor cone hypothesis is justified in the static limit of infinite cone and counter electrode of the idealised form, in contradiction to the conclusions of Sujatha et al. We find that Taylor's omission of the pressure difference term in the Laplace formula and his use of only a single term in the Legendre function expansion for the electrostatic potential are correct in this idealised case. In the non-ideal case of operating LMI sources there is both experimental and theoretical evidence to support a jet-like protrusion model of source shape which approaches a Taylor cone shape in the low current limit, or at large distances from the ion emitting region.

Acknowledgements - One of us (DRK) is grateful for financial support from a Royal Society University Research Fellowship. This work was partially supported by National Science Foundation Grant No. ELS-8303095

REFERENCES

*Permanent address from October 1984: VG Scientific, The Birches Industrial Estate, Imberhorne Lane, East Grinstead, Sussex, RH19 IUB, U.K.

1. Sujatha N., Cutler P.H., Kazes E., Rogers J.P. and Miskovsky N.M., Applied Phys.

A32 (1983) 55.

2. ~ a ~ l o G . ~ . , Proc. Roy. Soc. London (1964) 383.

3. Gomer R., Applied Phys. 19 (1979) 365.

4. Prewett P.D., Mair G.L.R. and Thompson S.P., J. Phys. D: Applied Phys. 15 (1982) 1339.

5. Gaubi H., Sudraud P., Tencd M. and Van de Walle J., Proc. 29th Int. Field Emission Symposium, GBteborg, Sweden, eds. H-0. Andren and H. Norden

(Almqvist and Wiksell, Stockholm, 1982).

6. Sudraud P., reported at 30th Int. Field Emission Symposium, Philadelphia, (1983) 7. Thompson S.P. and Prewett P.D., J. Phys. D: Applied Phys., to be published.

8. Aitken K.L., Jeffries D.K. and Clampitt R., UKAEA Culham Laboratory Report CLM/RR/E1/21, unpublished (1975).

9. Kingham D.R. and Swanson L.W., Applied Phys. A, to be published.

10. Kang N.K. and Swanson L.W., Applied Phys. (1983) 95.

note - The diameter of the tungsten needle shown in fig. 1 and shown coated in gold in fig. 2 was incorrectly reported at the symposium. DRK apologises for this error.

The correct value is a needle of diameter of 1 mm with the flat on the end of the

needle having a diameter of 0.25 m.