SOLUTION OF LAPLACE'S EQUATION FOR A RIGID CONDUCTING CONE AND PLANAR COUNTER-ELECTRODE : COMPARISON WITH THE SOLUTION TO THE TAYLOR CONICAL MODEL OF A FIELD EMISSION LMIS

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SOLUTION OF LAPLACE’S EQUATION FOR A RIGID CONDUCTING CONE AND PLANAR COUNTER-ELECTRODE : COMPARISON WITH

THE SOLUTION TO THE TAYLOR CONICAL MODEL OF A FIELD EMISSION LMIS

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

To cite this version:

M. Chung, P. Cutler, T. Feuchtwang, N. Miskovsky. SOLUTION OF LAPLACE’S EQUATION FOR A RIGID CONDUCTING CONE AND PLANAR COUNTER-ELECTRODE : COMPARISON WITH THE SOLUTION TO THE TAYLOR CONICAL MODEL OF A FIELD EMISSION LMIS.

Journal de Physique Colloques, 1984, 45 (C9), pp.C9-145-C9-152. �10.1051/jphyscol:1984925�. �jpa-

00224405�

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Colloque C9, supplément au n°12, Tome 45, décembre 1984 page C9-145

SOLUTION OF L A P L A C E ' S EQUATION FOR A R I G I D CONDUCTING CONE AND PLANAR COUNTER-ELECTRODE: COMPARISON WITH THE SOLUTION TO THE TAYLOR CONICAL MODEL OF A F I E L D EMISSION L M I S +

M. Chung, P.H. C u t l e r , T.E. Feuchtwang and N.M. MiskOvsky

Department of Physios, The Pennsylvania State University,, University Park, Pennsylvania 16802, U.S.A.

Résumé - Bien que le problème électrostatique d'un cône conducteur rigide et d'une contre-électrode plane n'ait jamais été résolu exactement, nous montrons, en accordant les solutions à courte et à grande distance, que le champ au voisinage de l'apex est donné approximativement par :

Il est aussi démontré que la solution de Taylor au problème électrostatique d'un cône rigide et d'une contre-électrode plane ne satisfait pas l'équation de Laplace pour les configurations réelles d'électrodes utilisées dans les sources d'ions à métal liquide et dans d'autres expériences sur des fluides conducteurs soumis à des champs électrostatiques.

Abstract - Although the e l e c t r o s t a t i c problem of a rigid conducting cone and i n f i n i t e planar counter-electrode has never been solved exactly, we show, by matching the near and far s o l u t i o n s , that the f i e l d in the v i c i n i t y of the apex i s approximately given by

I t i s also e x p l i c i t l y demonstrated that Taylor's solution t o the e l e c t r o s t a t i c problem of a r i g i d cone and non-planar counter-electrode model does not s a t - i s f y the Laplace Equation for the actual electrode configurations used in f i e l d emission liquid metal ion sources and other experiments on e l e c t r o s t a - t i c a l l y stressed conducting f l u i d s .

I . INTRODUCTION

in this paper we present a solution of the electrostatic problem of a conducting cone and infinite planar counter-electrode. Although this conical configuration was assumed by Taylor / l / and others /2,3/ to be the equilibrium shape of a conducting fluid in an electric field prior to and at onset of instability, the difficulty in solving Laplace's equation for the exact geometry and other constraints / 4 / , prompted Taylor to choose a different model for the counter-electrode. Using the conical shape for the electrostatically stressed fluid, Taylor invoked dimensionality arguments to show that the only field distribution consistent with a cone that satisfied his form of the Laplace condition is given by

(1) where notation is defined in Ref. 1. It then follows that the counter-electrode necessary to produce this field is

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Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984925

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C9-146 JOURNAL DE PHYSIQUE

The non-planar n a t u r e of t h i s e l e c t r o d e i s c l e a r l y i l l u s t r a t e d i n F i g u r e l a which is a r e p r o d u c t i o n of T a y l o r ' s a p p a r a t u s /l/ designed t o "produce" t h e c o n i c a l f i e l d g i v e n by Eq. ( 1 ) . F i g u r e l b i s a schematic of t h e s e e l e c t r o d e s ; n o t a t i o n f o r t h e a n a l y s i s t o be d e s c r i b e d i s g i v e n i n t h e f i g u r e . Also i l l u s t r a t e d i s T a y l o r ' s pro- posed c o n i c a l shape of t h e conducting f l u i d s u r f a c e j u s t p r i o r t o t h e o n s e t of in- s t a b i l i t y . I t i s important t o n o t e t h a t i n t h e experimental procedure used by Taylor, he f i l l e d t h e t o p of t h e t r u n c a t e d cone, denoted by C, with p r e c i s e l y t h e volume of f l u i d n e c e s s a r y t o complete t h e apex of t h e cone. This was done t o e n s u r e t h a t t h e r e would b e e x a c t l y t h e c o n i c a l shape t o s a t i s f y Eqs. ( 1 ) and ( 2 ) .

We now demonstrate t h a t T a y l o r ' s s o l u t i o n t o t h e e l e c t r o s t a t i c problem of a r i g i d c o n e a > d i d e a l i z e d c o u n t e r - e l e c t r o d e does not s a t i s f y t h e Laplace e q u a t i o n f o r t h e

a c t u a l experimental c o n f i g u r a t i o n s f o r f i e l d emission l i q u i d metal i o n s o u r c e s / S / and o t h e r experiments on e l e c t r o s t a t i c a l l y s t r e s s e d conducting f l u i d s / 6 / . The geometry common t o t h e s e experiments c o n s i s t s o f e s s e n t i a l l y an i n f i n i t e p l a n a r - c o u n t e r - e l e c t r o d e c e n t e r e d above t h e s o u r c e . T h i s i s i l l u s t r a t e d s c h e m a t i c a l l y i n F i g . 2a, t h e corresponding i d e a l i z a t i o n a s a r i g i d cone i s shown i n F i g . 2b. A q u a l i t a t i v e p l o t of t h e p o t e n t i a l d i s t r i b u t i o n i s a l s o d e p i c t e d i n F i g . 2b. Although T a y l o r ' s s o l u t i o n , given by E q . (1) ( h e r e a f t e r denoted b.y v T ( ~ , 8 ) E vT) s a t i s f i e s t h e D i r i c h l e t b o u n d a r y c o n d i t i o n V = 0 , on t h e i d e a l i z e d c o u n t e r - e l e c t r o d e , t h e p o t e n t i a l

vT

never s a t i s f i e s t h i s boundary c o n d i t i o n on a p 1 nar c o u n t e r - e l e c t r o d e . S i n c e t h e c o n s t a n t i n ~ q . ( 1 ) i n e a s i l y shown t o be - V ~ / R ~ ? ~ , i t follows from t h e a p p l i c a t i o n of t h e boundary c o n d i t i o n on t h e p l a n a r e l e c t r o d e t h a t vT = V. [l - p112 COS^) ~ e c ~ / ~ 8 l should be e q u a l t o z e r o f o r a l l 8. However' t h e b r a c k e t term cannot be z e r o s i n c e

i s never e q u a l t o s e c - 1 / 2 8 f o r any v a l u e o f 8. I n E q . ( 3 ) E and K a r e t h e e l l i p t i c i n t e g r a l s o f t h e 1 s t and 2nd k i n d r e s p e c t i v e l y / 7 / .

PLANAR COUNTER -ELECTRODE

R. l

Fig. l a . Diagram of T a y l o r ' s experimental a p p a r a t u s /l/. The s u r f a c e A i s t h e t r u n c a t e d cone of h a l f - a n g l e 4 9 . 3 " ; t h e c o u n t e r - e l e c t r o d e i s t h e metal s u r f a c e B.

l b . .Schematic comparison of t h e Taylor e l e c t r o d e c o n f i g u r a t i o n and t h e p l a n a r c o u n t e r - e l e c t r o d e model.

COUNTER-ELECTRODE

v,o

/

t -

FIELD LINES

Fig. 2a. Schematic of t h e e l e c t r o d e con- f i g u r a t i o n i n a t y p i c a l LMIS.

2b. The cone and p l a n a r e l e c t r o d e model of an LMIS.

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cone and p l a n a r c o u n t e r - e l e c t r o d e and show t h a t t h e r e s u l t i n g f i e l d s n e a r t h e apex d i f f e r s i g n i f i c a n t l y , q u a l i t a t i v e l y and q u a n t i t a t i v e l y , from T a y l o r ' s s o l u t i o n given by Eq. (1).

Consider f i r s t t h e g e n e r a l s o l u t i o n of L a p l a c e ' s e q u a t i o n i n s p h e r i c a l c o o r d i n a t e s and with azimuthal symmetry,

s u b j e c t t o t h e boundary c o n d i t i o n s V = 0 on t h e c o u n t e r - e l e c t r o d e and V = V0 on t h e cone. The model and c o o r d i n a t e s a r e d e f i n e d i n F i g . 3 . The g e n e r a l s o l u t i o n i s

V ( R , B ) =

C

( A ~ R ~

+

B ~ R - ~ - ~ ) P ~ ( c o s ~ ) ( 5 )

v

with Rev

>

^-

-2.

1 / 8 / .

S i n c e t h e e l e c t r o d e s do n o t correspond t o any of t h e c o o r d i n a t e s u r f a c e s i n t h e known s e p a r a b l e c o o r d i n a t e systems, t h e r e i s no c l o s e d form s o l u t i o n f o r t h e p o t e n t i a l V(R,8). However, t h e s p h e r i c a l c o o r d i n a t e system was s e l e c t e d i n t h i s problem because t h e c o o r d i n a t e s u r f a c e 8 = corresponds t o t h e s u r f a c e of t h e cone.

Fig. 3. schematic diagram d e f i n i n g t h e n o t a t i o n and domains i n t h e s o l u t i o n of Eq. ( 4 ) . To approximate t h e s o l u t i o n we u s e t h e

PLANAR COUNTER-ELECTRODE following procedure:

T E N T I * ~ 1. Solve L a p l a c e ' s e q u a t i o n e x a c t l y i n t h e i n s i d e domain o r t h e n e a r r e g i o n of t h e apex of t h e cone ( s e e F i g . 3 ) . 2. Approximate t h e s o l u t i o n i n t h e o u t s i d e

domain with t h e e x a c t s o l u t i o n of t h e

CONTACTE R "contacted" cone problem, which i s de-

s c r i b e d below.

3. Match t h e two s o l u t i o n s on t h e bound- a r y s u r f a c e between t h e two r e g i o n s t o determine t h e expansion c o e f f i c i e n t s f o r t h e s o l u t i o n i n t h e i n s i d e domain.

1 ^- -^ v^ I

-

^I

d d

/

/ ,F- CONE

i

Fig. 4a. A schematic i l l u s t r a t i o n of t h e "contacted" cone model.

4b. A schematic diagram i l l u s - t r a t i n g t h e boundary matching procedure.

I n t h e i n s i d e o r n e a r r e g i o n , where t h e i n f i n i t e s e r i e s i n i n v e r s e powers of R must v a n i s h , t h e e x a c t s o l u t i o n i s

where R&

,

which d e n o t e s t h e "boundary s u r f a c e , " i s d e f i n e d i n Fig. 4a. The V s ' s a r e s o l u t i o n s of t h e e q u a t i o n

Pvs(cos80) = 0 f o r 80 = n

-

a

,

( 7 1 and d e f i n e t h e z e r o s of t h e Legendre func- t i o n s . P v s l s , c f n o n - i n t e g r a l index / g / .

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C9-148 JOURNAL DE PHYSIQUE

This condition on the Vs's ensures that the boundary condition V = V0 is satisfied on the cone. In the outside or far regions, that is R > R&

,

the solution V(R.8) is approximated by the exact potential for a "contacted" cone, which is schematically shown in Figure 4b. As seen in the figure, the contacted cone is obtained by dis- placing the actual cone an upward distance R0 sec along the edge of the cone. It is assumed that the two "contacted" electrodes are separated by an infinitesimally thin insulating layer. The resulting configuration of electrodes in just a special case of the biconical antenna /10/ where the upper "cone" has been folded out with a half- angle of ~/2.

The exact solution to this problem, which is independent of R, is given by

with prime denoting the "contacted" cone coordinates and other symbols are defined in Fig. 4a. It is evident that for R'>>RO

,

this solution is a good approximation for the separated cone problem. Therefore, using the following relations between the coordinates for the contacted and separated cones

R'

-

R C O S ~ + R O

tan

-

₂⁼_{Rsine +}_Rotana _(9a)

and

we obtain

-

Rcose + R.

v'( B')

=

^V(R.0)⁼^Voln _Rsine₊_Rotancl

]An

^[t*

^g)

(10) To find the expansion coefficients in the solution for the near region, we match Eqs.

(6) and (10) on the "boundary surface" defined by R' = R&

.

The matching prodecure is done on different "boundaries" (i.e., different values of R&) and with different numbers of sample points until the coefficients are stable to within 0.1%. This procedure is illustrated schematically, with sample points, in Fig. 4b. The resulting stable solution is

With this solution, the boundary conditions are satisfied exactly on the cone, and are stable to within 0.1% on the infinite planar counter-electrode.

The field components corresponding to the potential given by Eq. (11) are

Ee =

-'p)

^R

^37,

⁼-[$)[0.46029 P-o-~(cos~)-cos p~.~(cOse)

sine

[k--O.q ^.

In order to compare with Taylor's results, we list below the corresponding fields for the potential, given by Eq. (l):

- - - =

-

0.5[$-)[ P-o.~(cos~)-cos Po.~(cos~) -0.5

R

a

sine

IM

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It is particularly important to note that although the first two terms of the solution for the planar counter-electrode model

LE^.

(11)) have the same functional form as the Taylor solution for the curved counter-electrode (Eq. (l)), the coefficients of the second terms differ by about 10%. Furthermore the higher order terms in Eq.

(11) lead to a different R-dependence and make significant contributions especially along the axis of the cone. The resulting field dependence is not simply R - ~ / ~ and this leads to both qualitative and quantitative differences in the two field distributions. This is illustrated first in Figures 5a and b which are numerical plots of the potential and field distributions for a cone with the Taylor half-angle a = 49.3".

These have been obtained using Eqs.(l2) and (13) for the planar electrode and Taylor model respectively. Most significant is the more divergent behavior of the field lines near the apex of the cone in the Taylor model. Thus apart from the question of whether the conical model is even a correct description of an LMIS /4,10/ the use or application of the Taylor fields in the calculations of, say space charge or ion tra- jectories in an LMIS, will lead to results which are inconsistent with those obtained from the model with a planar counter-electrode.

In Tables I and I1 we compare quantitively the fields, for the planar and Taylor models, along the exterior cone axis and

KO the surface of the cone respectively.

Distances are normalized in terms of the apex-extractor electrode distance Rg

,

and the fields expressed in units of (VO/~O). Along the direction of the axis the component E must, by symmetry, be zero, and only the radial components of the fields are compared. Similarly on the surface of the cone, defined by 80 =V-a, the radial component of the field is zero,

R ['h eOsed' so that only the 8-component is compared

in Table 11. In both cases the calculated fields differ from each other by about 10-25% over the range considered. These differences are further manifested in the calculation of the critical breakdown voltages, which can be experimentally determined.

I v-"

\

b Using the Taylor equilibrium condition

- EQUIPOTENTIALS /l/

---. FIELD LINES

R

Fig. 5. Numerical plots of the po- and ~ q . (l), in the form, tential and field distribution for

a. Taylor cone with planar V = Vo[l

-

Pl/2(~os8)

I ,

⁽¹⁵⁾

counter-electrode and

b. Taylor cone and idealized non-planar counter-electrode given respectively by Eqs. (11) and (12) and Eqs. (1) and (13).

the critical voltage for breakdown in the Taylor model, is given by

The corresponding expression for the planar electrode model, obtained using Eqs. (141 and (11) is

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JOURNAL DE PHYSIQUE

Table I. Comparison of Fields on Axis

Table 11. Comparison of Fields on Surface of the Cone

We first use Eqs. (16) and (171 for the case of transformer oil, which was studied experimentally by Taylor /l/. The results are given in Table 111. We note that in

Table 111. Comparison of the Critical Voltages for Breakdown

Taylor's model, the critical voltage is constant over the surface of the cone. By contrast, our calculated potential distribution predicts a position dependent breakdown voltage, which is in agreement with the local nature of the observed instabilities in both LMIS /12/ and other experiments on electrostatically stressed fluids /6/ including Taylor's own observations /l/. A though we obtain remarkably good agreement with Taylor's experimental value of V$'exp = 7.2-7.6 X 103 volts, the agree ment is probably more fortuitous than real. We have also calculated the critical voltages for liquid gold and gallium in the two models and again find differences of about 10-25% along the surface of the cone. However, both sets of the calculated values are about 2-3 times the experimental values /3,13/. This may be attributable to the fact that actual shape of the liquid metal source is not conical but probably cuspidal or cuspidal-like /12/, leading to higher fields and consequently lower breakdown voltages near the apex. Calculations of the critical breakdown voltages for the cuspidal model of an LMIS are now in progress. It is important to stress that the Taylor cone model and his resultant condition for equilibrium (see Eq. (1411 pre- dicts a constant value of the critical voltage over the entire surface of the cone.

i.e., global. This is in disagreement with experimental observations discussed earlier as well as a recent theoretical analysis of the stability of electrostatically stressed conducting fluids /14/ which clearly demonstrates that the onset of instability (i.e., breakdown) is a local phenomenon.

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conducting cone with an infinite planar counter-electrode can be summarized as follows:

1. It is never equivalent to Taylor's solution for the idealized non-planar electrode geometry.

2. The field dependence, in contrast to Taylor's model, is not simply R - ~ / ~ and consequently is incompatible with the Taylor stress condition for equilibrium.

Acknowledgements

The authors wish to express their thanks to Professor E. Kazes for helpful conversa- tions.

REFERENCES

+This work was supported in part by the Division of Materials Research, National Science Foundation. Grant No. DMR-8108829.

++Department of Physics, The Pennsylvania State University, Altoona Campus, Altoona, Pennsylvania 16803.

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-

b) C. T. R. Wilson and G. I. Taylor, "The Bursting of Soap Bubbles in a Uniform Electric Field," Proc. Camb. Phil. Soc.

22,

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C) W. A. Mackey, "Some Investigations on the Deformation and Breaking of Water Droplets in Strong Electric Fields," Proc. Rey. Soc.

E ,

565 (1931).

7. M. C. Gray, "Legendre Functions of Fractional Order," Quarterly of Appl. Math.

11, 311 (1953).

-

8. J. P. Jackson, Classical ~lectrodynamics, 2nd edition (J. ~ i l e y and Sons, Inc.,

New York, 1975). C

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C9-152 JOURNAL DE PHYSIQUE

9. R. N. Hall, "Application of Non-Integral Legendre Functions to Potential Prob- lems," J. Appl. Phys.

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S,

55 (1983).

12. H. Gaubi, P. Sudraud, M. Tence and J. van der Walle, "Some New Results About in situ TEM Observations of the Emission Region in LMIS,' in Proceedings of the 29th Int. Field Emission Symposium ed by H. 0.Andren and N. Norden (Almqvist and Wiksell International, Stockholm, 1982), p. 357.

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40, 440 (1982).

-

14. N. M. Miskovsky, P. H. Cutler and E. Kazes, "Derivation of the Condition for Onset of Instabilities of a Conducting Fluid Surface Under Electrostatic Stress:

Application to Liquid Metal Ion Sources," to be published in J. Vac. Sci. and Technol., 1984.