Universidade da Madeira Theory and modelling of plasma-cathode interaction in HID lamps: a review...

Universidade da Madeira

Theory and modelling of plasma-cathode interaction in HID lamps: a review

Theory and modelling of plasma-cathode interaction in HID lamps: a review

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M. S. BenilovDepartamento de Física, Universidade da Madeira, Portugal

Talk to be delivered at the Model Inventory COST 529 workshop

Funchal, April 13, 2005

Acknowledgements: M. D. Cunha, G. V. Naidis,

N. A. Almeida, M. Carpaij...

- 2 - Univ. MadeiraDept. Física

• Introduction• The model of non-linear surface heating - Solution on the plasma side - Solution inside the cathode body• Comparison with the experimental data • General features of the solution - Diffuse mode - Axially symmetric spot modes - 3D spot modes• Stability of the diffuse mode• Effect of metal halides - Variation of properties of the near-cathode plasma layer - Variation of the work function of the cathode surface

Contents of the talk


Cathodes of arc discharges: diffuse and spot modes

Cathode of an arc discharge in argon. W, R = 0.75 mm, p = 4.5 bar, I = 2.5 A. Courtesy of D. Nandelstädt, J. Luhmann, and J. Mentel.

The current is distributed over the front surface of the cathode in a more or less uniform way; the diffuse mode.

The current is localized in a region occupying a small fraction of the surface (cathode spot); a spot mode.

cathode

plasm a


History• Multiple modes of current transfer to hot arc cathodes were

first observed in 1951 (W. Thouret, W. Weizel, and P. Günther) and have been under the active theoretical investigation ever since.

• A self-consistent and universally recognized theory started to emerge 40 years later, in the 1990s.

• What are reasons of this lag? - It was realized relatively late that this is a self-organization problem.

- Measurements are difficult. Reliable data on the near-cathode voltage drop started to appear only in 1998. Reliable experimental data, referring to integral characteristics, allow more than one theoretical interpretation (at least, some people think so).

• Theoretical vs. applied physics approach: the theory which is generally recognized by now has been developed on a theoretical-physics basis.


Theoretical vs. applied physics approach

• A model should be based on fundamental physical principles and not rely on empirical parameters and/or arbitrary theoretical suppositions such as a "principle" of minimum voltage.

• All simplifications must be based on estimates.

• You cannot drop a term from equations just because you have not enough information or because it complicates a solution.

• If the model neglects important effects or is inconsistent with fundamental physical principles, it is a bad model

even if it generates beautiful pictures adored by the management and/or “agrees with the experimental data”.

• There has been a huge number of models violating these principles. Nearly all of them have been forgotten by now.


ions, plasma electrons

emitted electrons

BULK PLASMA

IONIZATION LAYER

SPACE-CHARGE SHEATH

CATHODE BODY

Structure of the near-cathode plasma region

Te = Th Ionization equilibrium ne = ni Bulk plasma No Yes Yes Ionization layer No No Yes Space-charge sheath No No No


Flux of plasma electrons from the quasi-neutral plasma to the cathode:

Flux of ions:

Why are models without the sheath inconsistent?Some authors disregard the near-cathode space-charge sheath: E. Fischer (1987), Flesch and Neiger (1998, ....)

A solution without a sheath does not satisfy boundary conditions for ni ,ne on the cathode surface, which are governed by different processesWithout a sheath, ni = ne in the whole plasma region right up the cathode surface.

A solution without a sheath is physically meaningless: the total electric current is directed from the cathode into the plasma

Je

J i m iTe

2 meT i Te exp eUD

kTe

Je nes

48kTe me

exp eUD

kTe

Ji neskT i Te

m i

UD 0, Ar, Te

T i 1. . . 10 : Je

J i 75. . . 100


The model of non-linear surface heating

• There is a considerable voltage drop in a thin near-cathode layer

• A considerable power is deposited in the near-cathode layer

• Part of this power is brought to the cathode surface by the ions and plasma electrons while the rest is taken away into the bulk plasma by the electrons

• The energy flux to the cathode surface is generated in a thin near-cathode plasma layer which is independent of the bulk plasma

• The plasma-cathode interaction is independent of the bulk plasma and may be modelled independently The model of nonlinear surface heating

electrons

ions, plasma electrons

BULK PLASMA

NEAR-CATHODE LAYER

CATHODE BODY

power input


• Solution on the plasma side: the 1D problem describing the current transfer across the near-cathode plasma layer is solved and all parameters of the layer are determined as functions of Tw and U. In

particular, functions q = q(Tw,U) and j = j(Tw,U) are found.

• Solution inside the cathode: the equation of thermal conduction is solved with the boundary condition q = q(Tw,U).

A complete solution can be found in two steps:


The model is self-consistent

What is specified is not a distribution of the energy flux from the plasma over the surface but rather a dependence of the energy flux density on the local surface temperature, this temperature being unknown apriori!

The model should have multiple solutions, describing different modes of current transfer from the plasma to the cathode.


• The model was apparently first suggested more than 40 years ago (W. L. Bade and J. M. Yos, Technical Report of Avco Corporation, USA, 1963) and was apparently re-discovered more than once. However, in general this model was half-forgotten and advances in theoretical description of the plasma-cathode interaction for many years have been limited.

• A self-consistent and universally recognized theory based on this model started to emerge only in the 1990s.

• Developments on the theoretical side: - A modern theoretical description of the near-cathode plasma layer was

developed (M. S. Benilov and A. Marotta 1995, R. Rethfeld et al 1996). - It was shown that the model gives multiple solutions describing different modes

of current transfer from the plasma to the cathode (M. S. Benilov 1998). Switching of mode mechanisms “by hand” became unnecessary.

• Developments on the experimental side: - Detailed electrical and thermal measurements have been performed (J. Mentel et

al 1998, … ). Near-cathode voltages of up to 50V were reported! - Theoretical results predicted by the model of non-linear surface heating have

been confirmed.



By now, this model has been successfully used by many authors; e.g.,


- J. Wendelstorf 1999;

- S. Coulombe 2000;

- R. Bötticher and W. Bötticher 2000, 2001a, 2001b; - T. Krücken 2001; - W. Graser 2001; - M. S. Benilov and M. D. Cunha 2002, 2003a, 2003b; - R. Bötticher, W. Graser, and A. Kloss 2004 (a reproducible diffuse-spot

transition was achieved in the experiment and was quantitatively correctly described by 3D non-stationary simulations);

- L. Dabringhausen 2004 (simulations of steady-state 3D spots);- S. Lichtenberg, L. Dabringhausen, J. Mentel, and P. Awakowicz 2004

(simulations of steady-state 3D spots);- M. S. Benilov, M. D. Cunha, and G. V. Naidis 2004 (a multiple-species MH

plasma);- M. Galvez 2004 (collisional sheath; simulations of the whole lamp); - K. C. Paul et al 2004 (collisional sheath; simulations of the whole lamp);

- G. M. J. F. Luijks, S. Nijdam, and H. A. v. Esveld 2004.


Solution on the plasma sideThe ionization layer: a hydrodynamics description

Equations

• Equation of motion of the ion fluid accounting for ion inertia, ion pressure gradient, electric field force, friction force due to elastic collisions of ions with neutral particles, friction force due to ionization of neutral particles.

• Equation of balance of the electron energy in the ionization layer.

Boundary conditions

• On the plasma side of the ionization layer: ionization equilibrium.

• On the edge of the space-charge sheath: the Bohm criterion.

A complete solution is still missing. Frank Scharf’s talk deals with this problem.

The space-charge sheath: a kinetic description

Equations

• A kinetic equation describing the motion of ions,

• The Poisson equation.

Boundary condition at the sheath edge: the Bohm criterion


Density of the energy flux from the plasma to the cathode surface vs. the surface temperature. Tungsten cathode in the argon plasma, p = 1 atm. From M. S. Benilov and M. D. Cunha 2002.

Solution on the plasma side

Tw (103 K )

qp (108 W /m 2)

0 1 2 3 4 5

0.0

0.5

1.0

1.5

2.0

U = 10 V

12

14

T w (103 K )

qp (108 W /m 2)

0 1 2 3 4 5 6

0

5

10

15

U = 15 V

50

25


Solution on the plasma side

• In the range U< 20V, the dependence of qp on Tw for fixed U includes a

growing and falling sections separated by a maximum

Reasons of the maxima are, respectively, non-monotony of the dependence of the ion current on the electron temperature which is caused by a deviation of the ion current from the diffusion value; and rapid increase of the plasma electron heating which is subsequently overcome by thermionic cooling.

• At higher U, the dependence changes: a plateau appears, which subsequently gives way to two maxima and a minimum

Reason of the non-monotony is overcoming of the increase of combined ion and plasma electron heating by an increase of thermionic cooling which occurs when the plasma approaches full ionization


• Inside the cathode body: the thermal conduction equation

• At the cathode surface: density of the energy flux from the plasma (or to the cold gas) is a given function of the local surface temperature and of the near-electrode voltage drop:

q = q(Tw,U)

Solution inside the cathode body

cathode

plasm a

Methods of solution:

• Home-made codes

• ANSYS

• Femlab


Current-voltage characteristics of the diffuse mode on a tungsten cathode in the argon plasma. p = 2.6 atm. Lines: modelling. Points: experiment.

1, : h = 24 mm, R = 0.75 mm. 2, : h = 25 mm, R = 0.5 mm. 3, : h = 15 mm, R = 0.3 mm.

0 2 4 6 8

0

10

20

30

40

50U (V )

I (A )

1

2

3

Comparison with the experimental data


Temperature in the center of a tungsten cathode operating in the diffuse mode in the argon plasma. R= 0.75 mm. p = 2.6 atm. Lines: modelling. Points: experiment.

: h = 24 mm.: h = 29 mm. : h = 19 mm.

0 2 4 6 8

2.6

2.8

3

3.2

3.4

h=24 m mh=29 m mh=19 m m

Tw (103 K )

I (A )



Power removed by heat conduction and irradiated from the cathode surface in the diffuse mode on a tungsten cathode in the argon plasma. R = 0.5 mm. p = 2.6 atm. Lines: modelling. Points: experiment. 1, 2: Power removed by heat conduction. 3, 4: Power irradiated from the cathode surface. 1, 3, , $: h = 30 mm. 2, 4, , : h = 20 mm.

0 2 4 6

0

5

10

15

20

25

I (A )

P (W )

1

2

4

3



D. Nandelsätdt, M. Redwitz, L. Dabringhausen, J. Luhmann,S. Lichtenberg, and J. Mentel 2002, Fig. 12

Comparison of the total power losses obtained by pyrometric measurements (points) with the model by Benilov and Cunha (lines). p = 2.6 atm argon; pure tungsten cathodes, h = 20 mm.

1, : R = 0.3 mm. 2, : R = 0.5 mm. 3, : R = 0.75 mm (the cathode operated in a spot mode!)


0 2 4 6

10

20

30

40

50

60

I (A )

P (W )

1

2

3


D. Nandelsätdt, M. Redwitz, L. Dabringhausen, J. Luhmann,S. Lichtenberg, and J. Mentel 2002, Fig. 13

Comparison of the measured cathode fall with the models by several authors. p = 10 atm xenon; pure tungsten cathode, R = 0.6 mm, h = 14 mm.


0 2 4 6

8

10

12

14

16

m easu rem en tsB en ilo v an d C u n h aS ch m itz an d R iem an nF lesch an d N e ig e rB ö ttich e r an d B ö ttich e rW en d e ls to rf

U (V )

I (A )


General pattern of various modes

Current-voltage characteristics of different modes of current transfer. W, R = 2 mm, h = 10 mm, Ar, 1 bar

101 102 103 104 105

10

20

30

40diffuse m ode1st 2D spot m ode1st 3D spot m ode2nd 3D spot m odebifurcation points

U (V)

I (A )


General features of the solution: the diffuse mode

a b

c d e

W, R = 2 mm, h = 10 mm, Ar, 1 bar

101 102 103 104 105

10

20

30

40diffuse m ode1st 2D spot m odeasym ptotic1st 3D spot m ode2nd 3D spot m ode

U (V)

I (A )

a

b c

e

d


General features of the solution: 1st 2D spot mode

c

a

d

b

W, R = 2 mm, h = 10 mm, Ar, 1 bar

101 102 103 104 105

10

20

30

40diffuse m ode1st 2D spot m ode1st 3D spot m ode2nd 3D spot m ode

U (V)

I (A )

a

b

c

d


General features of the solution: 3D spot modes

W, R = 2 mm, h = 10 mm, Ar, 1 bar

a

c d

b

101 102 103 104 105

10

20

30


U (V)

I (A )

a

b

c

d


General features of the solution: 3D spot modes

W, R = 2 mm, h = 10 mm, Ar, 1 bar

a

c

b

d101 102 103 104 105

10

20

30


U (V)

I (A )

ab

c

d


1 0 1 1 0 2 1 0 3 1 0 4 1 0 51 0

2 0

3 0

4 0diffuse m ode2D spot m ode1st 3D spot m ode2nd 3D spot m ode

U (V )

I (A )

Stability limit of the diffuse mode

Limit of stability

W, R = 2 mm, h = 10 mm Ar, 1 bar

• The first bifurcation point represents the limit of stability of the diffuse mode: the diffuse mode is stable at I>Ibif and unstable at I>Ibif .

• This point can be calculated in a relatively simple way.


• The stability limit decreases rapidly with a decrease of the cathode radius. The same trend is observed in the experiment (Neumann 1987).

1 0 1 1 0 2 1 0 3

8

16

24

32

40R = 0 .975 m mR = 0.985 m mR = 1.1 m mR = 1.5 m mR = 2 m m

U (V )

I (A )

h = 14 m m


Current-voltage characteristics of the diffuse mode and limits of its stability. Tungsten cathode. Argon plasma, p = 1bar.


1 0 1 1 0 2

8

16

24

32

40

U (V )

I (A )

13

14

14.45

14.4

h = 10 m m

R = 1 m m


• The stability limit decreases rapidly with an increase of the cathode height. The same trend is observed in the experiment (Neumann 1987).




3 6 9 12

15

20

25

30

35A = 4.55 eVA = 4.5 eVA = 4.45 eV

U (V )

I (A )

R = 1 mm, h = 14 mm

• A small decrease of the work function originates a strong decrease of the stability limit. The same trend is observed in the experiment (Neumann 1987).




8 16 24 32 40 48

9

12

15

18Ar, 1 a tmAr, 2 a tmAr, 5 a tmXe, 1 a tmH g, 1 atm

U (V )

I (A )

XeH g

Ar

• An increase of the argon pressure result in an increase of the stability limit. The same trend is observed in the experiment (Lichtenberg et al 2002).

• A change from argon to xenon and from xenon to mercury results in an increase of the stability limit. The latter is observed in the experiment (Thouret 1951).

Current-voltage character-istics of the diffuse mode and limits of its stability. Tungsten cathode. R = 1 mm, h = 14 mm.


Metal halide plasmas

The presence of metal halides in the plasma affects the current transfer in two ways:

• Through variation of properties of the near-cathode plasma layer due to the presence of metal halides in the gas phase

• Through variation of the work function of the cathode surface due to formation on the surface of an alkali metal monolayer


Near-cathode layer in multi-species plasmas• The equations are reformulated in terms of effective coefficients, the

most important of these coefficients being the effective rate constant of ionization of a neutral particle by electron impact and the effective mean cross section for momentum transfer in elastic collisions between a positive ion and a neutral particle.

• The effective coefficients are calculated by means of averaging the corresponding individual coefficients over plasma partial composition evaluated at the plasma edge of the ionization layer.

• Partial composition of a plasma at the edge of the ionization layer is determined from equations of local balance of production and loss of every plasma species in volume reactions for given elementary composition of the mixture and given values of the heavy-particle temperature Th, electron temperature Te, and gas pressure p.

• There is no detailed balancing between direct and reverse reactions at the edge of the ionization layer, therefore the calculation requires, in addition to thermodynamic data, also kinetic data (in contrast to the case of an LTE plasma, Th=Te).


MH plasma: composition

• The plasma-producing gas in metal halide (MH) lamps represents a mixture of molecular and atomic species composed of Hg and (some of) the following elements: Na, Tl, Dy, Sc, Cs, I, and Ar.

• The main neutral gas-phase species in MH lamps are: Hg, Na, Tl, Cs, Dy, Sc, I, NaI, TlI, CsI, DyI, DyI2, DyI3, ScI, ScI2, ScI3.

• Ion species produced in ionization of atoms are Hg+, Na+, Tl+, Dy+, Sc+, Cs+, I+. As the ionization potentials of metal iodides are substantially higher than those of corresponding metal atoms (e.g., 5.14 eV for Na and 7.54 eV for NaI, 6.11 eV for Tl and 8.47 eV for TlI, 3.89 eV for Cs and 7.25 eV for CsI), the presence of molecular ions is neglected.

• Attachment of electrons to atoms I results in appearance of ions I-.

• Thus, the present model takes into account the following species: Hg, Na, Tl, Cs, Dy, Sc, I, NaI, TlI, CsI, DyI, DyI2, DyI3, ScI, ScI2, ScI3, Hg+,

Na+, Tl+, Dy+, Sc+, Cs+, I+, I-, e.


MH plasma: diffuse discharge

Maximal temperature of the cathode surface. h=14,5 mm, R=1mm, p = 5 bar. MH plasma: Hg:Na:Tl:I=0.89:0.005:0.05:0.055.

• An addition of metal halides to the mercury plasma results in a small decrease of the cathode surface temperature.

1 10 100

2.8

3

3.2

3.4

3.6

I (A )

T w (103 K )

H g

M H


MH plasma: diffuse discharge

Current-voltage characteristics and the stability limit. h=14,5 mm, R=1mm, p = 5 bar. MH plasma: Hg:Na:Tl:I=0.89:0.005:0.05:0.055.

• The near-cathode voltage drop in the MH plasma,

UMH, is slightly lower than UHg, the latter being in turn slightly lower than UAr.

• The stability limit in Hg is higher than that in Ar, in accord with a general tendency (lower near-cathode voltages correspond to a higher stability limit).

• An addition of metal halides results in a considerable expansion of the range of stability of the diffuse mode. The reason: decrease of the derivative ∂q/∂Tw.

1 10 100

0

10

20

30

40

50

I (A )

U (V )

A r

M H

H g


Variation of the work function

Work function of tungsten covered by a monolayer of Na or Cs in Na-Hg or Cs-Hg plasmas. p=5bar. Calculation with the use of the data from J. Almanstötter, B. Eberhard, K. Günther, and T. Hartmann 2002 and Welton 2002

0 1 2 3 4 5

1

2

3

4

5

N a-H gC s-H g

ZN a=10 -210 -310 -9

ZC s=10 -2

10 -9

10 -3

A f (eV )

Tw (103 K )

• Even smalls amounts of the alkali vapor in the plasma, of the order of 1% and less, can produce a significant effect on the work function.

• The decrease of the work function produced by adding Cs to the mercury plasma is stronger than that produced by adding Na and comes into play at higher values of the surface temperature.


Na-Hg plasma: effect of variation of work function

Lines: maximal temperature ofthe cathode surface and current-voltage characteristics. Diffuse mode on a W cathode, p=5bar. Points: stability limit of the diffuse mode with ZNa=0.08% on a cathode covered by a monolayer of Na (the full circle); in the Na-Hg plasma with ZNa=0.08% on a clean cathode (the open circle); in the Hg plasma (the square).

• Formation of the sodium monolayer affects the diffuse mode of current transfer in the same direction that the presence of metal atoms in the gas phase does: Tw and U decrease, range of stability of the diffuse mode expands.

• Formation of the sodium monolayer produces only a moderate effect.

0 10 20 30 40 50

0

10

20

30

40

50 Tw

U

0

1

2

3

4U (V )

I (A )

Tw

(103 K )H g

N a-H g, Z N a=1%

N a-H g, Z N a=1%

H g


Cs-Hg plasma: effect of variation of work function

• Formation of the cesium monolayer changes the plasma-cathode interaction dramatically: the temperature of the cathode surface decreases very strongly, the diffuse-mode current-voltage characteristic becomes N-S-shaped.

• The question of stability of the diffuse mode in the Cs-Hg plasma requires an additional mathematical treatment.

0 10 20 30 40 50

0

10

20

30

40

50

0

1

2

3

4U (V )

I (A )

Tw

(103 K )

Tw : H g

Tw : C s-H g, Z C s=0.01%

U : C s-H g, Z C s=0.01%

U : H g

Maximal temperature of the cathode surface and current-voltage characteristics. Diffuse mode of discharge on a tungsten cathode.


Conclusions

Considerable advances in understanding the interaction of high-pressure plasmas with thermionic cathodes have been achieved during the last decade.

A self-consistent and universally accepted theory and simulation methods of plasma-cathode interaction have developed and validated by a comparison with the experiment.

The most important features of the modern simulation approach are:

• The approach is based on first physical principles and does not involve adjustable or empiric parameters;

• Different modes of current transfer and transitions between them can be simulated in a completely self-consistent way;

• Possibility of modelling of complex 3D geometries, also with account of non-stationary effects;

• Possibility of modelling of complex mixtures, such as MH plasmas.

Universidade da Madeira Theory and modelling of plasma-cathode interaction in HID lamps: a review...

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