Theory of cathode spot phenomena

Physica 104C (1981) 3-16 © North-Holland Publishing Company

T H E O R Y OF C A T H O D E SPOT P H E N O M E N A

E. H A N T Z S C H E Zentralinstitut fiir Elektronenphysik, Akademie der Wissenschaften der DDR, 1080 Berlin, DDR

Invited paper /~ general review is given of the state of affairs of the arc cathode spot theory. Two levels of this theory may be distinguished (with some intermediate stages): the first that considers the spot as an (essentially)stationary phenomenon of current transition between cathode and vacuum (or ambient gas) and that is well developed in several variants, and the second that includes non-stationary (dynamic)processes such as the formation of craters, the explosion of tips, the motion of the spot, and that is still in an early state of beginning. The ideas, the results, and the successes are briefly discussed, as well as the difficulties,the discrepancies, and the open problems.of the cathode spot theory. 1. Introduction

The exploration of vacuum discharges and, especially, of vacuum arc cathodes has been greatly intensified during the last two decades, motivated mainly by the new technical application in vacuum switches. But in addition, these discharges turned out to be very interesting objects of physical research, with a high complexity, extreme parameters, and largely obscure processes. The significance of these investigations exceeds the above-mentioned field of application because the cathode processes proved to be very similar in any kind of arc discharges, e.g. in the high pressure or the atmospheric arc, and this conclusion is already one of the results of this research. As to the cathode processes, the vacuum arc is 0nly the " p u r e s t " case. The efforts to obtain a theoretical interpretation of the arc cathode p h e n o m e n o n are impeded from the beginning not only by the high complexity of the system, but also by the dependence on many insufficiently known properties of the cathode material and, moreover, by the difficulties of measurements that result in considerable inaccuracies of all the quantitative basic data of the spot. In spite of these difficulties, many facts have been disclosed by the intensive experimental in-

vestigations, such as the distinction between two rather different types of arc spots (Rakhovsky et al. [1]), the cathode erosion rates (e.g. Kimblin [2], Plyutto et al. [3], Rondeel [4], Daalder [5], Kutzner et al. [6]), the careful measurements of the crater dimensions and the crater distributions (Daalder [7], Jtittner [8]); the investigations of the spot motion (e.g. Rakhovsky [1], Gundlach [9], J/ittner [8]), such as the velocities, the distances of single jumps and the directions of motion as well as their distributions; the intensity and distribution of the plasma light emission (Rakhovsky [1]) and the expansion velocity of the cathodic plasma, the distribution of the charge, the energy and the angles of the plasma ions (Kimblin [11], Daalder [12]), the distributions of ejected droplets (Utsumi et al. [13], Udris [14]), the energy input into the electrodes (Rondeel [4, 10]; [52]) among many others, for instance the simple current and voltage measurements and their time dependencies (noise frequencies and amplitudes), the characteristics of ignition processes, the lifetime distributions, the interaction with magnetic fields, and many auxiliary investigations relating to the emission processes, the surface diffusion, t h e Changes of the material parameters, etc. Many of these measurements are distinguished by very ingenious ideas and clever methods that are necessary to get the tiny arc spots under the

4

E. Hantzsche/Theory of cathode spot phenomena

control of measuring techniques. The references are selected examples only. Of course there are also many attempts that try to interpret the arc, to calculate its parameters and its behaviour, and to understand what happens there, especially in the cathode spot region. This, indeed, is the basic challenge for the theory: to answer the question of why the current transition between the metallic conductor and the vacuum (the metal vapour plasma) is concentrated in such a small cross section with such extreme conditions. Further problems that are closely related with the arc spot parameters are understanding the limited range of spot currents, the causes and properties of the spot motion, the connection between the arc spot mechanism and the erosion processes, the crater development and the droplet ejection, the plasma parameters, the different time scales and the stability of the arc, the energy dissipation, and many others.

2. General principles

The main principles of any theoretical interpretation are as follows. (1) The arc spot is comprehensible with well known physics by the application of the general laws of electrodynamics, thermodynamics, hydrodynamics, solid state and plasma physics. However, the problems are the complexity of the spot phenomenon, the interconnections and interactions of these processes, and the extreme range of the parameters within the spot that partially necessitates doubtful extrapolations. (2) The model of the spot and the presuppositions of the theory should be as simple as is feasible, at least in the first stage of the theory. Therefore conventional assumptions are: - n o ambient gas (one substance only); - a plane, smooth surface of the cathode; - a clean surface (no contaminations); - o n l y one circular arc spot; - c o n s t a n t parameters within the spot; - n e g l e c t i o n of processes outside of the spot; - no spot motion; - s t e a d y state conditions (no time dependence of the parameters).

A further essential point is the restriction on the cathode processes in most of the models. Indeed, the theory of the vacuum arc is almost identical with the theory of the cathode spot region - except the case of very high currents and contrary to the situation in the high pressure ambient gas arc. (As to the anode processes see [511.) There is a common physical conception of all of the theories that are founded on this simplified stationary situation. (1) Between cathodic plasma and cathode surface a space charge sheath is formed by an opposite flow of ions and electrons, with a dominating ion space charge. The consequences are a potential drop, corresponding suprathermal kinetic energies of the accelerated ions and electrons, and a high field strength at the surface. (2) Both the cathode surface and the plasma are sources of these charged particles, whereby the electrons are generated mainly by the ions and vice versa. (a) Because the conduction electrons within the metal have a lower potential (Fermi level) than the electrons in the vacuum gap or in the plasma, the emission of electrons is only possible either with a higher kinetic energy (i.e. a sufficiently high temperature of the metal) or with a lowering and shortening of the potential step (i.e. application of a sufficiently high field strength). Both effects are generated by the impacting ions (besides of additional resistance heating within the cathode). (b) The elevated temperature is also necessary for a sufficient evaporation to produce the plasma matter (at least, in the case of vacuum arcs). (c) The metal vapour is ionized by the accelerated electrons, mainly via collision deceleration, heating, and thermal ionization within an ionization zone adjacent to the space charge sheath, supported by resistance heating. Because of its high pressure this plasma cloud rapidly expands, and the ions and electrons are again accelerated towards the anode and the walls. However, the ions, also accelerated in the direction of the cathode, fall back, driven by their thermal velocities, diffusion, and drift in the plasma field.

E. Hantzsche/Theory of cathode spot phenomena

Altogether, there exists a self-consistent feedback mechanism for ion and electron production. The yield must be sufficient to warrant stationarity. 3. C o m m o n equations

5

- t h e energy balance of the plasma

Qp = Op(ji, je, Uc, Tp, j0, Tc, a, n,, a )

(7)

with the net power density input, Qv, that is to be carried away by the plasma expansion and heat conduction, resulting in the plasma temperature

F r o m this basic concept we obtain the following equations which will be written only in an implicit form because there are several or even many possibilities of a detailed description in most cases. Suitable equations for the spot models are: - t h e electron current density jr from thermofield emission (fig. 1)

(jp = plasma current density; Up= expansion velocity); - t h e degree of plasma ionization a from the Saha equation

A=j~(~,~),

a =

(1)

(T¢, F¢ cathode spot temperature and field strength, respectively); - t h e energy balance of the cathode spot

rp = rp(Op, ]p, Vp,p)

a(p,

Tp),

(3)

(a = spot radius, j = ji +j~); - t h e field strength from the Poisson equation

F~ = F:(ji, j,, U:, Tp, n,, Fp)

(4)

(Tp and Fp=plasma temperature and field strength, respectively; ne = plasma electron density); - t h e total spot current I: I = l ( j , a);

(5)

- t h e flow density of evaporation

jo = jo(p(T:), To) (p is the vapour pressure);

(10)

(2)

(Q~ is the net input of power density, ji the ion current density, U~ the potential drop, ]0 the flow density of evaporated atoms); - a solution of the heat conduction equation, taking resistance heating into account Tc = T~(Qc, a, j )

(9)

and finally the ion current density towards the cathode ji = ji(a, ne, Tp, Fp).

Qc = Oc(ji, je, U~, T,, Fc, jo)

(8)

(6)

In order to complete the system further equations may be applied for the particle and momentum balances of the plasma (to get the plasma current densities jep and lip at the anode side of the plasma ball), by hydrodynamic equations of the plasma expansion, the current transport by conduction and convection, diffusion equations within the plasma, the equation of state, or others. In some cases empirical relations are included to obtain definite results in a simpler way. 4. Differences and trends

In principle, the published theories of the stationary arc spot agree in the use of a set of equations similar to the above list. They differ in special suppositions, in the mathematical formulation, in the application of approximations, in the suppression of equations or the inclusion of additional equations and effects, in the different weight that is lent to alternating possibilities, or finally in the method of solution and the ways of interpretation. After the pioneering work of ~ and Greenwood [15] and still earlier efforts, further

E. Hantzsche/Theory of cathode spot phenomena

,,o 1

10

0

(eVcm2 s ) -t a -5

F-5.10?V/cm

-10

T-2.10

K/

'

'

-15

-5

,

,

,

i

,

i

~

i

i

i

10

i

i

I

I

5

i

I

-5

0

l

I0

I

lg ~ (Alcrn2eV) dc

b •

////~

F.5.10~V/cm

-5 10

5

0

-5

Fig. 1. Thermo-field emission of electrons (examples): (a) supply function N(T) and emission probability D(F) as a function of the electron energy e, with e = 0 at the Fermi level, work function ¢, = 4.5 eV ( T = temperature, F = field strength); (b) energy distribution function of the emitted electrons; a = pure thermionic emission, c = pure field emission, a + b + c = thermo-field emission: (1) if T = 2 x I(PK, F = 2 x 10~ V/cm and (2) if T = 5 x 103K, F = 5 x 107 V/cm.

E. Hantzsche/Theory of cathode spot phenomena

investigations were published by Koslov and Chvesyuk [16], Kulyapin [17], Djakov and Holmes [48], Beilis [18], Hantzsche [19] and Holmes [20]. An outstanding theory of this kind is that of Ecker [21]. Further new attempts are proposed by Harris and Lau [22], Kubono [23] and Nemtshinsky [24]. Some trends and improvements especially of the newer work are as follows. (1) The replacement of a part of the equations by inequations, in order to account for the uncertainties of the exact equations that appear to be too complicated or insufficiently known, and to get a more reliable basis of the theory and a more thorough judgement of the results, that become less precise but more trustworthy. Such inequations have been applied by several authors in some cases. However, this procedure has been developed into a complete method by Ecker [21,25] (compare, too, Beilis and Lyubimov [26]). In the form of "existence diagrams" limited regions in the parameter space (e.g. in a ] - Tc-plane) are determined, in which an arc spot (=a solution of the system of equations) possibly exists; outside of the boundaries of these regions the existence of a spot is impossible (if the equations and inequations are correctly chosen). The results are largely independent of the arbitrariness of special and imperfect suppositions, but they are less useful the larger the existence area becomes, and the full validity of the inequations used remains questionable if further processes are taken into account. (2) A more thorough discussion of the processes that happen in the expanding cathodic plasma, such as the diffusion of electrons and ions, the ion acceleration and the jet formation, space charge effects, particle and momentum balances, etc. (Beilis et al. [27], Lyubimov [28], Moisches et al. [29], Harris and Lau [22]; [53]). (3) The effect of Joule heating and possible thermal runaway processes (l-Iantzsche [30], Ecker [25], Daalder [7]). (4) The inclusion of minimum principles instead of badly known complicated equations as empirical means or as derivations from the general principle of minimum entropy production, mainly with the aim of determining the

7

total potential drop (Ecker [25], Nemtshinsky [24], Kubono [23], Hantzsche [31]). (5) Consideration of the effects of surface roughness on the spot parameters (Hantzsche [19], Ecker [25, 32]). (6) The inclusion of additional electron emission processes, e.g. by the impact of ions or excited atoms (Engel and Robson [33], Holmes [20]). Moreover, energy balances, the influence of parameter changes, and further questions are discussed. In some of the above-mentioned points the theory already goes beyond the simplest stage of spot models. Of course, there remain several rather crude approximations and doubtful assumptions; especially the material constants (such as work function, electric and thermal conductivities, accommodation coefficients, effective ionization potential, evaporation energy) are insufficiently known under these physical circumstances. Generally, the state of the theory is both complicated and rather elementary: it is mainly restricted to single particle behaviour and balance equations, and kinetic theory has not yet been applied. These simplified models of a stationary spot may be called "conventional" ones. 5. Results

In spite of these restrictions, the models have yielded many convincing successes, the most important of which are as follows. (1) The explanation of the physical processes and their cooperation in the arc spot, that enable the cathode to achieve the current transition. Of course this is an explanation only within the limits of the basic suppositions of these models. In particular quantitative yields of the production processes of new charged particles and calculation of the stationary feedback mechanism of the spot; substantiation of the necessity of a very high current concentration at the cathode. (2) The calculation of the parameters of the arc spot, such as To, Fc, a, ], it, ji, I, Uc, p, Tp, n,, a, energy flows, erosion rates, and possibly still others, that are - partially at least- in fair agreement with the real values, as far as they can

E. Hantzsche/Theory of cathode spot phenomena Table I Results from stationary arc spot model calculations, Cu cathodes.

Author Ecker [21] "mode 0" Ecker [21] "mode 1" Ecker [25] (improved model) Beilis [18] Beilis, Lyubimov [26] (alternat. calcul.) Koslov, Chvesyuk [16] Kulyapin [17] Holmes [20] Hantzsche [19] (one example) Harris, Lau [22] (improved model) Kub0no'[23] Kubono [23] Nemtshinsky [24]

Current density j (A/cm 2)

Spot surface temperature Tc (K)

Related to the spot current I

1.0 x 106

4000

~-50

1 x l0 s

6000

~-2000

1.5 x l 0 s 5.3 x 104

3600 3730

~30 ~-200

3.5 x 106 1 × 107 1 x 106 4.3 X 107

4170 (4200) 3860

~200

5500

---~15

2 x l0 s 1.2 × 10s 1.3 x 107 2.5 x 106 4.7 x los 5 x los

3700 5600 (5200) 4100 4300 3940

~20 ~9 ~80 ~10 ~100 ~-100

(A)

The results depend on the details of the model equations, on the assumed spot current, and on several material parameters.

I/A 3

,

*

i

*

,

i l l

i

101

,

,

J

,

,

i l l

102

i

,

i

,

l

i

* * I

103

Fig. 2. Spot current density j as a function of the current I ; results from model calculations of stationary arc spots on Cu cathodes, with. the indicated approximate range of possible values depending on the choice of several parameters; models: (1) Ecker [21]; (2) Beilis [18]; (3) Kubono [23]; (4) Hantzsche [19].

E. Hantzsche/Theory of cathode spot phenomena

be measured (table I, fig. 2). Particularly, such models may explain qualitatively and in a sense even quantitatively the existence of an upper and a lower limit of stable spot currents. Moreover, from an investigation of the existence areas we may conclude that a second "mode" of the arc may be possible with very high current densities and non-stationary behaviour [21]. Altogether, the cathode spot models of this kind have been proved to be an applicable first approximation of a complete theory of the arc cathode processes.

6. Problems There remain many questions and problems, as was to be expected, that are insufficiently solved or completely disregarded. If one looks at the flitting luminous dot on a cathode, o r - w i t h the aid of a scanning electron microscope - at the melting tracks left behind by the spot, some doubts are unavoidable as to whether the "stationary" models are indeed a good approximation. The most important of these problems are as follows. (1) The inhomogeneity of the real surface, mainly: its roughness, the existence of protrusions and ridges, or differences in the chemical state of the surface. By such influences the spot parameters are changed. This situation may be still included in the models discussed, at least in a statistical manner ("rough surface models" of Ecker [25, 32], spot model of Hantzsche [19]). The inhomogeneities may be connected with a substructure of the cathode spot, as was observed in some experiments (e.g. Kesayev [34]). (2) The random motion of the spot. Even in this case it is possible to include the consequences of this effect as corrections into the models of stationary spots (Ecker [21]: the only significant change is that of the energy balance). However, these models furnish no explanation of the causes and of the laws of the spot motion. (3) Still more important are the other timedependent processes in the arc spot (its motion is only a consequence): the random fluctuations of

9

voltage, current and light emission that indicate the rapid change of the surface state; the melting and the crater formation, i.e. the continuous and thorough reshaping of the surface; the destruction and production" of inhomogeneities; the surface erosion and the ejection of fast droplets; the possibility of a thermal runaway by growing resistive heating; the explosive destruction of sharp tips, at least during breakdown or initiating a jump of the spot, but possibly also within arc spots (e.g. Bugayev et al. [35], Kanzel et al. [36], Mitterauer [37], Hantzsche et al. [38]). Obviously these processes are clearly beyond the range of any kind of stationary spot models. (4) There are minor objections, for instance: no complete agreement between measured and calculated parameters even with consideration of the large limits of error (e.g. the spot radius becomes too large compared with the real crater radius), ambiguity in the choice of equations or alternative possibilities, sensitiveness of the results to such changes or to changes of the material constants (compare table I). (5) The experiments show the existence of two types of arc spots (Rakhovsky [1]) with rather different properties (as to the erosion rates, the velocities, the crater tracks, the stability). Within the models discussed there is no explanation of this fact. (6) Finally it may be proved that a strictly stationary spot is not possible on refractive cathodes (Hantzsche [39], compare Ecker [25], Beilis et al. [26]), and the same is true even regarding a slowly moving spot.

7. Non-stationary spot models -All the observations and measurements show that the arc spot is an exceptional non-stationary phenomenon in several respects. Therefore, any model of the "conventional" kind must be incomplete. It is necessary to develop a non-stationary or "dynamic" model of the spot that includes the above-mentioned effects and especially the time dependence of the processes. Such dynamic models make use of the experiences from the investigations of vacuum breakdown, for instance concerning the proper-

10

E. Hantzsche/Theory of cathode spot phenomena

ties of the plasma expansion and the occurrence of the very rapid production of metal vapour clouds and of flashes of electron emission, caused by fast heating (in about 10-1°s or even less) of sharp protrusions or of localized contaminations on the surface. The most extreme conception of a dynamic model reduces the arc spot to a succession of such explosive emissions: as soon as the expanding plasma cloud reaches a suitable point, producing a sufficient field strength, the next explosion and plasma production is triggered. Ideas of this kind are proposed by Mesyats and coworkers [35,40], Lyubimov and Rakhovsky [41], Mitterauer [37], and Bolanowski et al. [42]. In principle, these explosive processes are possible within the spot and they may be able to sustain an arc spot, at least qualitatively; and, doubtless, they explain its stochastic behaviour. However, there are some doubts as to whether such a model is satisfactory: the manifold relations that are described by the conventional models cannot be disregarded, the yield of the explosions may be quantitatively insufficient to maintain the arc without other processes (compare Ecker [25, 32]), and the production of new protrusions or other suitable sources of the plasma is questionable, though some possibilities have already been considered. Because of these arguments it seems more appropriate to use a combined model, i.e. to complete the conventional model (in a timedependent form) by the typical dynamic processes that are not at all confined to explosive ones, or to go the opposite way with the same aim. Probably the role of these dynamic processes in the spot mechanism will be different, depending on the cathode material, the surface state, and the current. The following aspects have to be taken into account. (1) Time-dependent heat conduction in the volume below the spot area (propagation of a heat wave), with the inclusion of resistance heat production (Ecker [25], Daalder [7]) that may lead to a thermal runaway if the current or the current density is sufficiently high (Hantzsche

[30]).

(2) Melting of the metal, propagation of a melting front, and time-dependent evaporation at the surface. (3) Acceleration of the molten layer by the plasma pressure (i.e. mainly by the ion pressure) in the radial direction, squeezing it out of the central spot region with the excavation of a crater, pushing the fused mass to bulges at the rims, and ejection of liquid droplets (Hantzsche et al. [38]). The electrostatic forces in the double layer, the viscosity, and the surface tension have to be taken into account. In a first approximation the combined problem of the melting front propagation and of the crater formation may be treated in a quasi-stationary manner (Hantzsche [43]). From points 2 and 3 we are able to make statements concerning the erosion rates in dynamic arc spots. (4) Joule and impact heating of protrusions (possibly even Nottingham heating), melting, rapid evaporation and increased electron emission, assisted by field enhancement, plasma density and ion current density enhancement, and by the reduction of heat conduction losses. All of these connected processes with feedback properties allow each intermediate stage between an almost steady state and an inertial-controlled explosion, depending mainly on the shape of the protrusion and the external conditions (Hantzsche [44]). This may occur with solid tips already existing near the spot centre, or with liquid fountains ejected out of the crater rims (the explosion will happen in a time short compared with the smoothing time by surface tension). Some kind of explosion by thermal runaway may even be possible in a smooth arc spot if the current density is sufficiently high (Ecker [35]). The calculation of heating of a protrusion is already a difficult task (Mitterauer et al. [45]), and this is all the more valid in view of the whole destructive process of the protrusion. (5) The opposite process, namely the generation of new protrusions. Here we may have: the formation of liquid tips by ejection out of the crater pool, caused by the plasma pressure, or by electrostatic forces; the formation of solid tips by condensation and crystallization from the vapour or the melt (growing of whis-

E. Hantzsche/ Theory of cathode spot phenomena

kers); surface diffusion enhanced by the high temperature, the high field strength and by ion impact; and finally the production of tips by the high-velocity impact of liquid or solid microparticles. Of course, these are only plausible suggestions, there is no quantitative theory. (6) The influence of such explosive processes (and of other less dramatic but time-dependent events affecting the local electron emission and evaporation) on the plasma parameters and the ion current, that become time and space dependent, too, with a feedback action on the cathode processes (by changing the ion current density, the field strength, etc.). (7) The causes of the spot motion must become comprehensible from these conceptions. At first glance the motion seems rather unintelligible because the spot has nowhere better conditions for existence than in the crater region. In principle there are two explanations: (a) the spot deteriorates the conditions by its burning, until it can no longer exist and must then jump to a new suitable place (e.g. Daalder [7], Ecker [25]), and (b) the spot produces in its neighbourhood still better conditions for existence and, therefore, changes to that place. Both explanations may be true, because experiments have shown (Jiittner and coworkers [46]), that two kinds of spot motion are detectable: normally it is a random walk with steps of the order of one spot diameter, but sometimes very quick motions or jumps occur bridging a larger distance. In any case, a breakdown-like process with an explosive plasma production seems to be the starting point. (8) Not directly connected with these dynamic surface processes is the question of a more thorough description of the plasma expansion, the potential distribution, the particle acceleration (explanation of the suprathermal ions escaping from the spot region) and the plasma jet formation (e.g. Lyubimov [28,53]) with regard to the time dependencies.

complicated n0n-linear differential equations of hydrodynamics, thermodynamics and electrodynamics must be included in the theory that is treatable only in a numerical manner, if at all. In spite of this, a few simple results are derived by approximative solutions of special problems, for instance: (a) the limit of thermal runaway by resistive heating in the cathode: I/a=2~X/Ktro

(11)

(Hantzsche [30, 47]), with the heat conductivity K = const, and the electric conductivity o" = tro/T, o'0 = const.; (b) the gross erosion rate X, x = VK/o'o arc

cos(To~Tin)

x [c In(Tm/To) + cm/Tm] -I

Altogether,

several

additional and

highly

(12)

(Daalder [7]), where Tm = melting temperature, To = temperature at large distances from the spot, c=specific heat capacity, Cm=specific melting heat; or X = (3moM/16kTc) u2. p/j

(13)

(Kimblin [2], Ecker [21]), where m0 = mass unit, M = relative mass of the ions; or X = UcD(1 + 'y) ( c ( r m - To) ~- Cm)]-1

(14)

(Hantzsche [47]), where y = je/ji. The net rates are between 10 and 30% of these gross rates; (c) the (maximum) spot radius a = l[21rX/~-o~oarc cos(To/T.)]-I

(15)

(Daalder [7]); (d) the mean random velocity of a spot t~,= (2/9). (IUc) u2

X [ap(l + y) (C(Tm - To) + Cm)At] -I/2 8. A few results

11

(16)

(Hantzsche [43]), with the density p and the time intervals of the spot observation At; or

12

E. Hantzsche/Theory of cathode spot phenomena Table II Some typical time parameters in the arc spot.

Designation

Formula

Typical value

Heating time constant

tl = pca2/K

~200 ns

Joule heating time constant

h = PCtro/J2

~200 ns

Acceleration time constant

t3 = a(p/j) 1i2 ( e / m U ) TM

~10 ns

Residence time constant

t4 = a p c T / j U

~10 ns

Smoothing time constant

t5 = (.oa3/s) ta

~1000 ns

Application heating and cooling of the arc spot controlled by heat conduction resistive heating of arc spots, thermal runaway removal of the molten layer, formation of splashes and droplets excavation of the crater, temporal step of spot motion smoothing of inhomogeneities by surface tension

p = density, c = specific heat capacity, a = spot radius (crater radius), r = heat conductivity, o, = or0/T = electric conductivity, T = temperature, j = current density, e/m = specific charge of the ions, U = potential drop, s = constant of surface tension. The formulae are given without numerical constants that depend on the model. The "typical values" relate to Cu cathodes and are order-of-magnitude values only, with a = 5/zm, j = 5 × 107 A/cm 2, U = 15 V. The application to the larger area of the spot plasma or to the smaller area of a substructure (e.g. protrusion) requires a change of both a and j. (tb t2) ~" ta is a further indication of the non-stationarity of arc spots: not the whole spot volume, but only a thin sheath is heated and removed after melting [43].

(17)

G = 2 K / c p • ( f i r ~ I ) tr2

(Ecker [25]); (e) the expansion velocity of the cathodic plasma t7v = [ 4 y ' / ( y ' - 1)" (e. +

ZeF)] -in

(18)

(Bugayev et al. [35]), where y ' = a d i a b a t i c exponent, es = specific heat of sublimation, Z = mean ion charge, eF = Fermi energy. This list may still be extended. It is not possible to discuss here the special suppositions of these formulae. Some further simple conclusions result from a comparison of the different time scales of the arc spot processes (table II) or from energy balances (table III).

9. State of things Although some of these results disagree, they are not in obvious contradiction with the measured values whereby the uncertainties and the probable margins of error must be taken into account. One may s t a t e - i n present-day c o n d i t i o n s that an agreement with experiment is almost in any case attainable (e.g. because many material parameters are very inexactly known), and therefore it cannot be considered as a sufficient proof that the theoretical model is correct. Moreover, the achievement of a more qualitative understanding of the physical processes that happen in an arc spot and of their mutual interactions, is a much more valuable result of the above-mentioned investigations than the special

E. Hantzsche/Theory of cathode spot phenomena

13

Table I I I Heating, evaporation and explosion of protrusions, summary (from [38]). T h e time constants, current densities and power inputs relate to Cu and a typical dimension of r0 = 0.1/.*m. I

Explosion/ Current (evaporation) density ume constant re Ins} i(A cm "2) Joule heot generotion dominont

Processes

Power

input {W)

Completely inertia-controlled explosion.

0'01

Explosive (dynamic} evaporation,

5x101° Lattice energy inessential Fast explosion, heat losses

decreasing time

inessential 7x 109

100

constont

I0 .._~'xplosion 'time time=heat c°nduci'°n ~ " - l -

0"1 'Slow' explosion, controlled by heat conduction Ion impact ondelectron emission heating

II 1(102) ]{10 3) (10~)

. . . . . , melting

5 xl08 -~imit of stotionority, very ropid evaporation (AT~ 3000 IO'~ Considerable stationary heating

dominont

I essential

S'tolionory evoporotion,

0"I (105) 3x107

Slow evaporation Inessential heating (AT
(10°)

results, which should not be overestimated. It opens the way for the future successful treatment of these processes. Doubtless, such a dynamic cathode spot model that includes all the time-dependent effects listed above is much more satisfactory and more realistic than the conventional models of a stationary spot. The existence conditions of the arc spot are improved not only by explosive processes, but by any kind of surface roughness, because the average yield of the vapour and electron emission is enhanced. However, the present difficulty is that until now no quantitative and satisfying theory for any one of the above-mentioned dynamic processes exists, only approximative discussions with semi-quantitative model character, not to speak of a theory of the non-stationary spot as a whole. The situation is still completely in an undeveloped

I increasing temperoture ond evoporotion rote

0 "01

state, contrary t o the case of the stationary spot models. The investigations concern several of the detail processes, partly with rather different assumptions and with conclusions that are even opposite in some cases. As an example, the explanation of the spot motion by Ecker [25] may be opposed by that of Daalder [7]: somewhat simplified, the spot must either move because it becomes too hot (and, therefore, the drop in voltage will become too large), or because it becomes too cool (since the excavation results in a drop in the current density in the spot and therefore of the resistive heating). Both seem to be possible causes, and there are still more explanations (e.g. [38, 24]). According to Rakhovsky [1] the spot moves because of the spreading plasma cloud that triggers new explosions of protrusions. Another view is that the random motion of the spot is

14

E. Hantzsche/Theory of cathode spot phenomena

caused by explosions of liquid splashes ejected out of the crater pool (Hantzsche et al. [38]), that shift the spot centre to this place, in addition to larger jumps to already existing solid tips or suitable contaminations, that may also be formed by droplet impact. Comparable discrepancies exist as to the role of resistive heating compared with the surface (ion impact) heating (e.g. [7], [25], [47]); as to the role of the individual explosive processes compared with the averaged (quasi-stationary) processes (compare [25]); as to the essential erosion processes (evaporation or particle emission); even with respect to the seemingly simple question of the mean current density where not only the results from the models are different (table I), but the measured values differ even more because it is questionable whether the cathode plasma cloud or the crater should be considered as the decisive area of the main current transition at the cathode surface.

10. Contaminated surface The results from the models of arc spots- both the stationary and the non-stationary o n e s show no indication of the existence of two completely different types of spots, as were observed by Rakhovsky [1, 41] and others. One possible explanation is the assumption that the high-velocity type-1 spot is essentially non-stationary and is generally maintained by explosive emission processes, while the slowly moving type-2 spot is based on almost stationary thermionic emission. However, from several other experimental investigations with contaminated surfaces (e.g. Guile et al. [49], Jfittner [50]) it seems more probable that the type-1 spot is the state of arc spots on a contaminated surface, i.e. on a cathode with a thin insulating layer (e.g. an oxide layer), while only the type-2 spot is the object described by the models, i.e. the spot on a clean surface. Indeed, there are already drastic differences in the phenomenology if such a layer exists: the craters become very small and very numerous, many of them are formed almost simultaneously (not one after the other as in the case of clean

cathodes), even between the craters there are indications of erosion, and the velocity of propagation is very high and corresponds to the expansion velocity of the plasma, while the erosion rate is small. This behaviour depends on the type and thickness of the layer. The theory of arc spots on contaminated surfaces is still more troublesome than that on clean surfaces, not only because of an additional substance with a further set of material constants, but mainly because of a very different mechanism: i.e. precipitating ions from the plasma form surface charges, resulting in electron emission by the tunnel effect (Malter effect). If the increase of the surface charge is sufficiently rapid (compared with its reduction by conduction losses or neutralization by plasma electrons) a local breakdown of the layer may occur, a hot conducting and expanding channel is formed, with a high current density and a sudden release of plasma. In this way microcraters and holes are generated in the l a y e r - a process similar to the explosive evaporation of a protrusion. The layer will be more or less destroyed, depending on its thickness, on the crater current, and on the length of time a spot exists. This, of course, is only a very rough and qualitative description; a theoretical treatment is largely an open question. It should be mentioned, however, that even on an almost clean surface localized contaminations may act as starting points for a new spot just like protrusions. As to the interpretation of the experiments, the knowledge of the state of the surface contaminations is essential, and its non-observance has caused some contradictions and confusion. 11. Open problems: Summary Many of the still unsolved problems have been mentioned above. There are further questions that should be the subject of future investigations, as follows. (a) The laws and causes of the spot motion as a random walk on a rough surface, the residence time scale and the distribution o f the step distances; also, the spot motion in a magnetic field (with regard to the retrograde and prograde motion there exist several models of this surpris-

E.

Hantzsche/Theory of cathode spot phenomena

ing effect, but none of them is convincingly proved or disproved). (b) The theory of the arc spot in a highpressure ambient gas (from our knowledge of the cathode spot processes we expect that there should be no essential difference compared with the vacuum are spot. Former work may be a starting-point, compare e.g. [55], [56]). (c) The stability of the single cathode spot and of the multi-spot arc (compare [54]), the transition to the high,current constricted form of the vacuum arc. (d) The non-stationary cathode plasma: its ambipolar expansion, the distributions of the space charge and the potential (existence of a potential hump?), the possible jet formation and the role of the self-magnetic field, the relation between conductive and convective current transport, and shock waves caused by explosive processes. (e) A fascinating general point of view is the interpretation of the arc spot as a dissipative structure in connection with the application and justification of minimum principles [31]. Again this is a list that may be extended considerably. However, it should not produce the impression of a disappointing view of ignorance. Indeed, we can state in summary that the arc spot as a quasi-stationary process is on the whole well understood and calculable, and that many of the non-stationary processes are well known, too, and partly even simple quantitative results are obtained. But the general valuation of non-stationarity is still questionable, and there are still too many possibilities for reasoning and too many variants of explanations; these must be eliminated step by step. Although no generally accepted theory of the whole of dynamic arc cathode phenomena has existed until now, and in spite of the many diverging ideas that have been proposed, a great number of encouraging results could be achieved, and the agreement of the views is growing in consequence of the cooperation of experimental and theoretical investigations. References [1] V.I. Rakhovsky, IEEE Trans. Plasma Sci. PS 4 (1976) 81.

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[2] C.W. Kimblin, J. Appl. Phys. 44 (1973) 3074. [3] A.A. Plyutto, W.N. Ryshkov and A.T. Kapin, ZETF 47 (1964) 494. [4] W.G.J. Rondeel, J. Phys. D 6 (1973) 1705. [5] J.E. Daalder, J. Phys. D 9 (1976) 2379. [6] J. Kutzner and Z. Zalucki, Int. Conf. on Gas Disch., London, 1970, p. 87. [7] J.E. Daalder, Thesis, Ein¢lhoven (1978). [8] B. J/ittner, Akad. d. Wigs. d. DDR, ZIE-Preprint 78-14 0978). [9] H.C.W. Gundlach, Proc. 5th Int. Symp. Disch. Electr. Insul. Vat., Poznan, 197.2, p. 249. [10] J.A. Foosnaes and W.G.J. Rondeel, J. Phys. D 12 (1979) 1867. [11] C.W. Kimblin, Proc. l l t h Int. Conf. Phen. Ion. Gases, Prague, 1973, p. 73. [12] J.E. Daalder and P.G.E. Wielders, Proc. 12th Int. Conf. Phen. Ion. Gases, Eindhoven, 1975, p. 232. [13] T. Utsumi and J.H. English, J. Appl. Phys. 46 (1975) 126. [14] Y. Udris, Int. Conf. on Gas Disch., London, 1970, p. 108. [15] T.H. Lee and A. Greenwood, J. Appi. Phys. 31 (1961) 916. [16] N.P. Koslov and V.I. Chvesyuk, Z. Tekhn. Fiz. 41 (1971) 2135. [17] V.M. Kulyapin, Z. Tekhn. Fiz. 41 (1971) 381. [18] I.I. Beilis, Z. Tekhn. Fiz. 44 (1974) 400. [19] E. Hantzsche, Beitr. Plasmaphys. 14 (1974) 135, and IPH AbschluBbericht, 1975, Anlage 1. [20] A.J.T. Holmes, J. Phys. D 7 (1974) 1412. [21] G. Ecker, General Electric Rep. 71-C-195 (1971), 73 CRD 053/056 (1973). [22] L.P. Harris and Y.Y. Lau, General Electric Rep. 74 CRD 154 (1974). [23] T. Kubono, J. Appl. Phys. 49 (1978) 3863. [24] V.A. Nemtshinsky, Z. Tekhn. Fiz. 49 (1979) 1373. [25] G. Ecker, Forschungsbericht d. Ruhr-Universit/it Bochum 79-02-001 (1979), and in: J.M. Lafferty, ed., Vacuum Arcs (John Wiley, New York, 1980) p. 228. [26] I.I. Beilis and G.A. Lyubimov, Teplofiz. vys. Temp. 13 (1975) 1137. [27] I.I. Beilis, G.A. Lyubimov and V.I. Rakhovsky, Dokl. Akad. Nauk 203 (1972) 71. [28] G.A. Lyubimov, Z. Tekhn. Fiz. 47 (1977) 297. [29] B.J. Moisches and V.A. Nemtshinsky, Z. Tekhn. Fiz. 50 (1980) 78. [30] E. Hantzsche, Beitr. Plasmaphys. 12 (1972) 245. [31] E. Hantzsche, Dissertation Akad. cl. Wigs. d. DDR. (1978). [32] G. Ecker, IEEE Trans. Plasma Sci. PS 4 (1976) 218. [33] A.v. Engei and A.E. Robson, Proc. Roy. Soc. 243 (1958) 217. [34] I.G. Kesayev, Cathode Proc. of the Electr. Arc, Moscow, 1968. [35] S.P. Bugayev, E.A. Litvinov, G.A. Mesyats and D.I. Proskarovsky, Uspekhi Fiz. Nauk 115 (1975) 101. [36] V.V. Kanzel and V.I. Rakhovsky, Proc. 6th Int. Symp. Disch. Electr. Insui. Vacuum, Swansea, 1974, p. 265.

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[37] J. Mitterauer, Acta Phys. Austr. 37 (1973) 175. [38] E. Hantzsehe, B. Jiittner, V.F. Puchkarov, W. Rohrbeck and H. Wolff, J. Phys. D 9 (1976) 1771. [39] E. Hantzsehe, Phys. Lett. 50 A (1974) 219. [40] E.A. Litvinov, G.A. Mesyats, A.F. Shoubin, L.M. Buskin and G.N. Fursey, Proc. 6th Int. Syrup. Disch. Eleetr. Insul. Vacuum, Swansea, 1974, p. 107. [41] G.A. Lyubimov and V.I. Rakhovsky, Uspekhi Fiz. Nauk 125 (1978) 665. [42] B. Bolanowski and A. Sobieszczuk, Proc. 2nd Int. Syrup. Switch. Arc Phen., Lodz, 1973, p. 11. [43] E. Hantzsche, Beitr. Plasmaphys. 17 (1977) 65. [44] E. Hantzsche, Proc. 12th Int. Conf. Phen. Ion. Gases, Eindhoven, 1975, p. 237. [45] J. Mitterauer, P. Till and E. Fraunschiei, Inst. f. Industr. Elektr. TH Wien, Rep. IE-75-01 (1975).

[46] A.I. Bushik, B. Jiittner and H. Punch, Beitr. Plasmaphys. 19 (1979) 178. [47] E. Hantzsche, Akad. d. Wiss. d. DDR, ZIE-Preprint 79-10 (1979). [48] B.E. Djakov and R. Holmes, J. Phys. D 4 (1971) 504. [49] A.E. Guile and A.H. Hitchcock, Arch. Elektrotechn. 60 (1978) 17. [50] B. J/ittner, Beitr. Plasmaphys. 18 (1978) 265. [51] H.C. Miller, IEEE Trans. Plasma Sci. PS 5 (1977) 181. [52] J.E. Daalder, J. Phys. D 10 (1977) 2225. [53] M.P. Sekzer and G.A. Lyubimov, Z. Tekhn. Fiz. 49 (1979) 3. [54] J.C. Sherman, R. Webster, J.E. Jenkins and R. Holmes, J. Phys. D 11 (1978) 379. [55] C.W. Kimblin, J. Appl. Phys. 45 (1974) 5235. [56] E.W. Gray, IEEE Trans. Plasma Sci. PS 6 (1978) 384.