in defence of the taylor cone model: application to …
TRANSCRIPT
IN DEFENCE OF THE TAYLOR CONE MODEL: APPLICATION TO LIQUID METAL
ION SOURCESSubmitted on 1 Jan 1984
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
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/r2 ) - p = \ eQ 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
2 surface, p is the excess pressure inside the surface and the term & E E is the
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 -QofS 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 reason f o r i t s i n c l u s i o n i s no t c l e a r because i n n e i t h e r t h e experiments of Taylor nor i n normal ope ra t ion of LMI sources 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. Tay lo r ' s apparatus 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 ressu re d i f f e r e n c e s , bu t t h i s does not amount t o a volume c o n s t r a i n t . I n normal LMI sources t h e f l u i d i s aga in supp l i ed from a r e s e r v o i r a t t h e ambient p ressu re and t h e smal l s i z e of t h e source means t h a t g r a v i t y can be neglected. The volume c o n s t r a i n t , which is c o r r e c t f o r a n i s o l a t e d drop is inappropr ia t e f o r a f l u i d i n con tac t wi th a r e se rvo i r . By inc lud ing t h i s volume c o n s t r a i n t i t i s apparent t h a t Su ja tha e t a l . a r e cons ide r ing an 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 considered by Taylor.
Su ja tha e t a 1 c la im 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 equat ions by applying them t o t h e case of a charged d rop le t . Not s u r p r i s i n g l y they f i n d t h e equ i l ib r ium shape is a sphere , but t h i ~ t r i v i a l case i s h a r d l y a proper v e r i f i c a t i o n of t h e i r eq. 12 when a l l bu t two of t h e terms a r e zero. Furthermore t h e i r eq. 9 was apparen t ly der ived assuming a cons tan 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 drop i s no t connected t o any b a t t e r y , but has cons tan t charge ins t ead .
3. APPLICATION TO, AND EXPERIMENTAL OBSERVATIONS OF, LIQUID METAL I O N SOURCES
The t r ea tmen t s of both Su ja tha e t a l . and Taylor only c la im t o be v a l i d f o r t h e s t a t i c equ i l ib r ium of a conducting f l u i d i n an e l e c t r i c f i e l d and ca re should be t aken when making comparison with opera t ing LMI sources . Su ja tha e t a l . suggest t h a t obse rva t ions of o p e r a t i n g sources /5,6/ i n d i c a t e " t h a t t h e equ i l ib r ium shape be fo re t h e onset of i n s t a b i l i t i e s , i . e . c u r r e n t flow, i s no t w e l l r ep resen ted , o r even genera l ly dep ic ted by a Taylor 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 equ i l ib r ium shape can e x i s t and Thompson and Prewett 171 suggest t h a t t h e onse t of i o n emission occurs be fo re t h e ion 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 Taylor cone, o r any o t h e r s t a t i c model shape, can only be meaningfully compared wi th an opera t ing 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 ion emi t t ing region. Observations of f rozen- in LMIS cones /a / , and they a r e indeed cones a p a r t from some rounding nea r t h e apex, do show a half -angle remarkably c l o s e t o Tay lo r ' s p red ic ted 49.3O. These obse rva t ions may be c r i t i c i s e d because t h e l i q u i d shape cannot be accura te ly maintained dur ing t h e f r e e z i n g process and t h i s l e d Sudraud and co-workers /5,6/ t o use i n - s i t u e l e c t r o n microscopy f o r LMIS observat ion. I n r e f . 5 they show a s e r i e s of p i c t u r e s of a gold LMIS from befo re onse t t o a cu r ren t i n excess of 100 uA. The opera t ing LMIS appears b a s i c a l l y c o n i c a l wi th a p ro t rus ion a t t h e apex which inc reases i n s i z e wi th inc reas ing 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 appear s l i g h t l y convex, but a s t h e c u r r e n t inc reases , t h e p ro t rus ion grows, t h e s i d e s become concave and t h e shape i s cusp- l ike . Su ja tha e t a l . claim t h a t " these images suggest a cusp a s t h e most probable conf igura t ion of t h e l i q u i d surface" , but a convincing cusp shape is only shown a t h igh c u r r e n t s , f a r from t h e equ i l ib r ium s i t u a t i o n t h a t they c la im t o be consider ing.
We have observed an LMIS of Au on a needle c o n s i s t i n g of a 49' cone on t h e end of a l mm wire wi th t h e end of t h e cone t runca ted i n a c i r c l e of d iameter 0.25 mm. This shape of need le was chosen t o g ive approximately t h e r i g h t boundary cond i t ions f o r t h e Taylor cone hypothesis 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 obse rva t ion of t h e l i q u i d shape. The ba re needle i s shown i n f i g . 1 and t h e opera t ing LMIS i n f i g . 2. The LMIS shape i s very c l o s e indeed t o a Taylor 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 not resolved.
Kingham and Swanson /9/ have developed a simple f l u i d dynamic t reatment of LMIS i n an a t tempt t o go beyond s t a t i c equ i l ib r ium models. The i r work favours a model of LMIS shape 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 Taylor cone, wi th the p ro t rus ion inc reas ing i n s i z e a s t h e cu r ren t inc reases . Th i s i s n i c e l y i n agreement wi th Sudraud's obse rva t ions /5,6/. The j e t - l i k e p ro t rus ion model i s c e r t a i n l y cusp-like a t h igh c u r r e n t s , but d i f f e r s s i g n i f i c a n t l y from Su ja tha e t a l . ' s conclusions because it reduces t o t h e Taylor cone model i n t h e s t a t i c l i m i t . We no te t h a t the j e t - l i k e p ro t rus ion model overcomes t h e problem, mentioned by Su ja tha e t a l , t h a t t h e Tayfor cone shape cannot s u s t a i n t h e ion c u r r e n t s observed i n LMIS. This problem had previously been addressed by Kang and Swanson /10/ who suggested a
C9-142 JOURNAL DE PHYSIQUE
Fig . 1 - A tungsten needle made from a 1 m diameter wire ground t o a cone of ha l f angle 49' w i t h a 0 .25 mm f l a t on the end.
Fig. 2 - An operating Au l iquid metal ion source supported on the needle shown i n f i g . 1. The sca le i s the same i n both f igures.
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.
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
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/r2 ) - p = \ eQ 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
2 surface, p is the excess pressure inside the surface and the term & E E is the
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 -QofS 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 reason f o r i t s i n c l u s i o n i s no t c l e a r because i n n e i t h e r t h e experiments of Taylor nor i n normal ope ra t ion of LMI sources 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. Tay lo r ' s apparatus 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 ressu re d i f f e r e n c e s , bu t t h i s does not amount t o a volume c o n s t r a i n t . I n normal LMI sources t h e f l u i d i s aga in supp l i ed from a r e s e r v o i r a t t h e ambient p ressu re and t h e smal l s i z e of t h e source means t h a t g r a v i t y can be neglected. The volume c o n s t r a i n t , which is c o r r e c t f o r a n i s o l a t e d drop is inappropr ia t e f o r a f l u i d i n con tac t wi th a r e se rvo i r . By inc lud ing t h i s volume c o n s t r a i n t i t i s apparent t h a t Su ja tha e t a l . a r e cons ide r ing an 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 considered by Taylor.
Su ja tha e t a 1 c la im 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 equat ions by applying them t o t h e case of a charged d rop le t . Not s u r p r i s i n g l y they f i n d t h e equ i l ib r ium shape is a sphere , but t h i ~ t r i v i a l case i s h a r d l y a proper v e r i f i c a t i o n of t h e i r eq. 12 when a l l bu t two of t h e terms a r e zero. Furthermore t h e i r eq. 9 was apparen t ly der ived assuming a cons tan 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 drop i s no t connected t o any b a t t e r y , but has cons tan t charge ins t ead .
3. APPLICATION TO, AND EXPERIMENTAL OBSERVATIONS OF, LIQUID METAL I O N SOURCES
The t r ea tmen t s of both Su ja tha e t a l . and Taylor only c la im t o be v a l i d f o r t h e s t a t i c equ i l ib r ium of a conducting f l u i d i n an e l e c t r i c f i e l d and ca re should be t aken when making comparison with opera t ing LMI sources . Su ja tha e t a l . suggest t h a t obse rva t ions of o p e r a t i n g sources /5,6/ i n d i c a t e " t h a t t h e equ i l ib r ium shape be fo re t h e onset of i n s t a b i l i t i e s , i . e . c u r r e n t flow, i s no t w e l l r ep resen ted , o r even genera l ly dep ic ted by a Taylor 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 equ i l ib r ium shape can e x i s t and Thompson and Prewett 171 suggest t h a t t h e onse t of i o n emission occurs be fo re t h e ion 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 Taylor cone, o r any o t h e r s t a t i c model shape, can only be meaningfully compared wi th an opera t ing 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 ion emi t t ing region. Observations of f rozen- in LMIS cones /a / , and they a r e indeed cones a p a r t from some rounding nea r t h e apex, do show a half -angle remarkably c l o s e t o Tay lo r ' s p red ic ted 49.3O. These obse rva t ions may be c r i t i c i s e d because t h e l i q u i d shape cannot be accura te ly maintained dur ing t h e f r e e z i n g process and t h i s l e d Sudraud and co-workers /5,6/ t o use i n - s i t u e l e c t r o n microscopy f o r LMIS observat ion. I n r e f . 5 they show a s e r i e s of p i c t u r e s of a gold LMIS from befo re onse t t o a cu r ren t i n excess of 100 uA. The opera t ing LMIS appears b a s i c a l l y c o n i c a l wi th a p ro t rus ion a t t h e apex which inc reases i n s i z e wi th inc reas ing 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 appear s l i g h t l y convex, but a s t h e c u r r e n t inc reases , t h e p ro t rus ion grows, t h e s i d e s become concave and t h e shape i s cusp- l ike . Su ja tha e t a l . claim t h a t " these images suggest a cusp a s t h e most probable conf igura t ion of t h e l i q u i d surface" , but a convincing cusp shape is only shown a t h igh c u r r e n t s , f a r from t h e equ i l ib r ium s i t u a t i o n t h a t they c la im t o be consider ing.
We have observed an LMIS of Au on a needle c o n s i s t i n g of a 49' cone on t h e end of a l mm wire wi th t h e end of t h e cone t runca ted i n a c i r c l e of d iameter 0.25 mm. This shape of need le was chosen t o g ive approximately t h e r i g h t boundary cond i t ions f o r t h e Taylor cone hypothesis 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 obse rva t ion of t h e l i q u i d shape. The ba re needle i s shown i n f i g . 1 and t h e opera t ing LMIS i n f i g . 2. The LMIS shape i s very c l o s e indeed t o a Taylor 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 not resolved.
Kingham and Swanson /9/ have developed a simple f l u i d dynamic t reatment of LMIS i n an a t tempt t o go beyond s t a t i c equ i l ib r ium models. The i r work favours a model of LMIS shape 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 Taylor cone, wi th the p ro t rus ion inc reas ing i n s i z e a s t h e cu r ren t inc reases . Th i s i s n i c e l y i n agreement wi th Sudraud's obse rva t ions /5,6/. The j e t - l i k e p ro t rus ion model i s c e r t a i n l y cusp-like a t h igh c u r r e n t s , but d i f f e r s s i g n i f i c a n t l y from Su ja tha e t a l . ' s conclusions because it reduces t o t h e Taylor cone model i n t h e s t a t i c l i m i t . We no te t h a t the j e t - l i k e p ro t rus ion model overcomes t h e problem, mentioned by Su ja tha e t a l , t h a t t h e Tayfor cone shape cannot s u s t a i n t h e ion c u r r e n t s observed i n LMIS. This problem had previously been addressed by Kang and Swanson /10/ who suggested a
C9-142 JOURNAL DE PHYSIQUE
Fig . 1 - A tungsten needle made from a 1 m diameter wire ground t o a cone of ha l f angle 49' w i t h a 0 .25 mm f l a t on the end.
Fig. 2 - An operating Au l iquid metal ion source supported on the needle shown i n f i g . 1. The sca le i s the same i n both f igures.
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.