Eur. Phys. J. Appl. Phys. 25, 169–182 (2004)DOI: 10.1051/epjap:2004007 THE EUROPEAN

PHYSICAL JOURNALAPPLIED PHYSICS

Transport coefficients of plasmas consisting of insulator vapours

Application to PE, POM, PMMA PA66 and PC

P. Andre1,a, L. Brunet2, W. Bussiere1, J. Caillard2, J.M. Lombard2, and J.P. Picard1

1 Laboratoire Arc Electrique et Plasmas Thermiques, Universite Blaise Pascal, 24 avenue des Landais, 63177 Aubiere Cedex,France

2 GIAT industries, Centre de Bourges, 7 route de Guerry, 18023 Bourges Cedex, France

Received: 10 February 2003 / Received in final form: 15 October 2003 / Accepted: 20 October 2003Published online: 21 January 2004 – c© EDP Sciences

Abstract. Calculated values of viscosity, thermal conductivity, electrical conductivity, density and specificenthalpy of plasma formed in five different insulators vapours (PE, POM, PMMA, PC and PA66) arepresented. The calculations, which assume local thermodynamic equilibrium, are performed for three pres-sures (1 atm, 10 atm, 100 atm) in the temperature range from 5000 to 30000 K. The results for PE, PMMAand PA66 are compared with those of other authors at atmospheric pressure. Significant discrepancies arefound; these are attributed both to differences in the collision integrals and to the formulation used incalculating the transport coefficients. We give all the data for the potential interaction and the formulationnecessary to obtain reliable values of transport coefficients.

PACS. 51.20.+d Viscosity, diffusion, and thermal conductivity – 52.25.Fi Transport properties – 52.25.KnThermodynamics of plasmas

1 Introduction

The transport coefficients and the thermodynamic proper-ties of high-temperature plasmas formed in insulating ma-terials are indispensable inputs in the modelling of plasmaprocesses as in circuit breaker or in electrothermal chemi-cal gun [1,2]. The thermodynamic properties, such as en-thalpy, specific heat and density are relatively easily cal-culated [3,4] since the fundamental data upon which thecalculations are based are relatively accurately known (seethe discussion in [5]). In contrast, the calculation of thetransport coefficients is not reliable because the uncer-tainty values of the potential interactions and not wellknown charged transfer processes. This is still discussedeven in well studied plasmas [6,7].

Concerning the transport coefficient calculation ofplasma formed in insulating material (CαHβOγNθ), wehave found a few number of papers: PA66 and PMMAby Kovitya [8,9], PE, PETP, POM, PA66 and PMMA byKoalaga et al. [10–12], methanol by Kappen [13], PE andPMMA by Jordan [14]. This list is not exhaustive, it isonly the list of the papers that we have at our disposal atthe present time.

In Section 2, we recall the formulation used to calculatetransport coefficients with the Chapman-Enskog method

a e-mail: pascal.andre@laept.univ-bpclermont.fr

and with an order of approximation usually used. Thedata of interaction potentials and the charge transfer pro-cesses are given in Section 3. In Section 4, we compareand discuss our results with other authors for plasmaformed in PE, PMMA and PA6-6 at atmospheric pres-sure and in the temperature range from 5000 to 30000 K.Then, we present and discuss ours results of viscosity, ther-mal conductivity, electrical conductivity, density and spe-cific enthalpy of plasmas formed in five different insulatorvapours (PE, POM, PMMA, PC and PA66) and for threepressures: 1 atm, 10 atm and 100 atm.

2 Theoretical formulations

2.1 Electrical conductivity

The electrical conductivity σ at thermal equilibriumis obtained with the third-order approximation of theChapman-Enskog method [15,16]:

σ =3 e2

2n2

1

√2π

m1 k T

∣∣∣∣ q11 q12

q12 q22

∣∣∣∣∣∣∣∣∣∣∣q00 q01 q02

q01 q11 q12

q02 q12 q22

∣∣∣∣∣∣∣(1)

170 The European Physical Journal Applied Physics

where e is the electronic charge, n1 the electron numberdensity, k the Boltzmann constant, ε0 the vacuum permit-tivity, m1 mass of an electron and T is the temperature.The parameters qij are given in Annexe A.

2.2 Coefficient of viscosity

The viscosity can be considered mainly due to the heavychemical species because of the mass ratio between elec-trons and heavy chemical species. We used the formulationgiven in [17,18]:

η =

∣∣∣∣∣∣∣H11 · · · H1υ x1

· · ·Hυ1 · · · Hυυ xυ

x1 · · · xυ 0

∣∣∣∣∣∣∣∣∣∣∣∣∣H11 · · · H1υ

· · ·Hυ1 · · · Hυυ

∣∣∣∣∣∣(2)

where

Hii =x2

i

ηi+

υ∑k = 1k = i

2 xixk

ηik

MiMk

(Mi + Mk)2

(5

3A∗ik

+Mk

Mi

)

Hij = −2xixj

ηij

MiMj

(Mi + Mj)2

(5

3A∗ij

− 1

)(i = j)

with A∗ij =

Ω(2,2)i j

Ω(1,1)i j

, ηi = 516

1

Ω(2,2)i,i

√k TNaπ

√Mi, ηij =

516

1

Ω(2,2)i,j

√k TNaπ

√2 MiMj

Mi+Mj. xi and Mi are the molar fraction

and molar mass of i chemical species, Na Avogadro num-ber Ω

(l,m)

i j are collision integrals given in Section 3 and υis the number of heavy chemical species.

2.3 Total thermal conductivity

The total thermal conductivity λtot can be separated intofour terms with a good accuracy [15,19]:

λtot = λetr + λh

tr + λ int + λ react (3)

where λ etr is the translational thermal conductivity of elec-

trons, λ htr the translational thermal conductivity of heavy

species particles, λint the internal thermal conductivityand λ react the chemical reaction thermal conductivity.

Thermal conductivity of electrons at the third approx-imation order is given by [15,16]:

λetr =

758

k n21

√2πRT

M1

q22

q11q22 − (q12)2. (4)

The translational thermal conductivity due to the heavyspecies in the second-order approximation can be written

as [18,20]:

λhtr = 4

∣∣∣∣∣∣∣L11 ... L1υ x1

...Lυ1 ... Lυυ xυ

x1 ... xυ 0

∣∣∣∣∣∣∣∣∣∣∣∣∣L11 ... L1υ

...Lυ1 ... Lυυ

∣∣∣∣∣∣(5)

where

Lij =2 xi xjMiMj

(Mi + Mj)2A∗

ij kij

(554

− 3B∗ij − 4A∗

ij

)for i = j

Lii = −4(xi)

2

kii

−υ∑

k = 1k = i

2 xi xk

(152 M2

i + 254 M2

k − 3B∗ikM2

k + 4A∗ikMiMk

)(Mi + Mj)

2A∗

ik kik

with kij = 7564k3/2

√NaT

π

√Mi+Mj

2 MiMj

1

Ω(2,2)ij

, A∗ik = Ω

(2,2)ik

Ω(1,1)ik

and

B∗ik = 5 Ω

(1,2)ik −4 Ω

(1,3)ik

Ω(1,1)ik

. The internal conductivity due to

the effect of internal degrees of freedom is taken into ac-count with the Eucken correction [16–19]:

λint =N∑

i=1

λint i

N∑

j=1

Dii (1)Dij (1)

xj

xi

−1

(6)

with the internal conductivity of the i chemical speciesλint i = PDii(1)

T

(Cp i

R − 52

)and the use of the first approx-

imation for the binary diffusion coefficients:

Dij (1) =38

(kT )3/2 1

PΩ(1,1)

ij

√Na (Mi + Mj)

2π (MiMj). (7)

The formulation of the chemical reaction thermal conduc-tivity was developed by Butler and Brokaw [21] and iswritten as:

λreact = − 1R T 2

∣∣∣∣∣∣∣A11 ... A1µ ∆H1

...Aµ 1 ... Aµµ ∆Hµ

∆H1 ∆Hµ 0

∣∣∣∣∣∣∣∣∣∣∣∣∣A11 ... A1µ

...Aµ 1 Aµµ

∣∣∣∣∣∣(8)

with A ij =υ−1∑k=1

υ∑l=k+1

R TP Dkl(1)

xk xl

(aik

xk− ail

xl

)(ajk

xk− ajl

xl

)i reaction can be written as

∑Nk=1 aikBk = 0 where Bk

is the symbol of the k chemical species. The µ reactionthat we have to take into account must be linearlyindependent. For i reaction the enthalpy variation ∆Hi

is ∆Hi =∑ν

j=1 aijHj where Hj is the specific enthalpyof the j chemical species.

P. Andre et al.: Transport coefficients of plasmas consisting of insulator vapours 171

3 Collision integrals, choice of the potentialinteraction

The average cross section Q(l,s)

ij and the parameter

Ω(l,s)

ij (T ) = Q(l,s)ij (T )

π are used for the transport coefficients

calculation (Sect. 2). The average cross section Q(l,s)

ij de-pends on the transfer cross section Ql

ij (ε):

Q(l,s)

ij (T )=2(l + 1)

(s + 1)! (2l + 1 − (−1)l)

∞∫0

xs+1e−xQlij (ε) dx

(9)where x = ε

kT with T the temperature, k the Boltzmannconstant, ε the kinetic energy in the center of mass. Thetransfer cross sections depend on the type of interactionbetween the particles and have to be chosen.

3.1 Electrons-neutral collisions

The values on the momentum-transfer cross section Q11j(ε)

can be found in literature or can be estimated (Tab. 1).By integration of the relation (9) with a 24 pointsGauss-Legendre method, we obtain the average cross sec-tions Q

(1,1)

1j . The other average cross sections Q(1,s)

1j neededfor the electrical conductivity calculation are deducedfrom the following recursive relation [17]:

Q(1,s+1)

1j = Q(1,s)

1j +T

s + 2∂ Q

(1,s)

1j

∂ T. (10)

The average cross sections Q(1,s)

ij , with s included be-tween 1 and 5, are fitted versus temperature in the tem-perature range from 1000 to 30000 K by a function writtenas [30]:

Q(1,s)

1j =

a1 ln(T ) + a2 + a3T + a4/T + a5T−2 + a6T

−3 + a7T−4.

(11)

We give all the ai coefficients that we use in Table 2.

3.2 Charged-charged collisions

The collisions (repulsive or attractive) between twocharged particles are described by a shielded Coulombianpotential:

V (r) =1

4πε0

ZiZje2

rexp(−r/λ d) (12)

where e is elementary charge, Zi is the number of ele-mentary charges of the i chemical species, r the distancebetween the two particles and λd is the Debye length.

To determine the parameters Ω(l,s)

ij and the average

cross sections Q(l,s)

ij , we used the table of Mason et al. [31]completed by the one of Devoto [32].

Table 1. Origin of the momentum-transfer cross sec-tion Q1

1j(ε) between electrons and neutral particles necessary

to calculate the Q(1,1)

1j . When we do not find data of Q11j(ε), we

used three relations: Relation T1 Q(1,1)

1 x2= π 2/3

√2

(Q

(1,1)1 xπ

)1.5

where x is the monatomic chemical species x and x2 is

the diatomic chemical species x2; Relation T2 Q(1,1)

1 x3=

π 2/3

√3

(Q

(1,1)1 xπ

)1.5

where x is the monatomic chemical

species x and x3 is the tri-atomic chemical species x3; Relation

T3 Q11 x (ε) = Q1

1 H2O (ε)(

Dipole moment of xDipole moment of H2O

)2

[24].

Dipole moment

C [22]

C2 Relation (T1)

C3 Relation (T2)

CH 1.46 [25] Relation (T3)

CH4 [26]

CO [23][24]

CO2 [23][24]

CN 1.45 [29] Relation (T3)

H [22][26]

H2 [23][26]

H2O 1.8546 [25] [23][24]

N [27]

N2 [23][28]

N2O [26]

NH 1.39 [25] Relation (T3)

NH3 [26]

NO [24]

NS 1.81 [25] Relation (T3)

O [24][26]

O2 [24][26]

O3 Relation (T2)

OH 1.655 [25] Relation (T3)

3.3 Neutral-neutral collisions

Many types of potential exist (Hulbert Hirschfelder,Buckingham. . . ) to describe the interaction between neu-tral particles [17,33,34]. They are used, successfully in sev-eral plasmas as O2, N2, Air, Ar, H2 [6,7,30,35]. In plasmaformed in insulating vapours many potentials of interac-tion are not known precisely or even not known at all.To obtain the same formulation of the interaction poten-tial we used a (12, 6) Lennard-Johns potential for eachcollision between neutral particles.

V (rij) = 4εij

[(σij

rij

)12

−(

σij

rij

)6]

. (13)

172 The European Physical Journal Applied Physics

Table 2. Coefficients of the Q(1,s)

1j fitting available in temperature range from 1000 to 30000 K.

j s a1 a2 a3 a4 a5 a6 a7

C 1 4.45125 −28.2984 −0.000196162 −4948.28 9.09029e+006 −4.65162e+009 7.97332e+0112 4.45118 −26.814 −0.000261548 −3299.3 3.03054e+006 −211131 −2.65741e+0113 4.45122 −25.7016 −0.000326937 −2474.28 1.51513e+006 1866.02 −7.92614e+0064 4.45121 −24.8112 −0.000392323 −1979.55 909222 −73105.5 1.11916e+007

5 4.45114 −24.0687 −0.000457708 −1649.98 606461 −174486 2.60003e+007

C2 1 7.06587 −44.9204 −0.000311386 −7855.26 1.44303e+007 −7.38417e+009 1.26572e+0122 7.06583 −42.5647 −0.00041518 −5237.06 4.81032e+006 −103061 −4.21889e+011

3 7.06583 −40.7983 −0.000518975 −3927.85 2.40524e+006 −69514.3 7.6903e+0064 7.06577 −39.3845 −0.000622769 −3142.83 1.44383e+006 −390555 6.7342e+0075 7.06581 −38.2073 −0.000726565 −2618.79 962332 −91762.7 4.78223e+006

C3 1 9.25897 −58.8629 −0.000408034 −10292.9 1.89086e+007 −9.67575e+009 1.65851e+0122 9.25899 −55.7768 −0.000544046 −6861.78 6.30274e+006 51909.1 −5.52847e+0113 9.25895 −53.4617 −0.000680056 −5146.61 3.15168e+006 −144147 2.75511e+007

4 9.259 −51.6104 −0.000816068 −4116.9 1.89058e+006 152818 −3.03767e+0075 9.25903 −50.0675 −0.00095208 −3430.63 1.26033e+006 85102.1 −9.78076e+006

CH 1 7.84639 −76.4691 −0.000123425 113373 4.12729e+006 −9.02949e+008 8.53369e+0102 7.84634 −73.8531 −0.000164566 75581.4 1.37657e+006 −622437 −2.83021e+0103 7.84633 −71.8914 −0.000205707 56686 688477 −324817 4.73036e+0074 7.84634 −70.3223 −0.000246848 45348.8 413081 −147911 1.4898e+007

5 7.84633 −69.0145 −0.000287989 37790.6 275541 −190893 3.19442e+007

CH4 1 −0.835182 7.62865 0.00067527 −2312.15 1.36835e+006 3.6845e+007 −3.04076e+010

2 −0.835169 7.35013 0.000900359 −1541.37 456080 6872.76 1.01361e+0103 −0.835183 7.14146 0.00112545 −1156.08 228075 −10033.1 1.43302e+0064 −0.835179 6.97438 0.00135054 −924.835 136818 79797.96 −1.59877e+0065 −0.835215 6.83553 0.00157563 −770.931 91446.4 −108258 1.86237e+007

CN 1 7.73918 −75.4243 −0.000121737 111824 4.07196e+006 −8.91346e+008 8.43293e+0102 7.73914 −72.8441 −0.000162315 74549.3 1.35756e+006 −110998 −2.80843e+010

3 7.73918 −70.9097 −0.000202895 55912.4 678162 387819 −9.3521e+0074 7.73917 −69.3617 −0.000243473 44729.8 406979 126422 −1.59264e+0075 7.73922 −68.0724 −0.000284054 37275.3 270783 347209 −5.5255e+007

CO 1 −43.1477 435.39 0.00108332 −295269 2.64778e+008 −1.16117e+011 1.86373e+0132 −43.1476 421.006 0.00144442 −196845 8.8259e+007 54818.8 −6.21243e+0123 −43.1476 410.219 0.00180552 −147634 4.41291e+007 203177 −3.50816e+007

4 −43.1476 401.589 0.00216662 −118107 2.64773e+007 190218 −2.73446e+0075 −43.1476 394.399 0.00252773 −98422.8 1.76521e+007 −176702 3.81202e+007

CO2 1 17.8812 −161.72 −0.000494035 108425 −5.52543e+007 1.81411e+010 −2.47065e+012

2 17.8812 −155.76 −0.000658713 72283.1 −1.8418e+007 −104403 8.23573e+0113 17.8812 −151.289 −0.000823392 54212.5 −9.20916e+006 76931.7 −2.10173e+0074 17.8812 −147.713 −0.000988071 43370.1 −5.5256e+006 58660.3 −6.59116e+006

5 17.8813 −144.733 −0.00115275 36142 −3.68413e+006 292194 −5.7439e+007

H 1 −10.2706 112.064 7.84224e−005 669.129 −7.63464e+006 3.9099e+009 −6.42619e+0112 −10.2706 108.64 0.000104561 446.779 −2.54582e+006 548903 2.14099e+011

3 −10.2705 106.072 0.0001307 335.421 −1.27325e+006 297293 −2.55177e+0074 −10.2706 104.019 0.000156843 267.803 −763328 −201632 5.11504e+0075 −10.2706 102.307 0.000182982 223.347 −509086 19252.2 −8.20652e+006

H2 1 −11.0639 124.628 1.72427e−005 −80664.3 7.05361e+007 −3.06238e+010 4.90042e+0122 −11.064 120.941 2.29926e−005 −53776.7 2.35125e+007 −169377 −1.63345e+0123 −11.064 118.175 2.87399e−005 −40332.3 1.17559e+007 166717 −4.07814e+007

4 −11.064 115.963 3.44899e−005 −32266.1 7.0537e+006 57745.7 −1.92978e+0075 −11.064 114.119 4.02386e−005 −26888.4 4.70245e+006 55030.6 −1.48946e+007

H2O 1 12.6611 −123.392 −0.000199165 182939 6.65912e+006 −1.4571e+009 1.37794e+0112 12.6609 −119.17 −0.000265547 121958 2.22172e+006 −1.13667e+006 −4.57178e+0103 12.661 −116.005 −0.000331938 91469.3 1.1095e+006 625826 −2.06622e+0084 12.6611 −113.475 −0.000398329 73176.7 664000 1.27783e+006 −2.41481e+008

5 12.6612 −111.365 −0.000464719 60981 442003 1.10984e+006 −1.93544e+008

P. Andre et al.: Transport coefficients of plasmas consisting of insulator vapours 173

Table 2. Continued.

j s a1 a2 a3 a4 a5 a6 a7

N 1 7.38224 −60.0688 −0.000206395 14636.4 −6.01151e+006 1.49942e+009 −1.52749e+011

2 7.38223 −57.6079 −0.000275192 9757.51 −2.00379e+006 −21777.6 5.09197e+010

3 7.38221 −55.7622 −0.00034399 7318.06 −1.00183e+006 −35372.4 4.86424e+006

4 7.3822 −54.2857 −0.000412788 5854.4 −601065 −24963.2 2.50628e+006

5 7.38217 −53.055 −0.000481584 4878.43 −400482 −119932 1.8969e+007

N2 1 −0.522468 27.2337 −0.000492379 −83126.2 1.12183e+008 −5.95981e+010 1.06723e+013

2 −0.522491 27.0597 −0.000656505 −55417.6 3.73945e+007 −106628 −3.55743e+012

3 −0.522519 26.9294 −0.00082063 −41563.4 1.86975e+007 −144746 2.1199e+007

4 −0.522507 26.8248 −0.000984756 −33250.7 1.12184e+007 −26020.2 −2.37494e+006

5 −0.522553 26.7381 −0.00114888 −27709.3 7.47935e+006 −216765 3.55951e+007

N2O 1 −9.54326 108.821 0.0001482 −100025 1.19526e+008 −5.8906e+010 1.00859e+013

2 −9.54332 105.64 0.000197602 −66683.6 3.98429e+007 −422239 −3.36188e+012

3 −9.54331 103.254 0.000247002 −50012.6 1.99213e+007 −77875.6 −2.44149e+006

4 −9.54331 101.345 0.000296403 −40010.1 1.19528e+007 7649.43 −1.37659e+007

5 −9.54329 99.7547 0.000345802 −33341.6 7.96837e+006 71336.7 −1.48759e+007

NH 1 7.11175 −69.3095 −0.000111864 102760 3.7436e+006 −8.19803e+008 7.75911e+010

2 7.11158 −66.9373 −0.000149147 68505.2 1.24972e+006 −1.16977e+006 −2.56111e+010

3 7.11161 −65.1597 −0.000186434 51379.1 624594 −130872 −3.22171e+007

4 7.11164 −63.7376 −0.000223722 41103.5 374512 63459 −2.32491e+007

5 7.11163 −62.5523 −0.000261009 34252.9 249566 119386 −2.61582e+007

NH3 1 −6.58574 59.1452 0.000420202 −30033.5 2.50841e+007 −1.00417e+010 1.69556e+012

2 −6.58574 56.95 0.000560269 −20022.4 8.36141e+006 −14694.2 −5.65182e+011

3 −6.58574 55.3035 0.000700336 −15016.7 4.18067e+006 15617.2 −3.52022e+006

4 −6.58574 53.9864 0.000840403 −12013.4 2.50842e+006 −2728.56 914291

5 −6.58572 52.8886 0.00098047 −10011.1 1.6722e+006 33741.3 −5.62876e+006

NO 1 −17.9272 174.792 0.000700659 −38902 −2.54366e+007 2.66491e+010 −5.74319e+012

2 −17.9273 168.816 0.000934213 −25934.9 −8.47857e+006 −163169 1.91443e+012

3 −17.9273 164.335 0.00116777 −19451.5 −4.23892e+006 −220444 3.12659e+007

4 −17.9274 160.75 0.00140132 −15561.4 −2.54315e+006 −181919 2.15646e+007

5 −17.9274 157.763 0.00163488 −12968.4 −1.69476e+006 −475922 8.23583e+007

O 1 1.43773 −8.29051 −1.29663e−006 92.6581 642415 −3.37606e+008 5.57895e+010

2 1.43772 −7.81122 −1.72865e−006 61.7462 214166 −14468.8 −1.85939e+010

3 1.43771 −7.45172 −2.16049e−006 46.2757 107101 −6924.55 −106482

4 1.43771 −7.16413 −2.59232e−006 36.9901 64287.6 −14413.9 1.86186e+006

5 1.4377 −6.92443 −3.02407e−006 30.7642 42927 −41745.7 6.50959e+006

O2 1 −6.62572 68.458 0.00027205 −39437.4 3.00018e+007 −1.16964e+010 1.72966e+012

2 −6.62571 66.2493 0.000362732 −26291.5 1.00005e+007 39182.4 −5.76562e+011

3 −6.62571 64.5929 0.000453416 −19718.6 5.00026e+006 11731.9 −373591

4 −6.62571 63.2677 0.000544099 −15774.9 3.00013e+006 17704.8 −2.67366e+006

5 −6.62565 62.1629 0.000634779 −13145.4 1.9998e+006 142001 −2.33852e+007

O3 1 2.99066 −17.2456 −2.69979e−006 193.147 1.33588e+006 −7.02057e+008 1.16015e+011

2 2.99065 −16.2486 −3.59948e−006 128.687 445390 −54438.8 −3.86614e+010

3 2.99069 −15.5013 −4.50084e−006 96.693 222543 51869.6 −1.0472e+007

4 2.9907 −14.9033 −5.40126e−006 77.4567 133442 54630.5 −7.43381e+006

5 2.99078 −14.4056 −6.30545e−006 64.9208 88650.7 162773 −2.45116e+007

OH 1 10.0829 −98.2651 −0.000158615 145684 5.29854e+006 −1.15811e+009 1.09363e+011

2 10.0828 −94.9037 −0.000211485 97122.2 1.76691e+006 −535247 −3.63255e+010

3 10.083 −92.3847 −0.000264363 72843.1 881586 949651 −2.06504e+008

4 10.0831 −90.369 −0.000317238 58275.3 527761 1.17911e+006 −2.26728e+008

5 10.083 −88.688 −0.000370109 48562.4 352413 245737 −5.96131e+006

174 The European Physical Journal Applied Physics

Table 3. (12, 6) Lennard Johns parameters used in our calculation.

sigma (A) Eps (K/particle) References

C C 3 100 [36]

CH4 CH4 3.8 144 [17]

CH4O CH4O 3.585 507 [13]

CO CO 3.60 100 [17]

CO2 CO2 4 200 [17]

C2H C2H 4 235 [13]

C2H2 C2H2 4.221 184 [17]

C2H4 C2H4 4.232 205 [17]

C2H6 C2H6 4.418 230 [17]

C3H4 C3H4 4.602 274.5 [13]

C3H6 C3H6 4.592 276.34 [13]

C3H8 C3H8 5.061 254 [17]

C4H6 C4H6 5.086 309 [13]

C4H8 C4H8 5.228 306.9 [13]

C2H4O C2H4O 4.373 325.45 [13]

C2H4O2 C2H4O2 4.672 449.65 [13]

C2H6O C2H6O 4.455 391.0 [13]

C3H6O C3H6O 4.994 378.35 [13]

C3H8O C3H8O 5.069 425.5 [13]

C4H8O C4H8O 5.42 404.80 [13]

H H 2.6 40 Evaluate from [30]

H2 H2 2.968 37.30 [13]

H2 H 2.8 40 Evaluate from [30]

O O 2.8 117 [37]

O2 O2 3.499 100 [13]

O2 O 3.011 107.3 [37]

O2 NO 3.479 111.65 [37]

O NO 3.150 120.5 [37]

O3 O3 3.756 100 [13]

N N 2.98 119 [37]

N2 N2 3.68 91.5 [17]

N2 N 3.33 104.5 [37]

N2 O2 3.557 101.85 [37]

N2 NO 3.6 109.14 [37]

NO NO 3.53 105 [17]

NO2 NO2 4.06 97 [17]

N2O N2O 3.85 219 [17]

We give the parameters of (12, 6) Lennard-Johns po-tential that we use and their sources in Table 3. For theunknown interaction potential, we use the following em-pirical combining laws (p. 168 in [17]):

εij =√

εii εjj , σij =12

(σii + σjj) . (14)

If the parameters (σii, εii) are not known, case ofCN for example, we estimate them by the following re-

lation [10]:

εCN CN =√

εCC εNN ;

σCN CN =((σCC)3 + (σNN )3

)1/3

. (15)

When the parameters of (12, 6) Lennard-Johns potentialare determined the parameters Ω

(l,s)

ij are determined fromthe table of Klein and Smith [38].

P. Andre et al.: Transport coefficients of plasmas consisting of insulator vapours 175

3.4 Charged-neutral collisions

For an M−M+ interaction the resonant charge exchangeoccurs when l = 1. From Dalgarno [39] the transfer crosssection can be written versus the relative speed g of thetwo interacting particles and two parameters A and B.The average cross section Q

(l,s)is then written as [40]:

Q(l,s)

= A2 − ABx +(

Bx

2

)2

+Bζ

2(Bx − 2A)

+B2

4

(π2

6−

s+1∑n=1

1n2

+ ζ2

)+

B

2[B (x + ζ) − 2A] ln

(TM

)

+(

B

2ln(

TM

))2

(16)

where M is the mass of a mole (g/mol), R is the perfectgas constant (erg/mol/K), and ζ = 0.99784 for s = 1.

The parameters A and B are not well known. So, evenin well studied plasma some discrepancies between authorsappear in the results of transport coefficients [6,7,35,41].In Table 4, we give the values of A and B parameters thatwe used in our calculation.

For the other collisions between neutral and chargedparticles, we consider them as elastic and that the chargeparticle engenders a dipole in neutral particle during thecollision. Then the interaction potential is written as:

V (r) =−α Z e2

4 π ε0 r4(17)

where r is the distance between the two particles, Z isthe number of elementary charges of the charged particleand α is the polarisability of the neutral particle.

From the paper of Kihara et al. [44], the relation (17)and the definition of Ω(l,s)(T ) (p. 525 of [17]), we obtainfor the parameter Ω

(l,s)

ij :

Ω(l,s)

ij =

4(l + 1)(s + 1)!(2∗l + 1 − (−1)l)

√4

kT

√α Z e2

4πε0Γ

(s +

32

)A

(l)4

where Γ ( ) is Euler gamma function (American definition),A

(l)4 are parameters given in [44]. The α polarisabilities of

neutral particle that we used in our calculation are givenin Table 5.

4 Results and discussions

In the calculation of transport coefficients the initial stepis the determination of the equilibrium composition of theplasma. We use the principle of minimization of the Gibbsfree energy as described in [3] to determine the composi-tion and the Debye length λd written as:

λd =ε0 k

e2

N∑i=1

T

Z2i ni

. (18)

Table 4. Parameter A and B that characterized resonantcharge exchange.

A B References

C C+ 20.1 0.864 Evaluate from [42][43]

H H+ 28.69 1.3 [30][34]

N N+ 26.61 1.27 [37]

N2 N+2 24.5 1.032 [37]

NO NO+ 23.61 1.095 [37]

O O+ 19.5 0.832 [37]

O2 O+2 24.05 1.132 [37]

Table 5. Polarisability values of neutral species used in ourcalculation code.

Chemical species α (10−30m3) References

H 0.666793 [25]

C 1.76 [25]

N 1.10 [25]

O 0.802 [25]

CO 1.95 [25]

H2 0.8059 [25]

N2 1.7403 [25]

NO 1.70 [25]

O2 1.5812 [25]

OH 1.470 [13]

CH 2.270 [13]

C2 3.200 [13]

CO2 2.911 [25]

H2O 1.45 [25]

HCN 2.525 [25]

C2H 3.870 [13]

N2O 3.03 [25]

NO2 3.02 [25]

CH2 2.940 [13]

CH3 2.76 approximation

C2H2 3.63 [25]

O3 3.21 [25]

C3 4.9 approximation

CH2O 3.740 [13]

CH4 2.593 [25]

We consider the following species in the compositioncalculation: C−, C, C+, C++, C+++, H−, H, H+, N, N+,N++, N+++, O−, O, O+, O++, O+++ for monatomicspecies, C−

2 , C2, C+2 , CH−, CH, CH+, CN−, CN, CN+,

CO−, CO, CO+, H−2 , H2, H+

2 , N−2 , N2, N+

2 , NH−, NH,NH+, NO−, NO, NO+, O2, O−

2 , O+2 , OH−, OH, OH+

for the diatomic species, CH2, CH3, CH4, CN2, CO2,C2H, C2H2, C2H4, C2N, C2N2, C2O, C3O2, CHO, CHO+,CHN, CH2O, CHNO, C2H4O, CNN, CNO, CO2, CO−

2 ,C3, C4, C4N2, C5, HNO, HNO2(cis), HNO2(trans), HNO3,HO2, H2N, H2O, H2N2, H2O2, H3N, H3O+, H4N2, NCN,NO2, NO−

2 , NO3, N2C, N2O, N2O+, N2O3, N2O4, N2O5,N3, O3 for the polyatomic species and the electrons e−.

176 The European Physical Journal Applied Physics

During the ablation, the plasma constituting theelectric discharge is considered to be composed mainlyof insulator wall vapours. That is a common hypothe-sis [3,4,8–12] and that is verified by first molecular dy-namic simulations [45,46].

The measurements made during the interaction be-tween argon plasma and an insulating material atatmospheric pressure show that the plasma is out ofthermal equilibrium [47,48]. That is why several authorspropose the calculation of plasma composition and ther-modynamic properties out of thermal equilibrium in theplasmas formed in insulating vapours [49,50]. Neverthe-less, pressure measurements made in circuit breakers [51]or in an igniter of electrothermal chemical gun [52] showthat the pressure quickly reaches a high value (until100 bars). Thus, we can suppose that the plasma is inthermal equilibrium and we made the calculation for threepressures: 1, 10 and 100 atm.

During the cooling when the plasma reaches lower tem-peratures it condenses near the interacting wall. This phe-nomenon allows transmitting of a heat flux to the wall [55].If the interaction is made in closed vessel the volume be-tween the two electrodes is filled up by solid carbon. Thisphenomenon increases the electrical conductivity in thevessel and then can produce either a re-arcing or the dis-charge of a capacitive bank [52,56]. This is function of theelectrical set up. Solid carbon appears at low temperatureand disappears for a temperature higher than ∼4500 Kfor PE, PMMA PA6-6, PC and higher than ∼2000 K forPOM [53,54]. So, we calculate the transport coefficientsfor a temperature range from 5000 to 30000 K.

The chemical equilibrium compositions at atmosphericpressure of the five considered insulating are shown in An-nexe B in the temperature range of 5000 to 30000 K. Itis noted that most of the species considered in this tem-perature range have concentration lower than 1020 m−3.Most of the molecular chemical species have dissociatedby a temperature around 9000 K at atmospheric pressureand by a temperature around 15000 K for a pressure of100 atm (Fig. B6 in Annexe B).

4.1 Comparison with authors

Figure 1 shows the temperature dependence of the elec-trical conductivity of plasma formed in PE, PA6-6 andPMMA insulator vapours with the results of other au-thors. We find a good agreement with the results ofKoalaga [10–12] with a difference less than 4%. The dis-crepancy of our results with those of Kovitya is due to themethod of calculation of Debye length. As a matter of factwe can calculate it taking the ionic contribution into ac-count (18) or taking only the electronic contribution intoaccount. The latter method has been chosen by Kovitya.A discussion of these two methods is given in [32].

The temperature dependence of the viscosity of plas-mas formed in PE, PA6-6 and PMMA insulator vapourswith the results of other authors is shown in Figure 2.In Figure 2a, we note a large discrepancy between our re-sult and those of Koalaga and Jordan. As a matter of fact,

(a)

(b)

(c)

Fig. 1. (a) Electrical conductivity versus temperature at at-mospheric pressure of a plasma formed in PE vapours. (b) Elec-trical conductivity versus temperature at atmospheric pressureof a plasma formed in PMMA vapours. (c) Electrical conduc-tivity versus temperature at atmospheric pressure of a plasmaformed in PA6-6 vapours.

Koalaga et al. [10–12] used an approximation relation withan empirical coefficient equal to 1.385 (p. 533 in [17]) andJordan [14] used a coefficient equal to 2 (p. 532 in [17]).We observe the same discrepancies with Koalaga’s resultsfor PA6-6 and PMMA (Figs. 2b and 2c). For the viscos-ity calculation, Kovitya has used the same formulation asus (Eq. (2)), so our results have to be similar. We find adiscrepancy up to 13% (Figs. 2b and 2c). This can be ex-plained by the choice made by Kovitya of an exponential

P. Andre et al.: Transport coefficients of plasmas consisting of insulator vapours 177

(a)

(b)

(c)

Fig. 2. (a) Viscosity versus temperature at atmospheric pres-sure of a plasma formed in PE vapours. (b) Viscosity versustemperature at atmospheric pressure of a plasma formed inPMMA vapours. (c) Viscosity versus temperature at atmo-spheric pressure of a plasma formed in PA6-6 vapours.

(a)

(b)

(c)

Fig. 3. (a) Thermal conductivity and components of thermalconductivity versus temperature at atmospheric pressure of aplasma formed in PE vapours. (b) Thermal conductivity andcomponents of thermal conductivity versus temperature at at-mospheric pressure of a plasma formed in PMMA vapours.(c) Thermal conductivity and components of thermal conduc-tivity versus temperature at atmospheric pressure of a plasmaformed in PA6-6 vapours.

178 The European Physical Journal Applied Physics

Fig. 4. Electrical conductivity versus temperature at atmo-spheric pressure of plasmas formed in studied insulator vapours(Tab. 6).

repulsive potential for the encounters between two atomsor by the parameters of the Lennard Johns potential usedby Kovitya.

In Figure 3, we compare our thermal conductivityresults with two authors for plasmas formed in PE,PMMA and PA6-6 insulators vapours. The total thermalconductivity is the sum of four terms: translation thermalconductivity due to electrons, translation thermal conduc-tivity due to the heavy particles, reactive thermal conduc-tivity and the internal thermal conductivity (Eq. (3)). Themain discrepancy occurs at temperatures around 15000 Kfor which the reaction thermal conductivity λreact associ-ated with the ionization of C and H is important (Fig. 3).So the charge exchange cross section for C+-C interactionand H+-H interaction are the main factor in determiningof λreact. No values of charge exchange cross sections aregiven in papers of Kovitya [8,9] and for C+-C interactionKoalaga assumed it as a polarization potential [10–12]. So,we observe a discrepancy up to 25% between our resultswith these two authors (Fig. 3).

4.2 Influence of the pressure

We present in Figure 4, the influence of pressure onthe electrical conductivity for the five studied insulators(Tab. 6). The electrical conductivities are similar in val-ues for the five studied insulators. This is due to the factthat the electrical neutrality is made between the samechemical species: e−, C+ and H+ [3,10]. Since the ion-ization appears at higher temperature when pressure ishigher, the electron density is lower at low temperature.Consequently, the electrical conductivity is lower. Whenthe pressure is higher and when the plasma is fully ionizedthe electrons density follows the Dalton law and is thenhigher. So, we observe higher electrical conductivity.

We present in Figure 5, the influence of the pressureon the viscosity for the five studied insulators (Tab. 6).All the results show an increase followed by a decrease

Table 6. Name and chemical formulae of the studied insula-tors.

Common name Abbreviation Chemical formulae

Perch 1000 PE CH2

Delrin POM CH2O

Plexiglas PMMA C5H8O2

Lexan PC C16H14O3

Nylon 6-6 PA6-6 C12H22O2N2

(a)

(b)

Fig. 5. (a) Viscosity versus temperature for three pressures ofplasmas formed in PC, POM and PE vapours. (b) Viscosityversus temperature for three pressures of plasmas formed inPA6-6 and PMMA vapours.

versus temperature. The decrease is due to the appari-tion of electrons and ionized chemical species. So, sincethe ionization depends on the pressure the flat part of vis-cosity depends on the pressure. The flat part of viscosityappears at ∼16000 K for a pressure of 1 atm, ∼22000 Kfor a pressure of 10 atm and higher than 30000 K for apressure of 100 atm. The viscosity reaches a maximumfor all studied insulators around 10000 K for a pressure of1 atm, 12000 K for a pressure of 10 atm and 15000 K for apressure of 100 atm. The plasma that reaches the highest

P. Andre et al.: Transport coefficients of plasmas consisting of insulator vapours 179

(a)

(b)

Fig. 6. (a) Thermal conductivity versus temperature for threepressures of plasmas formed in PC, POM and PE vapours.(b) Thermal conductivity versus temperature for three pres-sures of plasmas formed in PA6-6 and PMMA vapours.

viscosity is formed in POM vapours (1.9 × 10−4 Pa.s forP = 1 atm, 2.1×10−4 Pa.s for P = 10 atm, 2.4×10−4 Pa.sfor P = 100 atm). The plasma formed in PE vapours hasthe lowest peak in viscosity (1.3×10−4 Pa.s for P = 1 atm,1.45 × 10−4 Pa.s for P = 10 atm, 1.7 × 10−4 Pa.s forP = 100 atm). Finally, three plasmas formed in PC,PMMA and PA6-6 vapours have the same compounds(∼1.6 × 10−4 Pa.s for P = 1 atm, ∼1.8 × 10−4 Pa.s forP = 10 atm, ∼2 × 10−4 Pa.s for P = 100 atm).

The thermal conductivity is shown in Figure 6 for thefive studied insulators and for three pressures. We canassociate peaks in thermal conductivity to chemical reac-tions. The peaks appearing for a pressure of 100 atm at6000 K are due to the dissociation of H2 and C2H for PE,PMMA, PC and PA6-6. For POM, there is the dissocia-tion of H2 around 4800 K for a pressure of 100 atm. Sowe observe a part of the peak at 5000 K. The peaks ap-pearing at a temperature around 7000 K for a pressure of

(a)

(b)

Fig. 7. (a) Density versus temperature for three pressures ofplasmas formed in POM and PE vapours. (b) Density versustemperature for three pressures of plasmas formed in PC, PA6-6 and PMMA vapours.

1 atm, around 7900 K for a pressure of 10 atm and around9000 K for a pressure of 100 atm, are due to the dissoci-ation of CO. Indeed these peaks do not appear in plasmaproduced in PE vapours. Furthermore the concentrationof CO is higher for POM so the peak is higher for thisinsulator [3,10]. The peak appearing around 15000 K at apressure of 1 atm, around 18500 K at a pressure of 10 atmand around 22500 K at a pressure of 100 atm is due tothe ionization of monatomic species mainly H.

In Figure 7, we present the densities versus the tem-perature for the five studied insulators. The plasmas hav-ing the highest density are those produced in POM andPC vapours. The plasma having the lowest density is theplasma produced in PE vapours.

In Figure 8, we represent the specific enthalpy versustemperature of the five plasmas formed in the insulatorvapours studied. This thermodynamic property increases

180 The European Physical Journal Applied Physics

(a)

(b)

Fig. 8. (a) Specific enthalpy versus temperature for three pres-sures of plasmas formed in PC, POM and PE vapours. (b) Spe-cific enthalpy versus temperature for three pressures of plasmasformed in PA6-6 and PMMA vapours.

with the temperature because the dissociations and ion-izations appear when the temperature increases. Seeingthat the dissociation and the ionization appear at highertemperature when the pressure is higher, we observe atranslation of the enthalpy to the higher temperatures.

5 Conclusion

A database of thermodynamic properties and transportcoefficients has been set up allowing us to make hydrody-namics models of plasmas produced in insulator vapours.The material properties of PE, POM, PC, PMMA andPA6-6 are presented in the temperature range from 5000to 30000 K and for three pressures (1, 10 and 100 atm).Such materials are widely used in circuit breakers and inigniter of electrothermal chemical guns.

We have given in details the formulation of transportcoefficients and all the data used for their calculation.That will allow other authors to compare their calcula-tion method and their data set with ours. We have com-pared our results with those of other authors and triedto explain the discrepancies that we met. The main un-certainty in the calculations remains that resulting fromuncertainty in the values of collisions integrals.

We have compared and explained the results of trans-port coefficients for the five studied insulators and forthree pressures. We found similar results of electrical con-ductivity for the five studied insulators. PE plasma hasthe lower viscosity and POM plasma has the highest vis-cosity. PMMA and PA66 plasmas have similar viscosity.PE plasma has not the same chemical reaction in temper-ature range of 6000 to 8000 K so we do not observe thesame thermal conductivity behaviour than the other fourinsulators in this temperature range.

Annexe A

q00 =8∑

j

n1njQ(1,1)1 j

q01 =8∑

j

n1nj

[52Q

(1,1)1 j − 3Q

(1,2)1 j

]

q11 =8√

2n21Q

(2,2)1 j + 8

∑j

n1nj

×[254

Q(1,1)1j − 15 Q

(1,2)1j + 12Q

(1,3)1,j

]

q02 =8∑

j

n1nj

[358

Q(1,1)1j − 21

2Q

(1,2)1j + 6 Q

(1,3)1,j

]

q12 =8√

2n21

[74Q

(2,2)1 1 − 2 Q

(2,3)1 1

]+ 8

∑j

n1nj

×[17516

Q(1,1)1j − 315

8Q

(1,2)1j + 57 Q

(1,3)1j − 30 Q

(1,4)1,j

]

q22 =8√

2n21

[7716

Q(2,2)1 1 − 7 Q

(2,3)1 1 + 5 Q

(2,4)11

]

+ 8∑

j

n1nj

[122564

Q(1,1)1j − 735

8Q

(1,2)1j

+3992

Q(1,3)1j − 210 Q

(1,4)1,j + 90 Q

(1,5)1,j

].

P. Andre et al.: Transport coefficients of plasmas consisting of insulator vapours 181Annexe B

Fig. B.1. Chemical composition versus temperature at atmo-spheric pressure of a plasma formed in PE vapours.

Fig. B.2. Chemical composition versus temperature at atmo-spheric pressure of a plasma formed in POM vapours.

Fig. B.3. Chemical composition versus temperature at atmo-spheric pressure of a plasma formed in PMMA vapours.

Fig. B.4. Chemical composition versus temperature at atmo-spheric pressure of a plasma formed in PA66 vapours.

Fig. B.5. Chemical composition versus temperature at atmo-spheric pressure of a plasma formed in PC vapours.

Fig. B.6. Chemical composition versus temperature at a pres-sure of 100 atm of a plasma formed in PC vapours.

182 The European Physical Journal Applied Physics

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