RECENT PROGRESS IN LMIS THEORY

HAL Id: jpa-00229899

https://hal.archives-ouvertes.fr/jpa-00229899

Submitted on 1 Jan 1989

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.

RECENT PROGRESS IN LMIS THEORY

R. Forbes, N. Ljepojevic

To cite this version:

R. Forbes, N. Ljepojevic. RECENT PROGRESS IN LMIS THEORY. Journal de Physique Colloques,

1989, 50 (C8), pp.C8-3-C8-8. �10.1051/jphyscol:1989801�. �jpa-00229899�

COLLOQUE DE PHYSIQUE'

Colloque C8, supplement au n o 11, Tome 50, novembre 1989

RECENT PROGRESS IN LMIS THEORY

R.G. FORBES and N.N. LJEPOJEVIC

University of Surrey, Department of Electronic and Electrical Engineering, GB-Guildford, Surrey, GU2 5XH, Great-Britain

Abstract

-

1. Introduction

We are currently attempting to develop an improved numerical model of liquid-metal ion source ( L W S ) behaviour. This paper is an interim account of work in progress, and reports on some preliminary issues. Three topics are discussed: (a) the incorporation of the effects of viscous flow along the shank into the "shape condition" for the liquid-metal cone; (b) Some investigations with the Kingham-Swanson (KS) method /1/ for modelling LKCS behaviour: and (c) progress towards a possible variational formulation of the problem.

1.1 m e approach of Kingham and swanson

We begin with a brief reminder of how an operating LMIS is modelled in the context of the KS program. The shape of an operating LMIS has long been known experimentally to be a "cusp on a cone" /2-4/. (And, of course, similar shapes have been observed with other charged conducting liquids, e.g. Zeleny / 5 / ) .

In its simplest mode of use, the KS program models the cusp apex as shown in Fig.1. The apex itself is treated as an exact hemisphere, and the "cusp"

section of the emitter joins smoothly onto the hemisphere, at the plane A A ' . Let F denote the surface electric field, and p the hydrostatic pressure in the liquid. Ft and pt denote values at the liquid surface in the plane A A ' , rt is the radius of the hemisphere, and vt is the liquid speed as it crosses the plane A A ' . Assuming that the liquid speed is constant across the cross-section of the emitter, and making the simplifying assumption that all the material is emitted in the form of singly-charged ions, we obtain an expression for the

emission current I as: Figure 1: LMIS Apex

I = (Volume p. u .time across AA' ) x (No. of atoms p .u .volume) x (Charge on atom)

where p is the liquid-metal density, m is its atomic mass, and e is the elementary charge.

It is assumed that at large distances from the apex the shape of the emitter becomes conical, with half-angle a. The shape of the intermediate cusplike section is then modelled in terms of a simple approximation (see /1/ for details ); for the purposes of this discussion, we take this approximation as involving two parameters, the limiting half-angle a and a parameter ( Q ) that relates to the length of the cusp. In the work reported in Ref./l/, a is set to the Taylor angle 49.3O, but this choice is not intrinsic to the method.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989801

A second relationship involving rt and vt is needed. KS use the equation:

where €0 is the electric constant, and y the surface tension of the liquid. The derivation of this equation is discussed later; for t h present, note that combining (1 ) and (2 ) gives an explicit relationship between I and rt, if the value of Ft is specified.

Two "matching requirements" can now be introduced. The first is that at the plane AA' the surface field Ft should be equal to the evaporation field for the liquid metal. The second is that at some reference point d distant from the apex, at a position where liquid speed v is effectively zero, the electric field Fd be given by the stress balance condition:

where cl and c2 are the principal curvatures of the liquid surface, and cld and c2d are their values at the reference point d.

Given the relevant tip-shape parameters, the core of the KS program Solves Poisson's equation to determine the values of surface field. The overall KS approach operates in outline as follows :

(1) A value of I is chosen and this predicates a value of rt.

( 2 ) Values of the applied voltage V and length P are chosen.

(3) The value of V is adjusted until the matching requirement at P is satisfied.

(4) The value of P is adjusted until the matching requirement at AA' is satisfied.

Appropriate ion source characteristics can then be calculated. In reality, the operation of the program is somewhat more sophisticated than this, in order to achieve as much self consistency an possible (see /1/ for details).

2. A more aeneral s h a w condition

Equations (2 ) and ( 3 ) are examples of what we wish to call a "shape condition". Essentially, the "shape condition" is the boundary condition at the free liquid surface, taking into account the possibility of pressure variation within the liquid. We now consider the shape condition appropriate to a llquid-metal ion source in which the liquid supply moving along the supporting shank is subject to viscous drag. It is known experimentally that an ion source where the supply is subject to viscous drag has a different slope to its I/V characteristic, and there is a question as to h w this could emerge fmm modelling.

The shape condition comes from the combination of two equations. The first is the standard pressure difference equation for the excess pressure Ap immediately inside the liquid surface, relative to the pressure immediately outside it, namely:

where F is the field at the liquid-metal surface. In the case of a liquid metal ion source operating in vacuum, the external pressure is zero, so _we may write:

The second equation needs to relate to the hydrodynamics of liquid flow. If irrotational flow is assumed, then EBrnouilli's equation can be used /1/, i.e.:

p

+

The question now is: what is the liquid pressure in the region where the flow velocity is zero, i.e. in the region near the base of the supporting cone. Depending on their design (and perhaps on the conditions of fabrication), ion sources can be relatively free of the effects of liquid viscous drag along the shank, or be strongly influenced by it. Kingham and Swanson were attempting to model the former, and hence took the constant equal to zero. W e

argue that, if viscous drag effects are operating, then there will need to be a pressure drop along the shank in order to pull liquid towards the apex, and consequently the pressure pb at the base of the supporting cone should be taken as non-zero. So eq.(6) is best written in the form:

Under steady operating conditions, the base pressure pb will presumably be slightly negative and a function of the mass of liquid metal emitted, and hence of the current I

.

Combining equations ( 5 ) and ( 7 ), we obtain:

This is a generalised shape condition. The first two terms correspond to the shape condition used by Taylor /6/, and the first three terms to that used by Kingham and Swanson.

me next question is how big is pb likely to be. For the sake of discussion we use a value suggested by Sudraud (private communication) some years ago, namely that pressure drop along the shank might be of the order of an atmosphere or more. Tbat is, consider pb = - 105 Pa.

When viscous drag effects are taken into account, the matching requirements used by KS have to be replaced by applying eq; ( 8) at the plane A A ' and at the reference point d. At the plane A A ' , Ft needs to be equal to the evaporation field, which for gallium is about 15 V / m .

This makes the fleld stress term in eq.(8) of the order of - 109 Pa. Thus the inclusion of

pb is irrelevant here.

To deal with the reference point matching requirement, suppose that the liquid cone is supported on a wire of radius 0.1 mm. Since y for gallium is about 0.71 N/m, the surface tension term is of order 10-4 Pa. If pb were to be of order 1 atmosphere in size, then this term could have a significant effect in the matching requirement, if the reference point were taken well towards the base of the cone. Further, it would no longer be self-consistent to assume a exact Taylor cone as the limiting shape, because Taylor's analysis requires Ap = 0 at all points on the surface.

3. Explorations with the KS Program

subsequent to the above analysis, we carried out investigations with the Ks program. This is written in FWKPRAN, and is being run at Surrey on a MicroVax. Parts of the code have been rewritten, to achieve faster execution.

We have explored the effect of altering the assumed limiting angle of the cusp, with the following results:

For limiting angles below about 4 9 O the program does not operate reliably, for reasons that we do not fully understand, but appear to relate to the way space-charge effects are handled.

We also explored the effects of changing the position of the reference point at which the second matching requirement is implemented. Here we found that the predicted value of P could change by a factor of as much as 2, depending on the reference point chosen.

Our conclusion was that, to make further progress, the KS program needed to include a rather better approximation for the shape of the liquid cusp.

3. A variational derivation of the pressure difference formula 3.1 Intraduction

Although an analytical approximation for cuspidal shape has been given /7/, and its applicability under "static" conditions has been discussed /8/, this result is not without its difficulties and is not necessarily appropriate under dynamic conditions. Consequently we decided to look again at the possibility of deriving the liquid shape from a variational approach.

As a first step, it has seemed necessary to establishec! ?learly how to derive the standard pressure difference formula from a variational approach There have been various previous attempts in this direction /8-11/. Refs. /10/ and /11/ have probably taken the argument furthest, but none of these has been completely successful in providing a rigorous proof.

We believe that we have now been successful in this, using a argument based on thermodynamic principles. Pull details will be submitted for publication elsewhere. This section gives the main structure of the proof.

3.1 The equilibrium condition and virtual changes

For a system in equilibrium with its surroundings, a small virtual change in the system produces no change in the entropy of the universe. From this requirement, it may be shown /12/ that:

T denotes the thermodynamic temperature, S the entropy, and U the internal energy of the active system; 6s and SU denote infinitesimal virtual changes in S and in U; 6w is the infinitesimal amount of work done on the system: and A is defined by this equation.

The system we shall consider consists of the liquid metal, its supporting electrodes and the surrounding counter-electrode, and appropriate connections. A battery is connected between the electrodes, but this is considered to be part of the "surroundings" and does work on the system in transferring charge from one electrode to the other.

The virtual change _we shall consider is a small variation in the shape of the liquid electrode, so that it moves forward from a position I: to a position Z'. In this change, each element ds of the surface C moves forward by a distance 6n to become an element ds' of the surface

e ' .

we allow 6n in principle to be an arbitrary well-behaved function of position in surface C.

In such a change, the increase in area of the element ds is (cl+cp)6n.ds, where cl and cp are the principal curvatures of the element, and the volume enclosed between elements ds and ds'

is 6n.ds.

This virtual change in the liquid shape will produce:

( 1 ) A virtual change in the bulk internal energy of the liquid;

( 2 ) A virtual change in the (non-electrical) surface energy:

( 3 ) Electrical effects, which give rise to a term &l considered below:

(4) Gravitational effects, which are small and can be neglected.

Items (1) and (2) can be dealt with more-or-less in the conventional fashion (e-g. Ref ./la/), and we thus obtain the formula:

3.2 The electrical term

Now consider the electrical contributions. The electrical energy Uel stored in the system is the capacitive energy, so:

where VA is the voltage maintained by the battery, and q is the charge on the positive electrode. Alternatively, this can be expressed as a field-energy integral:

"el = 1/24vA =

(

R

where dr is the element of volume and the integral is taken over the whole region of space R between the electrodes.

The change in the shape of the liquid electrode produces changes both in the charge on the electrodes and in the field configuration. Thus the change 6Uel can be written in the following forms :

Now consider the operation of the battery. In order to transfer from the negative to the positive electrode, the charge 6q must pass through the battery. Thus the battery does work

6wel on the system equal to 6q-VA. Hence:

6wel 64 V~ = 6

f

R

There is no entropy change associated with the charge transfer, so we obtain:

= 6wel

-

f

R

Hence, on substituting into eq.(lO), we obtain as an equilibrium condition:

3.3 Transformation of the fteld-enerav intearal

Now concentrate attention on the field-energy integral in eq.(16). The change- in this integral can be split into two parts. In the volume between the surfaces C and E ' the fleld has been reduced from its original value to zero. Since the elementary volume swept out by an element ds in moving a distance 6n is 6n.ds, this first contribution to is:

A1 =

- /

I:

The second contribution comes from the region of space R' outside f'. The field distribution here has changed, so:

We now need to transform eq.(18) to a surface integral. Express A 2 as the sum of three components. The x-component is:

Azx =

/

/

R' R'

and the y- and z-components are similar.

Let V denote electrostatic potential, and let 8V be the change jn V caused by the movement of the liquid-metal surface from to

r ' .

Hence we obtain:

*zx

- ^/

^-

^-

R'

Azy and AzZ have similar definitions, so summing these we obtain:

A2 =

- (

+

where C denotes summation over the three coordinates. Laplace's equation applies to the volume R'; hence we have V ~ V Z O , and the first term in the integrand vanishes. For the second term, we use Green's Lemma /14/ to get:

L

Note that a change of sign is involved here, due to the way that we have defined the direction of the normal. P o is the electric field vector at the position of the element ds' (but with the liquid surface in its original position C), and 6V' is the change in potential at the position of the element ds' resulting from the change in the liquid surface position.

Now consider a surface formed by the elements ds and ds' and the quasi-cylindrical surface joining them, and apply Gauss' theorem. Because ds lleS in the liquid metal surface, the field is normal to ds, so there is no contribution from the cylindrical sldes. Hence:

where F here denotes the fleld value at the element ds. Further, the change 6V' is given by:

SV' = F 6n (26)

Hence we may rewrite eq.(24) in the form of an integral over the surface C :

Hence, on combining this with eq.(17), we get:

=

j

So the equilibrium condition becomes:

This condition must be true for any arbitrary set of infinitesimal variations Sn, so it follows that the integrand must be identically zero, and hence that at every element on the liquid surface :

This is the standard formula for the local pressure difference across a charged conducting liquid surf ace.

4 . Discussion

Since we have started from a well established formula of thermodynamics, the above argument constitutes a formal proof of the pressure difference formula. The older approach, of adding the m e 1 1 stress term to the well-established formula for pressure difference in the absence of a fleld, is in our view an adequate theoretical derivation in its own right.

However, in view of recent debate, we think it useful to have provided a direct first-principles derivation. It adds to our confidence in the overall self-consistency of the theory, and provides an illustration of a variational derivation. Obviously, a next step is to attempt a more general variational formulation of LMTS theory.

We thank Dr D R Kingham, of VG Ionex, for providing us with a copy of his program.

References

/1/ D.R. Kingham and L.W. Swanson, Appl. Phys. A34 (1984) 123.

/2/ K.L. Aitken, Proc. Field Emissjon Day, Noordwijk, 1976, p.23. (Noordwijk: European space Agency. )

/3/ R. Clampitt and D.K. Jefferies, Nucl. Instr. Meth.

149

/4/ H.Gaubi, P.Sudraud, M. Tence and J. Van de Walle, Proc. 29th Intern. Field Emission Symp., p. 357. (Stockholm, Almqvist and Wj.skell, 1982. )

/5/ J . Zeleny, Proc. Camb. Phys. Soc. 18 (1915) 71.

/6/ G.I. Taylor, Proc. R. Soc. .Lond.

- A280

/7/ N.M. Miskovsky, P.H. Cutler and T.E. Feuchtwang, Appl. Phys. A33 (1984) 205.

/8/ M. Chung, P.H. Cutler, T.E. Peuchtwang, E. Ka?x%3 and N.M. ~ i s k o v s k ~ , J. de Physique - 45 (1984), Colloque C9, 153.

/9/ R.G. Forbes, J. de Physique

45

/lo/ J.E. Allen, J. ~ h y s . D: ~ p p l . ~ h y s . 18 (1985) ~ 5 9 .

/11/ P.H. Cutler, T.E. Feuchtwang, E. ~ a z G , M. Chung and N.M. Miskovsky, J. Phys. D: Appl. Phys.

19

/12/ J.K. Roberts and A.R. Miller, Heat and Thermodynamics. (London, Blackie, 1%0.)

/13/ J.W. Gibbs, Trans. Conneticut Acad. (1875) 108 & 343. Reprinted in The Scientific Papers of J. Willard Gibbs. Vol. I. Thermodynamics. (New York, Dover, 1961. )

/14/ H. Jeffreys and B. Jeffreys, Methods of Mathematical Physics. (Cambridge University Press, 1956. )