Cutoff criteria of electronic partition functions and

Cutoff criteria of electronic partition functions and transport properties of atomic hydrogen thermal plasmas D. Bruno, M. Capitelli, C. Catalfamo, and A. Laricchiuta Citation: Phys. Plasmas 15, 112306 (2008); doi: 10.1063/1.3012566 View online: http://dx.doi.org/10.1063/1.3012566 View Table of Contents: http://pop.aip.org/resource/1/PHPAEN/v15/i11 Published by the AIP Publishing LLC.

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PHYSICS OF PLASMAS 15, 112306 共2008兲

Cutoff criteria of electronic partition functions and transport properties of atomic hydrogen thermal plasmas D. Bruno,1 M. Capitelli,1,2 C. Catalfamo,2 and A. Laricchiuta1 1

Istituto di Metodologie Inorganiche e dei Plasmi del CNR, via G. Amendola 122/D, 70126 Bari, Italy Dipartimento di Chimica, Università di Bari, via E. Orabona 4, 70126 Bari, Italy

2

共Received 29 July 2008; accepted 13 October 2008; published online 12 November 2008兲 Transport coefficients of equilibrium hydrogen plasma have been calculated by using different cutoffs of electronic partition functions and different sets of transport cross sections of electronically excited states. The selection of both the cutoff criterion and transport cross sections deeply affects the transport coefficients of the H, H+, e plasma mixture in the temperature range of 10 000– 50 000 K and in the pressure interval of 1–1000 atm. © 2008 American Institute of Physics. 关DOI: 10.1063/1.3012566兴 I. INTRODUCTION

The role of electronically excited states 共EESs兲 in affecting the transport properties of thermal plasmas is a topic of renewed interest as attested by numerous contributions appearing in the literature.1–7 EESs modify the composition and the thermodynamic 共internal energy, specific heat兲 properties of the plasma, both quantities entering in the relevant transport equations. Moreover they present transport cross sections which dramatically increase as a function of the principal quantum number n determining unusual effects in the transport equations. Previous work presented by our group on atomic hydrogen plasma 关H共n兲, H+, and electrons兴 has shown these effects either in a parametric form1,2 or by comparing the results obtained by the so-called ground state 共GS兲 method with the corresponding ones obtained by the confined atom 共CA兲 method.3,4 The first method 共GS兲 completely disregards the presence of EESs by imposing an electronic partition function of atomic hydrogen equal to 2, i.e., to the degeneracy of the GS. As a consequence, the internal energy and specific heat of atomic hydrogen is zero in this approximation. The CA approximation inserts in the electronic partition function of atomic hydrogen all levels whose Bohr radius does not exceed the interparticle distance. This method can be considered well representative for describing high pressure–high temperature plasmas. Another method very often used in truncating the electronic partition function is the Griem method,8 essentially based on the Debye– Huckel theory of electrolytes, i.e., on the static screening Coulomb potential 共SSCP兲 model. The three models 共GS, CA, SSCP兲 will be extensively used in this paper to calculate the composition and the thermodynamic properties of atomic hydrogen plasmas to be inserted in the Chapman–Enskog formulation of the transport coefficients. At the same time the effect of inserting actual transport cross sections, i.e., cross sections depending on the principal quantum number of EESs in the three thermodynamic models, will be tested by comparing the results with the corresponding ones obtained by imposing the GS cross section to all EESs. The structure of this paper is as follows. Sections II and III briefly report in order the thermodynamic models and the 1070-664X/2008/15共11兲/112306/7/$23.00

transport cross sections of EESs. Section IV discusses the numerous results distinguishing the effects due to the choice of thermodynamic models with GS cross sections 共usual case, Sec. IV A兲 from those obtained by using the actual cross sections 共abnormal case, Sec. IV B兲. Section V presents conclusions and perspectives.

II. CUTOFF OF THE ATOMIC PARTITION FUNCTION

It is well known that the atomic partition function diverges if the summation is carried over all unperturbed energy levels. The explanation of this paradox is that atoms are not completely isolated and many body effects perturb the atomic excited states. Although methods have been developed to correctly account for nonideal effects in the thermodynamics of dense plasmas,9,10 these methods cannot be directly extended to a transport theory. Several simplified models of equilibrium thermodynamics are currently used in practice:11 共1兲 GS model. In this simplest case only the GS is considered. 共2兲 CA model. Those excited states whose size exceeds the interparticle separation are excluded from the summation. 共3兲 SSCP model. This model attempts to simulate the intraatomic effects of the Debye-shielded Coulomb interactions. The effect of the different criteria on the equilibrium composition of the plasma is shown in Fig. 1 for hydrogen plasma at different pressures. Reference results obtained from a sophisticated model12 are reported for comparison, indicating satisfactory agreement especially with the CA model. It is worth mentioning that the SSCP model, at variance from the other two, entails a pressure-dependent lowering of the ionization threshold. This has a deep influence on the equilibrium ionization degree, especially at large pressures where this effect becomes significant.

15, 112306-1

© 2008 American Institute of Physics

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105

p=1000atm

1026

10

4

10

3

H+-H(n=12)


2(1,1)*, A2

nH, m-3

1024 p=100atm 1022

H+-H(n=6) H+-H+(100atm) +

+

H -H (1atm)

102

p=1atm 10

20

1018 4 1 10

101

2 10

4

3 10

4

4 10

4

5 10

100 4 1 10

4

Temperature, K

IV. RESULTS

Results are reported for the viscosity, thermal conductivity, and electrical conductivity of equilibrium hydrogen plasma. These coefficients are obtained in the framework of the Chapman–Enskog method; the first nonvanishing approximation in terms of Sonine polynomials3,4 has been used to estimate the heavy particle contributions and the second for the electron component.13

4

H+-H+(1000atm) 4

3 10 4 10 Temperature, K

4

5 10

4

103 H+-H(n=6)

H+-H(n=12)

(2,2)*

,A

2

102

+ + H -H (1000atm)

2




The calculation of the transport coefficients is straightforward once the collision integrals describing the interaction among different plasma constituents are specified. In order to properly account for the presence of EESs, each EES of the hydrogen atom, H共n兲, n being the principal quantum number, is considered as a separate species. The collision integrals for the relevant interactions among H共n兲, H+, and electrons are the same used in our previous works.1,2 They present a strong dependence on the principal quantum number especially for collision integrals diffusion-type of H共n兲 – H+ collisions. A sample of results is reported in Figs. 2共a兲 and 2共b兲 where we show the temperature dependence of the H共n兲 – H+ collision integrals for selected values of the principal quantum number. These are compared to the Coulomb collision integrals H+ – H+ at three different pressures. We note that the latter depend on pressure through the Debye length which is a function of the electron concentration. The reported H共n兲 – H+ collision integrals at high temperatures are higher than the corresponding H+ – H+ values.

2 10

(a)

FIG. 1. Hydrogen atom number density for equilibrium hydrogen plasma at different pressures as obtained by different cutoff models 共solid line: GS; dashed line: CA; dotted line: SSCP; symbols: from Ref. 9兲.

III. COLLISION INTEGRALS OF ELECTRONICALLY EXCITED STATES

+ H -H(n=1)

10

1

H+-H+(100atm)

H+-H+(1atm) +

H -H(n=1) 100 4 1 10 (b)

2 10

4

4

3 10 4 10 Temperature, K

4

5 10

4

FIG. 2. 共a兲 Temperature dependence of diffusion-type collision integrals for H共n兲 – H+ interactions for selected values of the principal quantum number n and for the H+ – H+ interaction at two pressures. 共b兲 Temperature dependence of viscosity-type collision integrals for H共n兲 – H+ interactions for selected values of the principal quantum number n and for the H+ – H+ interaction at two pressures.

A. Influence of composition and thermodynamic properties „usual case for transport cross sections…

The first effect of the different cutoff criteria is to produce differences in the equilibrium composition of the plasma. In order to show the extent of this influence, transport coefficients are calculated by using for EES the same cross sections as the GS. Figure 3 reports the comparison of the translational thermal conductivity of heavy particles for different plasma pressures as calculated under the three cutoff models. The thermal conductivity of atoms grows monotonously with temperature and is obviously independent of pressure. It is the large charge-exchange and Coulomb cross sections 关see

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Phys. Plasmas 15, 112306 共2008兲

Cutoff criteria of electronic partition functions…

6

2 10

-4

5

p=1000atm

1,5 10-4

, Kg/(m*s)

3

p=1000atm

h

, W/m*K

4

2

5 10-5

p=100atm

1

p=100atm

p=1atm 0 4 1 10

4

2 10

4

1 10-4

4

3 10 4 10 Temperature, K

0 100 4 1 10

4

5 10

p=1atm 2 10

4

3 10

4

4 10

4

5 104

Temperature, K FIG. 4. Viscosity of equilibrium hydrogen plasma at different pressures. Usual values for the transport cross sections 共solid line: GS; dashed line: CA; dotted line: SSCP兲.

Figs. 2共a兲 and 2共b兲兴 that make the conductivity decrease as the proton concentration increases. When the plasma approaches the fully ionized limit, the coefficient grows again because the Coulomb cross sections decrease with temperature. The differences in the predictions of the three models are therefore explained by the differences in ionization degree. While at p = 1 atm the differences are small, at higher pressures they can be very important. The SSCP model produces a higher ionization due to the insertion of the lowering of the ionization potential and the predicted coefficient is smaller. The results of the GS model lie somewhat in between the other two: They follow the CA results at low temperatures and then tend to the fully ionized limit at high temperatures. The CA model instead predicts a larger atom concentration at high temperatures. The viscosity coefficient behaves essentially like the thermal conductivity of heavy particles. In Fig. 4 the plasma viscosity for different plasma pressures as obtained with the three cutoff models is reported. Also in this case, the differences are due to differences in the equilibrium composition. Again at high pressures the viscosity calculated with the GS method lies between the GS and SSCP methods. The thermal conductivity of electrons and the electrical conductivity are shown in Figs. 5 and 6, respectively. These coefficients depend on electron collisions; therefore the differences of the three models in electron concentration coupled to the differences between electron-atom and electron-proton cross sections explain the observed results. Finally in Figs. 7 and 8 we show the reactive thermal conductivity and the internal thermal conductivity of all components. Also in this case the cutoff criterion used in the calculation of equilibrium composition and of thermodynamic properties entering in the relevant equations for internal and reactive contribution determines strong differences in these transport coefficients.

In particular, the inclusion of the lowering of ionization potential in the SSCP method is such to anticipate the maximum in the reactive thermal conductivity as compared with the corresponding results obtained with the GS and CA methods 共see Fig. 7兲. On the other hand the GS and CA methods present maxima in the reactive thermal conductivity located approximately at the same temperature. Moreover the maximum occurring in the GS method is much higher than the corresponding maximum of the CA method. This is the consequence of the fact that the ⌬H of the ionization reaction

e

, W/m*K

FIG. 3. Translational thermal conductivity of heavy particles of equilibrium hydrogen plasma at different pressures. Usual values for the transport cross sections 共solid line: GS; dashed line: CA; dotted line: SSCP兲.

5 10

1

4 10

1

p=1000atm

3 101

p=100atm 2 10

1

p=1atm

1 101 0

0 10 1 104

2 104

3 104 4 104 Temperature, K

5 104

FIG. 5. Translational thermal conductivity of electrons of equilibrium hydrogen plasma at different pressures. Usual values for the transport cross sections 共solid line: GS; dashed line: CA; dotted line: SSCP兲.

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Phys. Plasmas 15, 112306 共2008兲

Bruno et al.

8 104

2,5 100

7 104

0

3 104

1,5 10-1 p=1atm

1 10-1

0

4

2 10

4

3 10

4

4 10

4

5 10

0 10

Temperature, K

2 10

4

4

3 10 4 10 Temperature, K

4

5 10

4

appearing in the reactive thermal conductivity is much higher in the GS method compared with the corresponding quantity obtained by the CA method. Concerning the internal contributions calculated according the CA and SSCP methods, we can note that they present a trend similar to that one discussed for the reactive thermal conductivity 共the internal thermal conductivity in the GS method is zero for definition兲. Again the maximum in the SSCP method anticipates that one of the CA method presenting at the same time smaller values as a result of the minor

6 p=1atm p=100atm 4 p=1000atm

3

r

, W/m*K

0

1 10

0 100 4 1 10

H共n兲 = H+ + e

2 1 0 1 104

2 10-1

5 10-2

p=1atm

FIG. 6. Electrical conductivity of equilibrium hydrogen plasma at different pressures. Usual values for the transport cross sections 共solid line: GS; dashed line: CA; dotted line: SSCP兲.

5

p=100atm

-1

4

0 100 4 1 10

2,5 10-1

5 10

2 104 1 10

1,5 100

, W/m*K SSCP

p=100atm

3 10-1

int

4 104

, W/m*K CA

5 10

4

3,5 10

p=1000atm

2 10

p=1000atm

int

6 10

4

-1

4 10

-1

e

, S/m

112306-4

2 104

3 104

4 104

5 104

Temperature, K FIG. 7. Reactive thermal conductivity of equilibrium hydrogen plasma at different pressures. Usual values for the transport cross sections 共solid line: GS; dashed line: CA; dotted line: SSCP兲.

FIG. 8. Internal thermal conductivity of equilibrium hydrogen plasma at different pressures. Usual values for the transport cross sections 共dashed line: CA; dotted line: SSCP兲.

number of levels inserted in the electronic partition function and in their internal specific heats. It is also worth noting that the differences in the two methods increase with pressure. As a whole these results confirm and extend earlier results describing the dependence of transport coefficients on the cutoff criterion used for terminating the electronic partition function in atomic nitrogen plasmas.14

B. Influence of EES cross sections

The effect of EES on the plasma transport coefficients is due to the large EES collision integrals. The largest increase pertains to H共n兲 – H+ diffusion-type collision integrals 关Fig. 2共a兲兴 and, to a minor extent, to H共n兲 – H+ viscosity-type collision integrals 关Fig. 2共b兲兴, whereas H共n兲 – e show a smaller dependence on the principal quantum number n. The highest effect is therefore expected on the internal and reactive contributions while the translational thermal conductivity of heavy particles as well as the viscosity should be less affected. These effects are then modulated by differences in the atom concentration and in the number of allowed EESs predicted by different models. In order to show the relative weight of these effects, the GS model results are taken as benchmark for comparisons. The GS model completely neglects the presence of EESs and is therefore not dependent on any assumption about EES cross sections. In the following, the results obtained by using a different cutoff model and appropriate EES cross sections 共abnormal values兲 are compared for equilibrium atomic hydrogen plasma at different pressures. Figures 9 and 10 report the translational thermal conductivity of heavy particles and the viscosity. In the CA model the differences produced by EES cross sections are very large and overcome the differences introduced by different thermodynamic models. We can note in fact that the CA curves are situated below the corresponding GS ones, reversing the behavior shown in Figs. 3 and 4. The SSCP model,

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Phys. Plasmas 15, 112306 共2008兲

Cutoff criteria of electronic partition functions…

6

5 101

5

4 10

1

3 10

1

2 10

1

, W/m*K

3

e

p=1000atm

h

, W/m*K

4

2

p=100atm

p=1atm

1 101

p=100atm

1

p=1000atm

p=1atm 0 4 1 10

0

4

2 10

4

4

3 10 4 10 Temperature, K

0 10 1 104

4

5 10

2 104

3 104 4 104 Temperature, K

5 104

FIG. 9. Translational thermal conductivity of heavy particles of equilibrium hydrogen plasma at different pressures. Abnormal values for the transport cross sections 共solid line: GS; dashed line: CA; dotted line: SSCP兲.

FIG. 11. Translational thermal conductivity of electrons of equilibrium hydrogen plasma at different pressures. Abnormal values for the transport cross sections 共solid line: GS; dashed line: CA; dotted line: SSCP兲.

instead, predicts a larger ionization and a smaller number of EESs. In particular, at high pressures, where the effect of EESs should become significant, the SSCP model predicts a large lowering of the ionization potential that affects the atom concentration together with the number of allowed EESs. Thermodynamics prevails on the effect due to abnormal cross sections, so in the SSCP model the effect of EESs is strongly reduced, never exceeding 5% of the GS value. The differences with Figs. 3 and 4 are indeed small. The electron transport coefficients weakly depend on EES cross sections and strongly on ionization degree. This is shown in Figs. 11 and 12 for the translational thermal con-

ductivity of electrons and the electrical conductivity. Comparison of the relevant curves with the corresponding results reported in Figs. 5 and 6 shows the effect of abnormal cross sections especially for the CA method compared with the GS one. The effect is indeed very small in the SSCP model where thermodynamic effects prevail. Finally in Figs. 13 and 14 we report the reactive thermal conductivity and the internal thermal conductivity calculated by using the abnormal cross sections. Comparison of the relevant curves of the reactive thermal with the corresponding values of Fig. 7 shows that only CA results are affected

8 104

2 10-4

7 104 p=1000atm

-4

6 104

, S/m

5 104 1 10-4

4 104

p=100atm

e

, Kg/(m*s)

1,5 10

p=1000atm

3 104 2 104

5 10-5 p=100atm 0 100 4 1 10

1 10 p=1atm

2 104

3 104

4 104

5 104

Temperature, K FIG. 10. Viscosity of equilibrium hydrogen plasma at different pressures. Abnormal values for the transport cross sections 共solid line: GS; dashed line: CA; dotted line: SSCP兲.

p=1atm

4 0

0 10 4 1 10

2 10

4

4

3 10 4 10 Temperature, K

4

5 10

4

FIG. 12. Electrical conductivity of equilibrium hydrogen plasma at different pressures. Abnormal values for the transport cross sections 共solid line: GS; dashed line: CA; dotted line: SSCP兲.

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6

V. CONCLUSIONS

p=1atm

5

p=100atm

, W/m*K

4 p=1000atm

r

3 2 1 0 4 1 10

2 10

4

3 10

4

4 10

4

5 10

4

Temperature, K FIG. 13. Reactive thermal conductivity of equilibrium hydrogen plasma at different pressures. Abnormal values for the transport cross sections 共solid line: GS; dashed line: CA; dotted line: SSCP兲.

by the insertion of abnormal cross sections, while the SSCP results are slightly modified. This is indeed due to compensation effects rather than to insensitivity of the reactive thermal conductivity on the abnormal cross sections. The situation completely changes for the internal contribution to the thermal conductivity. Comparison of the relevant curves of Fig. 14 with the corresponding ones of Fig. 8 shows dramatic effects on the influence of the abnormal cross sections in both the results coming from the CA and SSCP methods.

1 100 p=100atm

1

-1 -1

6 10-1 p=1000atm

4 10-1

1 10-1

, W/m*K SSCP

1,5 10

int

int

, W/m*K CA

8 10

5 10-2 p=1atm

0

0 10 1 104

0

2 104

ACKNOWLEDGMENTS

The present work has been partially supported by MIUR-PRIN 共Aspetti Microscopici della Reattività Chimica, 2008兲 共Project 2007H9S8SW_003兲.

2 10-1

2 10-1

Transport coefficients of equilibrium atomic hydrogen plasma have been calculated by the Chapman–Enskog theory in very extended temperature and pressure ranges. The contribution due to the atomic EESs has been properly included. In particular, the impact of different models for terminating the atomic partition function has been analyzed in detail. While the role of EES cross sections in affecting plasma transport has been studied and is now widely accepted, care must be taken to ensure a consistent procedure in determining plasma composition and EES populations since this choice largely affects the results. Further work is required in order to develop a fully consistent transport theory of the nonideal plasma15 accounting for EES contribution. Extension of these ideas to air plasmas is in progress in our laboratory. Contrary to the atomic hydrogen plasma, the nitrogen plasma contains atomic and ionic species which possess, in addition to EESs with n ⬎ 2, low lying excited states, i.e., states which have the same principal quantum number of the GS and different spin and angular momenta. These states, due to their low excitation energy, are strongly populated and can affect the plasma properties at temperatures of the order of 10 000 K. On the other hand, the high lying excited states, while presenting transport cross sections similar to those discussed for atomic hydrogen, have statistical weights much larger than the atomic system. They therefore should affect the transport properties of air plasmas more significantly than atomic hydrogen plasmas. To this aim the efforts made by our group to refine the transport cross sections of low lying16 and high lying EESs of atomic air species17,18 seem promising to strongly improve our earlier results for nitrogen plasmas.19

3 104

4 104

5 104

0 10

Temperature, K FIG. 14. Internal thermal conductivity of equilibrium hydrogen plasma at different pressures. Abnormal values for the transport cross sections 共dashed line: CA; dotted line: SSCP兲.

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