Growth of silicon ribbons from powder using a plasma spray torch

Thin Solid Films, 202 (1991) 359 372

CONDENSED MATTER FILM BEHAVIOUR 359

G R O W T H O F S I L I C O N RIBBONS F R O M P O W D E R U S I N G A P L A S M A SPRAY T O R C H

B. KAYALI, R. SURYANARAYANAN AND M. RODOT

Laboratoire de Physique des Solides de Bellevue, CNRS, I place A. Briand, F-92195 Meudon (France) M. VARDELLE AND A. VARDELLE

Laboratoire Cbramiques Nouvelles, C N R ~ U R A 320, Universitk de Limoges, 123 Av. Albert Thomas, F-87600 Limoges (France)

A . AMRI

CEA-Dkpartement d'Analyse de Sfiretb, F-92260 Fontenay-aux-Roses (France) (Received February 28, 1990; accepted February I 1, 1991)

Plasma spraying of silicon powder onto an alumina substrate allows the production of self-supported ribbons having good photoelectric quality in spite of their poor microstructure and purity. In the first part of this paper, the plasma particle interactions are modelled so as to determine the position, velocity, temperature and mass loss of a silicon particle as a function of the spraying distance;

measurements of the velocity and temperature are used to validate this simulation.

The second part is devoted to some local measurements of the impurity content:

most impurities are found to be included in microprecipitates, leaving the silicon matrix very pure, which explains its good photoelectric quality.

1. INTRODUCTION

In previous papers 1 3 the possibility of using plasma spray ("PLAST") deposition of silicon powder to obtain silicon ribbons of interesting electronic properties was demonstrated. Most recently 3 it was proved that large area (25 cm 2) self-supported ribbons could be obtained under optimized conditions and that, in spite of a high content of impurities such as copper, iron and sodium, this material could be highly photosensitive, sufficiently, in principle, for valuable solar cells to be obtained from it: a best value of the minority carrier (electron) recombination length of 69 ~tm was reported for a P L A S T ribbon recrystallized by electron gun annealing.

In the present paper we bring two important complements to this study.

First we shall present a simulation of the plasma-particle interactions and some in-flight measurements of the particle velocity and temperature which are compared with the calculated values. Between the torch in which a silicon particle is injected and the substrate on which it is collected, its time of flight is about 1 ms, during which it is deflected, accelerated, heated and partly vaporized. The model, well validated by the experiments, allows us to justify the optimized projection conditions which were empirically chosen, and in particular the powder size distribution, the injection velocity and the spray distance.

Second, we shall improve the knowledge of the impurity distribution in the

0040-6090/91/$3.50 ~) Elsevier Sequoia/Printed in The Netherlands

360 R. KAYALI et al.

ribbons, using local secondary ion mass spectrometry (SIMS) analyses. Most impurities are found to be included in complex oxide microprecipitates, which are formed during the 1 ms time of flight of the silicon particles. This observation allows us to explain the good photoelectric quality and to forecast the conditions for the future obtention of good solar cells from such materials.

2. MODELLING OF PLAST DEPOSITION OF SILICON PARTICLES

2.1. Mathematical model

In order to calculate the silicon particle trajectories and temperature evolution after they have been injected into the plasma jet, a two-dimensional mathematical model was developed, based on the following assumptions.

(a) The plasma is in local thermodynamic equilibrium, optically thin and having a cylindrical symmetry.

(b) Spherical particles with negligible internal temperature gradients are considered.

(c) Particle trajectories are controlled only by the gravitation and drag forces 4, 5

(d) For the plasma velocity and temperature fields, we use the experimental results of Vardelle et al. 6, reported on Fig. 1 (for a gas composition (83~o A t + 1T~o H2) and an arc power (20 kW) similar to those used in our experiments).

2 0 L Ar/H 2 PLASMA

15 ;Io) T E M P E R A T U R ~ 1500

12500 1(3000 6000

IO (b) AXIAL VELOCITY ~ O O m / $

Imrn}

150

0 I0 20 ~bO 40 50 60 70 80 90 oO0 I IO 120 Z(mm )

Fig. 1. Temperature and axial velocity isocontours for the A r - H 2 plasma 5. Plasma gas composition, 83% A r + 17% H2; arc power P = 29 kW.

2.2. Governing equations

On the basis of the above assumptions, the equations governing the momentum and heat transfers between the particle and the plasma can be written (see the definitions in Appendix A) as follows 7.

GROWTH OF Si RIBBONS FROM POWDER 361

2.2.1. Particle velocity

dVp,r/dt = g - ( 3 / 4 ) C D ( V v . , - Vf,r)[ Vg[pf/ppdp dVp,z/dt = --(3/4)CD(Vp, z - Vf,z)l VRlpf/ppdp

The drag coefficient Co is correlated with the Reynolds n u m b e r as follows:

C o = 24/Re = Coo Co = Coo(1 +0.178 Re) C o = Coo(1 +0.11 Re TM) Co = Coo(1 +0.189 Re TM)

(1) (2)

if Re < 0.2 (3)

if 0.2 < Re < 2 (4) if 2 < R e < 2 0 (5) if 20 < Re < 200 (6) Equation (3) is based on the Stokes solution for the flow around a sphere; eqn. (4) is based on the Oseen a p p r o x i m a t i o n ; eqns. (5) and (6) are based on the experimental data of Beard and P r u p p a c h e r (see ref. 8).

2.2.2. Particle temperature

The particle temperature was determined by an energy balance described by the following equations:

dTp/dt = Q/mCp (7)

Q = aph(Tf - rp) - apeaTv 4 (8)

Here the first term of eqn. (8) represents the conductive and convective heat transfer between the particle and the plasma, while the second term represents the heat lost by radiation from the surface of the particle.

After the particle temperature reached its melting point, it was assumed to remain constant, and the rate of change in the molten fraction x of the particle was calculated using

d x / d t = Q/mLm (9)

Once the particle was completely molten (x = 1), its temperature was allowed to rise again, governed by eqn. (7), until it eventually reached its boiling point where it would start to evaporate. The particle temperature was then maintained constant and its diameter allowed to decrease according to

d p / d t = - - 2 Q / p p L v r C d p 2 (10)

The heat transfer coefficient was estimated using the R a n z - M a r s h a l l equation 9:

h = 2rNu/dp (11)

N u = 2 + 0 . 5 1 4 R e °'5

Special attention was given to the effect of steep temperature gradients on the variation in the plasma properties across the b o u n d a r y layer of the particle. These were evaluated by taking the integral mean of each considered property between the particle surface temperature and the temperature of the plasma, calculated as

362 B. KAYALI et al.

follows (for the case of the thermal conductivity):

1 f T f - - - 2 d T

~'f T f T p T p

Similar equations were used to evaluate fif and/~f.

(12)

2.3. Computed results," model predictions

C o m p u t a t i o n s were carried out for silicon particles under plasma conditions near those used in the experimental work. The physical properties of silicon are the following: volumic mass Pv = 2 3 3 0 k g m 3; melting temperature T m = 1685 K;

vaporization temperature Tv = 2628 K. The properties of liquid silicon at T = T m are as follows: thermal capacity C = 9 6 0 J k g - 1 K l; heat of melting Lm_____l.8×106jkg 1; thermal conductivity £ = 5 8 . 5 W m I K 1. The heat of vaporization L v = 11.5 × 1 0 6 J kg 1

Using a Runge K u t t a numerical method, we obtained from eqns. (1)-(12) the trajectory, velocity, temperature and mass loss of a particle as a function of the distance along the plasma jet axis. These computations were carried out for both various particle sizes doo and various injection velocities Vro. Results are presented in Figs. 2 and 3. One can make, in particular, the following observations.

(1) If their injection velocity is too small, the particles cannot acquire enough kinetic energy to penetrate into the plasma core; if it is too large, the particles cross the plasma jet. In both cases, the particle velocity is decreased. With an injection velocity near 15 m s 1, one should obtain a maximal particle velocity at the impact on a substrate placed at an 8-10 cm distance from the torch.

(2) The small size particles stay near the axis for a longer time than the large size particles, so that their mass loss is much larger. They also reach their m a x i m u m velocity at a shorter distance from the torch. Thus an initial diameter of 40 tam or higher is preferable, in order to obtain both a smaller mass loss and a larger spraying distance (8 cm or more, which prevents the substrate from undergoing too large thermal shocks).

These conclusions are in agreement with the optimal spraying conditions quoted in ref. 3: a flow rate of 5 1 rain - 1 in the powder distribution corresponds to an injection velocity of 20 m s 1; the Wacker powder with large grain size (mean diameter, 80 lam) was found much easier to use than a finer powder (in Fig. 3 of ref. 3, we regret an inversion of Figs. 3(a) (which indeed refers to the K e m a n o r d powder) and 3(b) (which refers to the Wacker powder)) and the optimal spraying distance was 10 cm.

3. MEASUREMENTS OF PARTICLE PROPERTIES; MODEL VALIDATION

The flux density, the velocity and the surface temperature distributions of the particles within the plasma jet have been measured simultaneously with the experimental set-up shown in Fig. 4. The particle density was measured in different cross-sections of the plasma jet by counting during a given time the pulses resulting from the light scattered by the particles passing through the focus point of a laser beam. The mean particle trajectory is obtained by moving the measurement volume

'7, o "7

Trajectory Z==0.. Ro=-0,00Am. To=300)C. OPo=0.000040m S.0 i '. ~ - ~ : ~ i ! ' 4.0 ' -- _ . ; , 3"¢T~0 = 15 ==A '" i ! i ~ v~0. ~o =/, 2.0 ~ . . ~ i

0.0~

_o >

2S0 200 150 100 -

Axial velocity Ze=0.. Ro=-0.004m. T~=300K. OPo=0.0000e,0m SO ~KO - l0 =/s i i 6v~o - is =/, i i I ... ,, o O.OO 0.02 0.04 O.06 O.OB O.lO 0.12 0.14 0.16 (a) Axlalposltlon (m) (b) Axial position (rn) Temperature {K} Diameter Zo=0, . Ro=-0.O04m . To=300K , DPo=O.0000L0m Zo=0.. Ro=-0.004m . To=300K DPo=0.000DLOm =, ,ooo i. i .~ ,s.o ; [ ! ~ ~~ 2 F~ 1000 ~ 30.0 0 ~ i 25.0 ooo 002 004 006 008 o~o o~2 o~ 0.~5 ooo o oz 004 006 o o~ o to o~z o~ 0~6 (C) Axial position (rn) (d) Axial position {m) Fig. 2. Variations, along the plasma jet stream, in the particle position, velocity, temperature and diameter, as a function of the injection velocity Vro, for a fixed initial particle diameter dpo = 40 I.tm and fixed position (Z o, Ro) of the powder injector.

© =: © -n ;a © © © ~n ;=

Temperature (K) an=0. Ro=-0.004m To=300K , VRo=lSm/S 4000 100 si. 3000 90 P=29 Kw 8o vo=15 m/s ~ 70 ~ d,,o~=8 cm 2000 G 60 zo,ro =cte. i , -

,°o I!

1 = 40 ~ iooo ~ i ~ 1D,o .... on .... :~ ~0 i i i 12=Po- o oooozo =1 20 • i t 113 DFO - o oooo,o =1 lo i ! i o ~ ', it4 oFo - o.oooo:o =I 0.oo o.02 o.o4 0.os 0.os 0.10 o.12 0.14 o.15 20 40 60 80 (a) Axial position (m} (b) Initial diameter of the particle { ym ) Trajectory Axial velocity Zo=O.. Ro=-0.004rn . To=300K , VRo=lSm/S 0.010 Zo=0.. Ro=-0.004m . To=300K . VRo=ISm/S [I 300 iI "~,"/~i: ; i ii i': I

l

l,o > ,< >

G R O W T H OF Si RIBBONS FROM POWDER 365 VELOCITY

f

TEMPERATURE I I X ox~s

. . . - ] l ~

- - " PM I A PM

M O N O C H R O M A T O R F ~ i I ~T~7:7 J L _ __

I L ., -I BEAM

- S P L I T T E R

LENSES ~ LENSES

I

FLUX

Fig. 4. Schematic diagram of the experimental set-up used for the measurement of the particle number flux, velocity and surface temperature distributions.

along two orthogonal directions and determining the corresponding radial distributions. The position of the two m a x i m a corresponds to the mean trajectory in the plasma jet "slice" considered.

The particle velocity was measured by laser D o p p l e r a n e m o m e t r y with an interferential arrangement. The surface temperature distribution was determined by p y r o m e t r y (for details see ref. 5).

All these measurements are statistical and the values obtained for each p a r a m e t e r are mean values for n particles passing through the corresponding m e a s u r e m e n t volume.

We observed a rather large velocity distribution, reflecting the distributions of grain size in the Wacker powder: at z = 5 cm, it peaks at 150m s-1, but values as low as 100 m s - 1 and others as large as 300 m s - 1 are also found. The mean value of Vz along the jet stream as well as that of the particle temperature Tp are shown in Fig. 5: they fit reasonably well the theoretical curves traced for do = 40 Itm. This can give us a certain confidence in the results of the model.

All the assumptions (a)-(d) in Section 2.1, used to derive the model, are well justified. In particular, because of the high value Of 2p, the temperature homogeneity in a liquid silicon droplet is established after ^{c a .} 100 Its, whereas we know, from the values of V~, that a particle stays for ^{c a .} 1 ms in the plasma jet, for a projection distance of 10 cm. However, there are other assumptions, implicit in the equations of the model, that m a y be less justified.

(1) We did not consider the cooling of the plasma by the gas carrying the particles, which flows at a rate only 10 times less than that of the plasma gas.

(2) We did not consider the change in the transport properties of the plasma

3 6 6 B. KAYALI et al.

E

>-, i.j 0

>

3 0 0 -

250.

2 0 0

1 5 0

100

50

0

A x i a l ~ v e l o c i t y

Z o = 0 . , R o = * 0 . 0 0 4 m , T o = Z 0 0 K V R o = 2 0 m / S . D P o = 0 . 0 0 0 0 4 0 m

J ^{. it"

M o d e l . Measurement.

0 , 0 0 0,02 0,04 0,06 0,05 O,lO 0,12 0.14 0.16

(a) A x i a l p o s i t i o n (rn)

TEMPERATURE (K)

+m

=¢

[.. .¢

=.1 =., :m

4000.

3000

2 0 0 0 -

tO00.

Z o = 0 . R o = - 0 . 0 0 4 r n , T o - - - 3 0 0 K . V R o = 2 0 m / S , D P o = 0 . 0 0 0 0 4 0 m

_ _ ] _ _ t _ _

~. M o d e l .

-~-~¢ Measurement.

N

"

0

0 , 0 0 0 . 0 2 0,04 0,06 0,08 0,10 0,12 0.14 0.16

(b) A X I A L P O S I T I O N (M)

Fig. 5. Mean measured values of particle velocity and temperature during projection of Wacker silicon powder in a plasma similar to that of Fig. l ; comparison with the model results.

G R O W T H OF Si RIBBONS FROM POWDER 367 due to the presence of vaporized silicon, in spite of the probable decrease in the heat transfer resulting from that silicon vapour.

In conclusion, we can consider that our experiments validate correctly the model, if we duly consider the crude a p p r o x i m a t i o n s made in building the latter.

4. PROPERTIES OF THE SILICON RIBBONS

Let us recall the main properties reported in ref. 3.

Spontaneously detached from the alumina substrate (owing to the large difference in dilatation coefficient), the ribbons have several kinds of defects: pores and microcracks; small grain size (10-201am); non-uniformity of the d o p a n t distribution (boron added to the powder in the form of B z O 3 particles); a concentration of more than 1016cm -3 of copper and chromium, detected by neutron activation analysis, and of about 1017 cm 3 of calcium, iron, potassium and sodium, i.e. a large impurity content; nevertheless, the electron diffusion length L, in these as-grown ribbons was measurable (17-22 ~tm), perhaps because hydrogen was incorporated from the plasma gas and could passivate the grain boundaries.

Much better ribbons were obtained after zone melting using electron gun annealing3: decreased n u m b e r of pores and microcracks; large grain size (several millimetres long); uniform b o r o n distribution; hole mobility larger than 50 cm 2 V - 1 s 1; impurity content reduced by a factor of a b o u t 100 (except in the tail of the zone melting) but still large: 1016 c m 3 for iron, a b o u t 1015 c m - 3 for c h r o m i u m and

i!!!i!!~i~! ¸ ~

(a) (b)

t

(c) (d)

Fig. 6. SIMS imagl,g, with O~- ions (300 nA), of a region (diameter 150 ~tm) of an annealed Wacker ribbon: (a) 28Si image; (b) 23Na image; (c) 4°Ca image; (d) 27A1 image.

368 B. KAYALI et al.

nickel, a n d a b o u t 10 ~4 c m - 3 for m o l y b d e n u m , titanium a n d tungsten, all impurities k n o w n to be diffusion length killers at these levels; electron diffusion length as high as 2 8 - 6 9 p.m.

These two last series of d a t a are difficult to reconcile, unless we assume that the transition metal impurities are present n o t in the silicon matrix but in a precipitated second phase. This a s s u m p t i o n was m a d e in ref. 3, where it was a r g u e d that the P L A S T projection was m a d e in air, so that oxygen can, in principle, oxidize the impurities during the 1 ms time of flight of the silicon particles.

10 6-

56Si2 104 / ~..

W . . . . "~ . ~ - 40Ca 160

lO2 2 7 ,, ....

I' _{¢ '} liJ' I' _{', I '}

i i~ It", ' ', ,

0 lrO 20 30 4 0 5'0 60

erosion time (ran)

Fig. 7. SIMS profiling through a precipitate similar to that of Fig. 6. 30 min erosion with a 300 nA O~

probe corresponds approximately to an erosion depth of 1.5 lain.

(a) (b)

(c) (d)

Fig. 8. SIMS imaging ofanother region ofan annealed Wacker ribbon (same conditions as Fig. 6):(a) (d) asin Fig. 6.

G R O W T H O F Si R I B B O N S F R O M P O W D E R 369

CNR$-MELOoN

6 ~MS3F

HASS SPECTRUM 8 / ~ I / 9 0

PLAST MS! PAGE 1 / 2

1 0

1 0 4

\ o

o 3

o 1 0 2 c c

H 1

1 0

10 c~

1o-1 _{' ° ' ' i}

! 6

Si ++ 0

, , i | i , , i

1 1

C OH Na big?

1 6

Si~ Sill

i

2 1 MASS

F 2 A1

M

• | - i ,

2 6 3

AJ

i i J | | i i | ,

3 6

1 0

1 0 4

0

0 3

~ 1 0 2

0

H 1

1 0

1 0 0

1o-I

CNRS-HEUDOH

6 ~HS3F

MASS SP£CTRUM 8 / O 1 / 9 8

PLAST MS1 PAGE 2 / 2

s i o , siolF s i 2 , Si.2H

Ca

°

IUJ

4 0

"r[? Ni?

• i , , , o i ,

5 5 6 0 6 5

MASS

4 5 5 0

Si20, Si2oll

7 0 7 5 8 0

F i g . 9. S I M S s p e c t r u m in a z o n e o f d i a m e t e r 8 p m c o n t a i n i n g a p r e c i p i t a t e .

3 7 0 B. KAYALI et al.

1 0 5

1 0 4

\

o 3

co 1 0 I) E

H 1

1 0 0

- - 1 1 0

C N R 6 ~ M E d D O ~ [ H S 3 F

]

M~SS SPECTRUM

PL~ST NS2 P A G E 1 / 2

- =

St ++ 0

_5_

OIL

2 1 2 6

M R S S

3 S

1

1 0

C N R S - N E U D O N

6 I H S 3 F

H A S S S P E C T R U H 8 / 0 1 / 9 8

P L R S T M S 2 P A G E 2 / 2

1 0

1 0

\ 0

1 0 2 C

¢ C

1 0 1

1 0 0

--I 5

"4

3

.

Ni?

4 ~ ' 4 5 5 ~ 5 5 6 ~ 6 5 7 ~ 7 5

M A S S

, l l , 8 0

F i g . 10. S I M S s p e c t r u m i n a z o n e o f d i a m e t e r 8 ~tm c o n t a i n i n g n o p r e c i p i t a t e .

G R O W T H OF Si RIBBONS FROM POWDER 371 We now present three arguments able to confirm this assumption.

First, we note that a large proportion of the impurities are present at the surface of the silicon particles (sodium, potassium, calcium introduced by manipulation, iron introduced by the contact with the powder distributor). For such impurities, as well as for silicon itself, the time of reaction with oxygen at temperatures larger than 3000 K is of the order of some microseconds. Thus the formation of silicometallic oxides is very likely during the particle flight lasting 1 ms.

Second, with atomic diffusion coefficients in liquid silicon of the order of some 10 4 c m 2 s - x , even some of the bulk impurities (up to 10 lam below the particle surface) can reach the surface and be oxidized during the particle flight.

Third, we present the following results of SIMS analyses. Figure 6 shows the images made, with O ~- ions, of a zone (150 ~tm) deep in an annealed Wacker ribbon.

Clearly the 28Si image is reinforced, in many places, by the presence of oxygen (a well-known "exaltation" effect); the impurities sodium, calcium and aluminium (for which the SIMS analysis is the most sensitive) are concentrated in the same places.

One of these precipitates is further evidenced by the profiles of Fig. 7, showing the disappearance of the precipitate for an erosion depth estimated at 2 p.m. Another part of the same ribbon (Fig. 8) is similar, except that aluminium appears to be localized outside the oxide phase rich in sodium and calcium. Finally, Figs. 9 and 10 allow us to compare the impurity spectrum on a precipitate (Fig. 9) and outside it (Fig. 10); there the analysed zone has a diameter of 8 ~tm. We can conclude with much confidence that the impurities are localized essentially in microprecipitates and that the silicon matrix itself is of high purity.

5. CONCLUSIONS

The conditions for P L A S T spraying of silicon have now been optimized, except for the manual movement of the torch which should be replaced by a better- controlled movement. These conditions are well understood, i.e. our theoretical model of m o m e n t u m and heat exchanges between powder and plasma seems to be fairly accurate and well validated by experiments. The reason for the high photoelectric quality of the ribbons obtained after recrystallization has been found in the high purity of the silicon matrix, with most impurities being concentrated in oxide-rich precipitates. There seems to be a possibility of building valuable solar cells on such ribbons, especially by the use of"cold" processes.

A C K N O W L E D G M E N T

We thank C. Grattepain for the SIMS analyses.

REFERENCES

R. S u r y a n a r a y a n a n and G. Zribi, J. Phys. (Paris), Colloq. CI, 43 (1982) 375.

R. S u r y a n a r a y a n a n , G. Brun and M. Akani, Thin Solid Films, 119 (1984) 67.

M. Akani, R. S u r y a n a r a y a n a n and G. Brun, J. Appl. Phys., 60 (1986) 457; Thin Solid Films, 151 (1987) 343.

372 B. KAYALI e t al.

2 R. Suryanarayanan, M. Akani, J. E. Bouree, M. Rodot, G. Brun, R. M'Ghaieth, R. Gauthier and P. Pinard, Ann. Chim., 12 (1987) 411.

3 B. Kayali, M. Rodot, R. Suryanarayanan, Le Quang-Nam, R. M'Ghaieth, R. Gauthier and P. Pinard, Mater. Sci. Eng. B, 5 (1989) 51.

4 E. Pfender, Pure Appl. Chem., 57 (1985) 1179.

5 A. Vardelle, ThOse de Doctorat d'Etat, Universit+ de Limoges, 1986.

6 A. Vardelle, J. M. Baronnet, M. Vardelle and P. Fauchais, IEEE Trans. Plasma Sci., 8 (1980) 417.

7 M. 1. Boulos and W. H. Gauvin, Can. J. Chem. Eng., 52 (1974) 355.

8 M. Vardelle, A. Vardelle, P. Fauchais and M. I. Boulos, A1ChE J., 29 (1983) 236.

9 W.E. Ranz and W. R. Marshall, Chem. Eng. Prog., 48 (1952) 141.

A p p e n d i x A : N o m e n c l a t u r e

ap p a r t i c l e surface a r e a (m 2) C t h e r m a l c a p a c i t y (J k g - 1 K 1) dp p a r t i c l e d i a m e t e r (m)

h h e a t transfer coefficient ( W c m 2 K - 1 ) L m l a t e n t h e a t of fusion (J kg 1)

Lv l a t e n t h e a t of v a p o r i z a t i o n (J k g - 1) m p a r t i c l e m a s s (kg)

N u N u s s e l t n u m b e r

Q h e a t flow to the p a r t i c l e (W) Re R e y n o l d s n u m b e r

t t i m e (s)

T t e m p e r a t u r e (K) V~ r a d i a l velocity ( m s 1)

VR relative p l a s m a - p a r t i c l e velocity (m s - 1) V~ axial velocity (m s x)

e p a r t i c l e e m i s s i v i t y

2 t h e r m a l c o n d u c t i v i t y ( W m -1 K 1) /~ fluid viscosity (kg m 1 s - 1)

p d e n s i t y ( k g m 3)

S t e f a n - B o l t z m a n n c o n s t a n t

S u b s c r i p t s f p l a s m a fluid p p a r t i c l e