7RD-A149 258 ANALYSIS OF THE DISSIPATION MECHANISM WITHIN

Ni/INSTATIONARY CATHODE SPOT.-CU) STUTTGART UNIV (GERMANY

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AFCSR-82-0298

Analysis of the Dissipationmechanism

within an Instationary Cathode Spot

of Metalvapor Arcs

M01.0 H.Q. Schrade, H.L. Kurtz, M. Auweter-Kurtz

CNI and M. Kirschner

IRAINSTITUT FUR RAUMFAHRTANTRIEBE

UNIVERSITAT STUTTGART

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PFAFFENWALD RING 31

* 7000 STUTTGART 80 (VAIHINGEN)

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AFOSR-82-0298

Analysis of the Dissipationmechanism

within an Instationary Cathode Spot -.

of Metalvapor Arcs S

H.O. Schrade, H.L. Kurtz, M. Auweter-Kurtz

and M. Kirschner

0IRA 84i IB-6 ii

August 1984 nT 1.e

AInterim Scientific Report 0

Approved for public release; distribution unlimited

prepared for

Air Force Office of Scientific Research

Bolling AFB, D.C. 20332

and S

European Office of Aerospace Research

and Development

6 London, England0

REPORT DOCUMENTATION PAGE F: IN 7

rE P: r' C\ 7 AC E SSI:% NO V E'-Pi EN- % v i

AFOSR.TR. -4L & -2~7~~774 E &.' 5..bfm: F R F-6 rE

Analysis of the Dissipation-echanisnANiA

wi thin an Ins tati onary Cath,- e Spot 01A,1S : .c. lo f ME talvapcr A r cs 1ER00N GrEV rE

IRA - S. lB68 C~oN.PA;' Gr&N- %-~t

H .0. Schrade, H.L. Kurtz, M. Auweter-Kurtz AFOSR-S2-0-198and M. Kirschner

PC E %:L NN A,,: A:-1rESS PP :;Ak E-E%EN P .E. -A!-AZZ A V ':RK~

Institut fiir Raumfah-rtantriebe 61102FUniversit~t Stuttgart, Piaifenwaldring 317000 Stuttgart 80, West Germany - I_______

IRFORCE OFFICE CF SCIENTIFIC RESEIRCH,/N!jAAuut 17, 1B T LINO LFB D 233748I3-~Z

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'9 KEY *ADPS (Cor:Mr. r -t' r* - f3 re~es arl rdc jden'!(3 t- blc-h num-tr,

4 Cathode spots, vapor pressure, energy dissipation mechanism,

d ten h ca t t r ar. sfe r.

2~ A B,17 R A -'~t or, , e C. side If r,.. essar) and fdewjfs- bv block numnber)

The paper deals with the dissipation mrechanism,- of microspots oil

lo~w prEssure arc cathcdes.

Based cr. a nEw YFD stabili tv theorv and or, an exeILnal: un

em7pirical relation between spot current and crater radius orn purt.

copper and mnolybdenum cathodes, the time behavior of pressure and

DD ~~ 1473 EDION OF t 40V 65 IS OBSOLETE UCAS~E ___

Q-TV g~ CL jSSr' A'N Or ImIS PAG E a £rtrJ

SELRT Ty CL.ASS ,CAION OF TmIS PAGE.'14"IeV, Data nrervd9

temperature within the spot and the appertaining voltage drop

have been calculated. Emphasis has been placed on finding an

analytical solution for the time dependent heat flux into the

hemispherical crater surface as a function of the expansion

of the spot.

U:N(TL.'- SSI FIED

SECRe ITY CLA$SIFICATIOIN OF A - i E'A"' em :',&'f E

' - --V - - i - : - - / . . , . .. .

... * .I , " . . . . .. . . -" . . .. .. _ . _ . -

t7~ - ~-- --- ---- ---

-I-

PREFACE 0

This report deals with the time dependent energy

dissipation mechanisms which should occur in micro

spots on cold metal cathode surfaces of low pressure 0

arcs.

The Appendices A and B contain especially the

mathematical way of how the analytical solution of

the time dependent heat transfer equation and the

formula for the time dependent heat flux for a

growing spherical spot has been found.

0

This research has been sponsored in part by the Air

Force Office of Scientific Research/US Air Force

under Grant AFOSR 82-0298.

/

Ai

0 0'

-*E

* .

6 °

- .o

TABLE OF CONTENTS

Abstract 2

Survey of Pictures and Diagrams 3

List of Parameters 4

Introduction 7

Hemispherical Crater Model 8

Analysis 10

Results and Discussion 26

References 32

Appendix A 33

Appendix B 40

0

* 5°

0 0

-2-

ABSTRACT

The paper deals with the dissipation mechanism of

microspots on low pressure arc cathodes.

Based on a new MPD stability theory and on an experi-

mentally found empirical relation between spot cur-

rent and crater radius on pure copper and molybdenum

cathodes, the time behavior of pressure and tempera-

ture within the spot and the appertaining voltage

drop have been calculated. Emphasis has been placed -

on finding an analytical solution for the time depen-

dent heat flux into the hemispherical crater surface

as a function of the expansion of the spot.

*

* 0. ''

* S]

* 0j

0- I W."-. 7'

SURVEY OF PICTURES AND DIAGRAMS

Fig. 1: Hemispherical crater model 9

2: Zone I: Current carrying plasma channel 11

3: Zone II: Hemispherical crater 14

4: Vapor pressure curves 15

5: Zone II and III: crater and molten layer 17

6: Zone III and IV: Phase area liquid-solid 19

7: Zone IV: Solid metal cathode 21

8: Temperature distribution 23 "

9: Function y 24

10: Spot behavior (cold copper cathode) 28

11: Spot behavior (cold molybdenum cathode) 29 '

A 1: Temperature distribution around a ball with 33

the surface temperature Ts

~4

° .

LIST OF PARAMETERS

Ac Surface of the melting within the crater

A s Phase boundary area solid/fluid within the

crater

Az Crosc section area of the channel within the

plane of the cathode surface

Magnetic induction

c Evaporation constant

C 1 ,C 2 , Vapor pressure constants

C3

Cf Specific heat of the cathode material

EIntensity of electric field

ec Uit vector

ec z Unit vector

f Factor depending on current density

distritution 0

f(T) Function

h Enthalpy

I Current e

Current density

M Molar mass

rMassflow

m 0 Mass of an atom

P Pressure tensor

Pc:PD Vapor pressure above the melting

S

Pz Pressure within the area Az

p Ambient pressure

Qs Heat of fusion

Qv Heat of vaporization

q Heat flow

qc Heat flow into the area Ac

qf Heat flow out of the phase area As-tas9 Heat flow out of the melt into the phase

area As

R Universal gas constant

rs Crater radius of the phase area As

rso Radius of a sphere

s Velocity of the phase area As

ro Crater radius of the phase area Ac

S Radiation

Radiation into the melt

T Temperature

Tc Evaporation temperature = temperature of the

gas within the crater straight above the melt

Temperature at the surface of the sphere with

rs

Ts Melting temperature

T ,T* Cathode temperature at infinity

T*,Ts* Substituted temperature

t Time

Uk Voltage drop at the cathode

-6-

U Voltage drop in zone IIII .

UII I Voltaqe drop in zone III

V Volume

v Velocity

V c Exhaust velocity of cathode material on the

crater surface

v z Velocity component in z-direction within the

area Az

x,x I Mathematical variables

y Function

,- f Matter constant of cathode material

* x) Function

9-crit Stability functions

Heat conductivity of cathode material

..0 Magnetic permeability of free space

f Thermo diffusion constant of cathode material

Density *1c Gas density above the area Ac

Density of cathode material

Electrical conductivity of cathode material

Time

Potential function

U x) Error function

e Work function

* . S

INTRODUCTION

According to todays knowledge the attachment region

of a low pressure arc or a so-called metal vapor arc

consists of one or more bright "spots" of very high

current-density ( 1012 A/m 2 and even higher). A vapor

jet (containing even droplets) and a current-carrying

plasmachannel emanate from each of these microspots.

The discharge channel which electrically connects the

cathode with the anode or with the main inter-

electrode plasma region is therefore exposed to a

very strong axial flow.

I4

These microspots are highly nonstationary, they

propagate and disappear and, depending on the

electrode material (oxidized surface layer, surface

chemistry), surface roughness, overall temperature,

and type and pressure of gas, these microspots can

cluster together to macrospots of different sizes

witn different erosion rates.

This report presents an analytical investigation of •

the instationary energy dissipation mechanisms within

an individual microspot on a pure copper and

molybdenum cathode. It is based on the knowledge of

an experimentally found, empirical relation between

spDt current and spot size and assumes a strong

enough axii ow in order to maintain a stable

straight discharge channel.

* -5

HEMISPHERICAL CRATER MODEL

The following analysis considers a simplified model which,

*however, according t o experimental findings 3)4% i s

quite reasonable. At first the arc attachment region is

divided into five different zones (Fig. 1). 0

Zone I is the current carrying plasma channel which

electrically connects the low pressure plasma of the

interelectrode space with the crater zone II . This latter

zone contains a complex , turbulent mixture of evaporated

material like atoms, ions and electrons under fairly high

pressure. Adjacent to the high pressure zone II is the

molten layer zone III, which is connected to the solid

cathode zone IV through a hemispherical phase transition

area . Zone V is the gas and vapor region outside the

* current carrying channel and is characterized by the fact

that n o, or only an insignificant, current density is

* present here.

Cathode material is evaporated out of the crater which

forms a gas and vapor jet, causing an axial flow within

the current carrying plasma channel (zone I) and a

diverging flow in the non-, or only weakly, electrically

conducting ambient zone V. As often observed in experi-

ments,this jet can contain droplets which indicate that

* the surface of the melt within the crater is boiling and,

as will be shown later,that due to an explosively

increasing pressure in the crater (zone II), molten

m ma ter ial1 is pressed andJ e jec t ed out around t he crater

rim. The behavior is indicated by the protruding rims of

t he craters l e ft on the surface of cold metal cathodes

that have been struck by an arc.

, 4l

-9-

CURRENT DIRECTIONI "

EVAPORATED

CATHODE MATERIAL

,' DROPLETS

ZONE I CURRENT CARRYING PLASMA CHANNEL "

ZONE II HEM SPHERICAL CRATER

ZONE III MOLTEN METAL LAYERS _ I

ZONE IV SOLID METAL CATHODE

ZONE V NON CURRENT CARRYING GAS AND VAPOR REGION

S Fia. 1: Hemispherical crater model

- 10 -

In our analysis we assume a hemispherical crater, neglec-

ting this crater rim and the thrown out droplets. However, -

since these droplets have a minor effect on the energy

balance, and since the crater radius follows as a function

* of the spot current from empirical relations of experimen-

tal observations, one indirectly accounts for the material0

* loss due to this rim and droplet effect in zone III any-

* way.

Now we will investigate the zones separately and match the

0 boundary conditions in between. Thus we obtain an integral0

view of the energy dissipation and at the same time the

voltage drop Uk between the plasma channel (zone I) and

the solid cathode (zone IV). It will be shown that this

C voltage drop f its very well with the cathode drop of a

- , -spotty arc discharge. .1-

First we look at t he current carrying plasma channel,

-I.+

zone~~~~~ ~ ~ ~ ~ I (Fg 2) omrivsiaiosi r tblt

and f 0.8 w

which means that the axial impulse transport within the

discharge channel rve'aer has to be larger than

Z Jd 4* 0 4 "J

hpea of J

ta0bevtosoeidrcl acut o hemtra ,

CHANNEL AXIS

GAS AND PLASMA FLOW

xa

DISTRBUTIO

\ ii

THE ELECTRICALLY CONDUCTIVE CHANNEL is 1 KINK 1 UNSTABLE

AND BENDS MORE AND MORE IF:

WHERE /

AND JA',

Fi.2 Zone I: current carrying plasma channel

1% V I. .- V* - S

- 12 -

For a pure copper cathode, we know from experiments 7) ".i

that / 0.12 (m/A) . Using this we can neglect

the second term within the curved brackets as long as the

constant outer magnetic field is small enough (Bo _ 1 T)

so the criterion above is reduced to r O.

Utilizing the momentum equation for the crater zone II,

the vapor pressure Pc above the molten layer can be rela-

ted to the axial mass flow (Fig. 3). Neglecting the ambi-

ent pressure p , the criterion above can be expressed as

OF r A ~ (-3

which means that in the case of our hemispherical crater

model the vapor pressure above the melt has to be about

3 times the average magnetic pressure 1I

If this condition does not hold, a small disturbance, ]

which means a small curvature of the channel axis, willbend the channel more and more until it touches the

electrode surface at another point and a new spot is

created. Fig. 4 shows the vapor pressure curves for copper

and molybdenum with another ordinate, the product of

current and current density, Ij, with dimension A 2 /cm 2 .

For Ij above these curves the current carrying plasma

channel is unstable; correspondingly if Ij is less than

the value given by these curves we have a stable discharge

channel.

With the assumption that the mean evaporation rate fromthe fluid surface Ac within the crater corresponds to the I

vapor temperature Tc, which is the temperature of the gas

within the crater straight above the melt and is

77 (5)fr A - -

- 1 3 - - -

the criterion (4) becomes

with 0

c- 'S[ L ]7

M is the atomic or molecular mass of the evaporating

cathode material.

The minimal evaporating rates for one spot at a pure •

copper and a pure molybdenum cathode yield to

(6a)'foS

Pio ;r J, .60 bL34 ( 6b)

with I in amperes and T in Kelvin.

These results are now used for the analysis of the dissi- I

pation mechanism together with the justified assumption

that the voltage fraction of the spot is just high enough

to obtain a stable spot.

First the energy equation for the crater zone II is formu-

lated, which means that the power converted within zone II

given by the electrical power input*- -

" " 0V

Pao dAz

FLUID 4

CATHODE

CRATER S

4

- -_ - .0FORCE BALANCE: 4 ,D+ TdA =J'x ~

UNIT VECTOR

0Az Ac 1

P AND T ARE HOMOGENEOUS ON THE INNER SURFACE OF THE CRATER -

dIc e d A,& 022 1Z{ f- 7.

,V 2 aAz p -Ac - g - 0,227,

STABILITY CRITERION: S

Fig. 3: Zone II: hemispherical crater

1,[A 2/cmrn2 E 3 P CBARJ

C,: 2,2El 0C,=: -1,21

C -- 40 341.3

I E2

1E9

IEI

COPPER 1E8

* MOLYBDENUM

C4:: 9 429E10C2:=-1.148C3= -7 6 42 6.1

1 E- 1

1 E2

1E5

IE-3 *

2000. 3000. 4000. 5000. 5000. 7000.

T EK)

Fiz. 4: vapor pressure curves for copper and molybdenum

PC* C2 e(C 3/7)

accordi4ng to 10) and extrapolated for pressures

,LO above 1 bar

- 16 -

must be equal to the difference of energy which leaves and

which enters the crater zone per unit time. _

Within this calculation the purely convective part

and the parts due to heat transfer and radiation have to

be separated. The convective energy flow can be expressed 0

as inh, where ; is the evaporated mass through the crater

area Ac. Because of the continuity equation, m is equal to

the massflow through the circular crater orifice area Az .

-"h is the enthalpy difference of the plasma-vapor mixture

between the orifice area Az and the region directly above

the melting zone III. Because the mean pressure and

temperature at both areas are known, this enthalpy

difference can be calculated. In this evaluation the 0

resultant heat conduction and radiation loss through the

area Az can be neglected with respect to the convective

effects. The radiation and heat transfers (which include

ion bombardment) *hrouarh the area Act however, contribute 0

importantly to the heating up of the melting zone and to

the evaporation of the cathode material.

0 The energy balance of zone III can be written in a similar _

way. The electrical energy (I U III) within this zone

equals the difference of outflowing and inflowing energy

per second.

*.-0

* S

* 0

- 17 -

- CURRENT CARRYING

V PLASMA CHANNEL

A

A c : •7 A CATHODE

.. . A s( qs

ENERGY BALANCE 11I: q h (7-d ) + . ' ' " A f 'Z (

Ar_#A z Ac4,Z P

7h Ah + 2rr 7FS, 4 Sc/ -f v, gvdA4~z

ENERGY BALANCE IIl:; 7 4, 2 7(~4~ 2 712 + Sc.c

* 0

S U M I I A N D I I I : U = L 4' ,

F2 .ar an ml

0 Fig. 5: Zone II and III: crater and molten layer0

- 18 -

Two other terms have to be added in the energy balance of

zone III: mQv because of the transition of evaporated

material to zone II. and I e because of the work function

of the emerging electrons (Fig. 5). The energy of the

particles which flow over the crater rim and that of the

ejected droplets are neglected. It is a main fraction in

respect to the mass but a minor fraction in respect to the

energy in this zone.

The energy balance of the spotty cathode voltage drop is

the sum of the voltage drops of the crater zone II and the

liquid zone III (Fig. 5). This yields to

17r+

where (2 -r- s ) is the energy flow per second through the

phase area solid-fluid because of heat conduction.

The evaporated mass m cannot become larger than the molten

mass in the same time period; this leads to the require-

ment:

S(10)

From Fig. 6 the energy which causes the melting of thf

solid material can be calculated to

where Qs is the heat of fusion and qf the radial heat flow

directed from the phase area liquid-solid to the solid

zone IV.

V - SK _.__

- 19 -

• ;, .;.SOLID CATHODE•

SS

PHASE AREA r

2 Of

REQUEST: ,, 27r I25- --2., --

INTO THE ENERGY EQUATION

U r-; ) , t •

U A + 0t ) +

Fic. 6: Zone III and IV: phase area fluid-solid

200- 20 -..

With Eqs. (10) and (11) and the results from Fig. 5, the

heat flow in the liquid zone can be expressed with respect _

to the heat flow in the solid zone. Therefore the cathode

voltage drop follows as:

'rJ I ) # rJ f ) (12)

Before looking at the heat conduction qf within the solid

zone IV, the limitation of rb caused by the stability cri- •

terion (6) and the above requirement (10) will be added.

On a cold solid metal cathode the evaporation rate m with-

in a spot is limited by

To calculate the heat flow q , the heat flow equation S

within zone IV (Fig. 7) has to be solved. Therefore two

assumptions are made:

a) the heat transfer coefficient f respectively the ther-

mal diffusion coef fii'_ ent f within the solid is constant 0

and

b) the term fj 2 , which takes the ohmic heating within the

silid into account, can be approximated step by step by a

potential function with .f 1/r.

Hence it is

r

"

I - 21 -

qff

E SOLID CATHODEAREAS OF CONSTANT TEMPRAURE

HEAT CONDUCTION EQUATION OF ZONE IV:

T(r2 7rr 2fr a

C :CONST,. J z_

..pc Cf x (

INITIAL CONDITIONS:

0 (t 00

05 P

BOUNDARY CONDITIONS:

r r5 T'r T9 CONST.

r - ~T~r~ Tce, CONST,

Fic;. 7:- Zonec IV: solid metal cathode

- 22 -

in the region the potential function is

given as _

- r (rdE704s

S

Stin CAS I thr olw•eas

of g ¢c16)1

which is in fact no potential function. But within a small

r e go. I y' o r- -k the term (14) can be approximated by 0

where - is nearly realized This can be done within

all1 regions 1; 2 I23.. With this step by stepwhrich tio of f j2 by a potential function Bu the heatflow equation can be rearranged c

Replacing T by iii i i.!

where T* is te new independent variable bdn wh

the new differential equation follows as

T 7-

PT( 18

-. -- - -;-~'--*~---- -

-23-

SOLUTION T T (r,t) IN THE RANGE 0;

rs :s r< -.5-- o

T -Too F- ( T r. ) I " C ; :

r-~ r -5

S. 0

tz

X26 () t 2o x•TT-= _ _ rs_/~d _

= I

0t

6(t) fs -' x)iI2x]00

rrST

rs t, t t •

Fi.8 eprtuedSrbto

00.

450. . ......... .........

40... .. . .

.0 . ... . . . . . . . . . * . ... . . . . . . . . .

I20 . .. . . . .

10 . ... . . . . . . . . . . . . . . . . . . . . .

*0 . .... ... . . .

0.0 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Fig. 9 Function Y.

* 0

- 25 -

which is solved for the given initial and boundary con-

ditions in Appendix A. Equation (17) leads to the solution - 6

for the temperature T(r,t) which is shown in Fig. 8. Using

this solution T(r,t) the heat conduction loss can be cal-

culated by forming suitable gradients on the solid side of

3 the phase area fluid-solid (r=rs; exactly: r=lim (rs+Lr). 0

_r-O

In Appendix B the heat flow into the solid zone IV is cal-

culated taking care of the fact that at time t:O the ]

initial radius is rso=O;

N T {-19)

The function 2 ( t- .. 4 ) is displayed in Fig. 9. Taking

the rises of current I(t) and spot radius rs(t) from ex-

@1 periments 2) Eqs. ? allows the calculation of the time •

dependent heat flow into the solid electrode. With this it

is possible to determine the cathode drop voltage Uk

(without inductive part), the minimal vapor pressure pc

0 and the spot temperature Tc as a function of time using 0

Eqs. (12),(4) and vapor pressure diagrams (Fig. 4).

* .

*• i

*• 0

- 26 -

RESULTS AND DISCUSSION

It is known from experimental investigations 1),2),11)

that during the first 3 to 5 ns of a pulsed discharge with

a current rise of about 1 to 5 *109 A/s, a small circular

melting point occurs on the pure metal electrodes which

afterwards forms a crater; then the crater rim expands

with a velocity of 104 to 5"104 cm/s. After about 5 to 40

ns the final crater size is reached which means that the

current transportation is incured by a new spot.

On pure metal cathodes the crater radius rs increases with

the current I S . Experiments lead to the empirical

approach: (20)

with pure copper: ro=1,7 2:m and I*=83 A

with pure molybdenum: ro=1.2 ;im and I*=33 A 2).

It is now possible to analyze the time behavior of the

dissipation mechanism of a cathode spot and to calculate

the purely ohmic part of the cathode voltage as a function

of time if one takes the spot current I s as a given quan-

tity which linearly rises with time.

Figures 10 and 11 show the results of these calculations

for copper and molybdenum. In both cases the characte-

ristic traces of the curves are similar to each other for

the vapor pressure pc, the surface temperature Tc and for

the ohmic part of the voltage Uk.

During the initial phase the pressure rises explosively

and sooner or later, depending on electrode material and

current rise (dI/dt), it reaches a maximum. With copper

- 27 -

(dI/dT 2.67 109 A/s) this maximum of about 160 bar is

reached after about 30 ns; with molybdenum (dI/dt

8.109 A/s) it needs only about 3.7 ns to get to the maxi-

mum of about 55 bar. This very significant, explosive

pressure rise on the inner crater surface gives an expla-nation for the fluid metal which is ejected and which 1

forms a crown like rim.

Initially the temperature rises steeply, later on it rises

more slowly up to nearly 5000 K for copper, and nearly S

7000 K for molybdenum. Nevertheless the temperatures do

not reach the value which would be needed for a pure

thermionic emission even in the case of molybdenum.

To explain the very high electron emission on the cathode

G. Ecker postulated a combination of thermionic and field

emissions 12). This seems to be possible because new

experiments 13) show field amplifying "Taylor cones" at S

formerly plane fluid surfaces. With this explanation the

emission of the charged particles is not uniformly

distributed on the inner fluid crater surface but will

occur at one or more "Taylor cones" which emerge and S

vanish at the fluid surface. Therefore only the time

average of the emission is uniform over the inner crater

surface.

A complex, turbulent high pressure plasma exists in the

crater-zone where the Debye-length reaches quite possiblyonly 10 - 8 to 10-7m and therefore the electrical field at "

the tips of the "Taylor cones" could be some 109 V/m.

With that, the relatively high emissions of 108 A/cm 2

(and more) can be explained. A further possibility would

be the evaporation of negative ions out of the tip of such

Taylor cones, similar to those observed with positive ions S

- 28 -

UKIrVolti Fig. 10: Spot behavior (cold, pure copper cathode)

20K 5000 K-

20 T0

15 15-a

3500 K

10 100 bar THEORY

UK0

1 5 10 20 30 [ns)

J0

rSSrsI[A] 100

8,, gO rs : .:HA) I -

50

EXPERIMENT

* A- (GUILE-JUETTNER)

, 320

5 0 2 0 30 In [ s] ..

-29-

Uk[Volt] Fig. 11: Spot behavior (cold, pure molybdenum cathode)

[Volt]

20

15

10 -- 550OK'

10 bart [ns]

.01I

2 3 4SK 1 2 3 4s

[A] EXFERIMENTr (GUI LE-JUETTNER)

50

K40 r

30S

20

-10

1101

12 3 4 5

- 30 -

on anode tips 14),15). The life span of such negative ions

within the crater plasma is expected to be short; they de--

compose rapidly into free electrons and atoms; through

further impact the atoms are ionized into electrons and

positive ions. These considerations are only speculative,

but they could yield a plausible explanation of the0

* experimentally found high current densities of about 108

* A/cm2 and greater.

The purely ohmic part of the cathode voltage Uk above the

spot , first drops ste ply from relatively high v alu es ,

* passes through a minimum and then increases slightly. It

* is of interest, that the high voltages at the start are

* caused mainly by the high heat-conductivity losses into

* the electrode material at the beginning. In the first mo-

* ment after the ignition of the spot discharge, the tempe-

rature decay at the phase boundary layer solid-liquid is

very steep (see Fig. 8) and therefore the heat flux into 0

* the solid cathode is very high. The actual ignition is not

described by this analysis . The ignition voltage however

* has to be surely greater than or at least equal to the

* purely ohmic ot. voltage Uk, calculated here, to init-

iate a spot discharge which leads to a craterlike cavity.

Also, the time when the channel becomes unstable is not

* included in this evaluation - the assumption was that the

stability criterion (1) or (14) is just satisfied at any

time. This means that the product of current and current

density Ij would reach its maximum at the same time as the

* vapor pressure P.. However, it is improbable that with a

further increase of temperature, current and voltage, the

emission ~r the average current density j at the liquid

surface of t he crater would decrease. An increase of

- 31 -

,j would however lead to an immediate falling short of the

stability criterion (4) and therefore cause the kinking of

the discharge h al ine 1 5),16) . A s a consequence o f t h is

plausible discrepancy, we have to expect the extinction of

reaches its maximum. In our calculated examples this maxi-

mum lies at ca. 30 ns and at a current of 80 A for copper

and at ca. ~4 ns and 33 A for molybdenum.

- 32 -

REFERENCES

1) B. JUttner "Beitr. Plasmaphysik" Vol. 19 p. 25 (1979)

2) A.E. Guile and B. JU ttner "IEEE Transactions on

Plasma Science" Vol. PS - 8 No3 p. 259 (1980)

3) J.E. Daalder "Diss. Thesis" Univ. Eindhoven (1978) 0

4) J.M. Lafferty "Vacuum Arcs" John Wiley and Sons (1980)

5) M. Auweter-Ming and H.0. Schrade "J. Nucl. Materials

93 and 914" p.799 (1980)

6) H.O. Schrade,M. Auweter-Ming and H. Kurtz "Proc. XV

Int. Conf. on Phen. in Ion. Gases" P. 531 (Minsk 1981)

7) J.P. Datlev, A.E. Guile, B. Jattner "Beitr. Plasma-

phys." Vol. 21, No 2, p. 135 (1980)

* 8) J.E. Daalder "J. Phys. D", Vol 9, p. 2379 (1977)

9) D.T. Tuma et. al. "J. Appl. Phys. 49" p. 3821 (1978)

10) Hultgren, Orv., Anderson and Kelly "Selected Values of

Thermodynamic Properties and Alloys" John Wiley and

Sons Inc. (1963) 0

11) E. Hantzsche, B. Jittner, V.F. Puchkarov, W. Rohrbeck

and H. Wolff "J. Phys. D." Vol. 9, p. 1771 (1976)

12) G. Ecker "Ergebnisse d. exakten Naturwiss. 23" (1961)

13) G.N. Fursey "Local Elementary Processes at the For-

mation and Operation of Vacuum Arcs"*)

14) R. Clampitt and D.K. Jefferies "Proc. of the third

Int. Conf. on Ion Beam Analysis" Washington, D.C.

(1977)

15) J. Mitterauer "Liquid Metal Field Ion and Electron

sources"*)

16) H.O. Schrade, M. Auweter-Ming and H.L. Kurtz "Analy-

* sis of the Nonstationary Low Pressure Arc Cathode •

Proc. VIIth Int. Conf. on Gasdischarges and their

Applications" London (Sept. 1982)

17) L.D. Landau, E.M. Lifschitz "Lehrbuch der theoret.

* Physik VI; Hydrodynamik" Akademie Verlag Berlin S

p. 230 (1966)

*) presented at the Vth Workshop on Electrode Phen. Berlin

1982; organized by Zentralinstitut fUr Elektronenphysik,

* Academy of Science of GDR 0

Li __

*] - V -,, .-; , - - . . *- * *' .. . . .- r --

I 0

- 33 -

APPENDIX A

Before solving the heat flow problem as demonstrated in

Fig. 7, the following problem must be considered:

During t<O, the temperature within the homogeneous medium

surrounding a sphere with radius rso is set to be T . At

t 0, the temperature at the surface of the sphere jumps

suddenly from T. to T o = T s and for t>0 it increases

K steadily, corresponding to To(t.O) Ts+'Tso(t). Beginning

with t = 0, a homogeneous, radial current flows through

the outer medium, which leads to additional ohmic heat

sources. The heat flow equation describing the time

dependent spacial temperature behavior is given by

X& ( r 7v, (A 1

A (, A

-- I

II

T. W

Fig. Al: Temperature distribution around a ball with the

surface temperature Ts .

* 01

- 3 4 - 0

with the conditions:

ir 77

7T(r -- )P vo 1 7,-.r'-'_

Solu t ion:

- The term j 2 with is represented within each

hemispherical shell by the potential function

e- su"tsi t ut in: . ' '-,,)-T'(i/ J4J' tA3~ S.t

0

~ r ~z Ir2 (pe/ 'd

th! h a o on t n eqiat ion becomes

ra Pr.

witn n the t4mc ir -erval the conditions are

- < 7 -o T* ,.) . -- , ) .

o ,. < , T'( ., -T + 7-., €).

W; 7j- 4)h-

S-

- 35 -

The solution of this problem is found with Landau-Lif-

schitz 17) as _

7-A 0

where

00

With this approach the temperature field is determined if

the temperature T*(rso, ) on the surface of the sphere is

known at ar.y time. The location of the temperature front,

T., at any time is known by experiment. Therefore we try

to reduce the time dependent temperature behavior on the

surface of the sphere to the behavior of the temperature

front T s . 0

Choosin7

#.,(r" -- IAo

r-r,,

the solution witn - (A7)

results in

76*1 7o-.~~ 7T-r 4 J, ;T (tifzf .

(A8)7t ,,s A )</,_

.- 36 -

kL

where l ftc/y is the error function.

With the substitution-_

_ _ _ - _ _ _ r -r .t " d u - c) ' / 'Y JqC (l -) l t a z (A9)

and using (A7) r*rc'yi = r,# (Aloa)

the following solution is obtained:

(All)

Olt-

As can be shown analytically with the linear approach

A11

(A8) or (All) are the solutions of the problem. In the

same way, at any time r4.+4# ( small), the function

0 f(-.) can be approached with S

This yields also the equation (A8) or (All).

01

37 -

Substituting f(T ) in Eq. A(11) with respect to A(13) we

obtain

T( -r , f) 7o a T - Jo ,0 f& [ [ )- ,r

or by using r/ * S/I

and calculating ( ) with (A10)

(A15)

where

. ¢(A16)

and

I lit'

-38-

k o and Mi have to be chosen at any time with respect

to the condition that the location of the temperature -

front at time t in the surrounding medium is the experi-

mentally given one. So we obtain from

and with (A15):

/ (A17)

with

w h 7 (A18) - 2

Following from (A17), because of t 0, rs = rso, and Sxs o, -2J '

that is, ko = 0 (A 19)

In the same way we obtain with (A17) and (A19)

-. - _ -_ _ - -(A20 ) _0,lr(As) 4.L"?

Using this together with (A19), Eq. (A15) yields the

temperature distribution, at any time, and at any place "

* 0

too

0 f n0) rOwe _7"_- -

*j - "S

- 39 -

* where

XISS

((k ~~ 4)] 4 ~px) - x

4 #(x)~AfeX

- '. .. ; =. , .- % . -. r , - - " 5" : . - - ; -- o . " - . . ','.

400- '40 -

APPENDIX B

Calculation of the heat flow 2-r qf

The heat flow at time t through the surface of a half

sphere with radius r( > rs ) into the colder solid body

leads to

T(, (Bl)

The heat flow is obtained as a function of t with the

passage to the limit:

2r (e '') Tp2~ - l/ -T (B2)

T(r,t) can be calculated with Eq. (A21):r

~ Tri #() (Kj L-4 r)

Wi th L

..,,,- "-) 5e" (.4-1" °" • . .

L L

U.)

.~ .f. r. A LO..K d.f,-,<)]- Ae,.

In addition is

F-, ' £, ,fd .-, g,.

J/ 1.~q~J~/L ~4.Zi-

* . *. -... I

41

because off- (potential function) and therefore

ir L (- /A) [-- -Or 761 "p- ,P-/"'' ( ) - O 2 -,,"'<e" O.] ° o. - ,_I

for r- rS -p x-* x S results in

opt (7-7-)f2 /,,:'t, 7-/_^/, r.

z',, N g- .O"

KAW

"~ ~( 7-,," (7j, -/.- ./^-v, .•,- .. I /r - .

i"T} K, e ?CLIo -e6X)O (,- t6, (s - 7-

S c,7..) - A,.'e ,e• t

- 9 7j. , - ..-

_ [,, .7,, t . (<,----.j4)/

* */- ,'- 'i... -

4 .

.16.

(72 -70- 7- -- i

ge c

+1( - Or / -- -

j e

-- - "

'2(d ) -A)[ 4 L - - .rS

Therefore the hea" 'l"w follows as

+ Z I €C,<, - i + S7,< ) - /

- , -A , ,, o .

0

S S•

jl (7 -- 7o,)'+,('': ){<"° ""/'(B3)

Th~~ref'orer th et l'llor Ewe

,'., % m ", t & ,.)[,++ .,. z r7-j -0-c 1

a FILMED 4L 2-85

.i DTIc -*.~