A model for electrons in partially ionized plasmas that self-consistently captures non-Maxwellian electron energy distribution function (EEDF) effects is presented. The model is based on the solution of scalar and vectorial velocity moments up to the contracted fourth-order moment. The set of fluid (macroscopic) equations is obtained with Grad's method and exact expressions for the collision production terms are derived, considering the electron–electron, electron–gas, and electron–ion elastic collisions as well as for electron–gas excitation and ionization collisions. A regularization of the equations is proposed in order to avoid spurious discontinuities, existing in the original Grad's moment model, by using a generalized Chapman–Enskog expansion that exploits the disparity of mass between the electrons and the heavy particles (ions and atoms) as well as the disparity of plasma and gas densities, typical of gas discharges. The transport model includes non-local effects due to spatial gradients in the EEDF as well as the impact of the EEDF in the calculation of the elastic and inelastic collision rates. Solutions of the moment model under spatially homogeneous conditions are compared to direct simulation Monte Carlo and a two-term Boltzmann solver under conditions that are representative of high plasma density discharges at low-pressure. The moment model is able to self-consistently capture the evolution of the EEDF, in good quantitative agreement with the kinetic solutions. The calculation of transport coefficients and collision rates of an argon plasma in thermal non-equilibrium under the effect of an electric field is in good agreement with the solutions of a two-term Boltzmann solver, largely improving models with a simplified Bhatnagar–Gross–Krook collisional operator.
References
1.
F.
Taccogna
and G.
Dilecce
, “ Non-equilibrium in low-temperature plasmas
,” Eur. Phys. J. D
70
, 251
(2016
).2.
V.
Kolobov
and V.
Godyak
, “ Electron kinetics in low-temperature plasmas
,” Phys. Plasmas
26
(6
), 060601
(2019
).3.
J. P.
Boeuf
and L. C.
Pitchford
, “ Two-dimensional model of a capacitively coupled rf discharge and comparisons with experiments in the gaseous electronics conference reference reactor
,” Phys. Rev. E
51
, 1376
–1390
(1995
).4.
G.
Hagelaar
and G.
Kroesen
, “ Speeding up fluid models for gas discharges by implicit treatment of the electron energy source term
,” J. Comput. Phys.
159
(1
), 1
–12
(2000
).5.
M. M.
Becker
and D.
Loffhagen
, “ Enhanced reliability of drift-diffusion approximation for electrons in fluid models for nonthermal plasmas
,” AIP Adv.
3
(1
), 012108
(2013
).6.
R. E.
Robson
, R. D.
White
, and Z. L.
Petrović
, “ Colloquium: Physically based fluid modeling of collisionally dominated low-temperature plasmas
,” Rev. Mod. Phys.
77
, 1303
–1320
(2005
).7.
L. L.
Alves
, “ Fluid modelling of the positive column of direct-current glow discharges
,” Plasma Sources Sci. Technol.
16
, 557
–569
(2007
).8.
G.
Hagelaar
and L.
Pitchford
, “ Solving the Boltzmann equation to obtain electron transport coefficients and rate coefficients for fluid models
,” Plasma Sources Sci. Technol.
14
, 722
–733
(2005
).9.
A. T.
del Caz
, V.
Guerra
, D.
Gonçalves
, M. L.
da Silva
, L.
Marques
, N.
Pinhão
, C. D.
Pintassilgo
, and L. L.
Alves
, “ The LisbOn KInetics Boltzmann solver
,” Plasma Sources Sci. Technol.
28
, 043001
(2019
).10.
11.
R. E.
Robson
, R.
Winkler
, and F.
Sigeneger
, “ Multiterm spherical tensor representation of Boltzmann's equation for a nonhydrodynamic weakly ionized plasma
,” Phys. Rev. E
65
, 056410
(2002
).12.
G. K.
Grubert
, M. M.
Becker
, and D.
Loffhagen
, “ Why the local-mean-energy approximation should be used in hydrodynamic plasma descriptions instead of the local-field approximation
,” Phys. Rev. E
80
, 036405
(2009
).13.
L.
Alves
, G.
Gousset
, and S.
Vallee
, “ Nonequilibrium positive column revisited
,” IEEE Trans. Plasma Sci.
31
(4
), 572
–586
(2003
).14.
A. L.
Ward
, “ Calculations of cathode-fall characteristics
,” J. Appl. Phys.
33
(9
), 2789
–2794
(1962
).15.
J.-P.
Boeuf
, “ Numerical model of rf glow discharges
,” Phys. Rev. A
36
, 2782
–2792
(1987
).16.
C.
Busch
and U.
Kortshagen
, “ Numerical solution of the spatially inhomogeneous Boltzmann equation and verification of the nonlocal approach for an argon plasma
,” Phys. Rev. E
51
, 280
–288
(1995
).17.
U.
Kortshagen
, C.
Busch
, and L. D.
Tsendin
, “ On simplifying approaches to the solution of the Boltzmann equation in spatially inhomogeneous plasmas
,” Plasma Sources Sci. Technol.
5
, 1
–17
(1996
).18.
G. J. M.
Hagelaar
, G.
Fubiani
, and J.-P.
Boeuf
, “ Model of an inductively coupled negative ion source: I. General model description
,” Plasma Sources Sci. Technol.
20
, 015001
(2011
).19.
S.
Lishev
, L.
Schiesko
, D.
Wünderlich
, C.
Wimmer
, and U.
Fantz
, “ Fluid-model analysis on discharge structuring in the RF-driven prototype ion-source for ITER NBI
,” Plasma Sources Sci. Technol.
27
, 125008
(2018
).20.
D.
Zielke
, D.
Rauner
, S.
Briefi
, S.
Lishev
, and U.
Fantz
, “ Self-consistent fluid model for simulating power coupling in hydrogen ICPs at 1 MHz including the nonlinear RF Lorentz force
,” Plasma Sources Sci. Technol.
30
, 065011
(2021
).21.
M.
Becker
and D.
Loffhagen
, “ Derivation of moment equations for the theoretical description of electrons in nonthermal plasmas
,” Adv. Pure Math.
3
(3
), 343
–352
(2013
).22.
S.
Dujko
, A. H.
Markosyan
, R. D.
White
, and U.
Ebert
, “ High-order fluid model for streamer discharges: I. Derivation of model and transport data
,” J. Phys. D
46
, 475202
(2013
).23.
R.
Futtersack
, “ Modélisation Fluide du Transport Magnétisé Dans Les Plasmas Froids,” Ph.D
thesis (Universite de Toulouse III
, 2014
).24.
A. H.
Markosyan
, S.
Dujko
, and U.
Ebert
, “ High-order fluid model for streamer discharges: II. Numerical solution and investigation of planar fronts
,” J. Phys. D
46
, 475203
(2013
).25.
N. A.
Garland
, D. G.
Cocks
, G. J.
Boyle
, S.
Dujko
, and R. D.
White
, “ Unified fluid model analysis and benchmark study for electron transport in gas and liquid analogs
,” Plasma Sources Sci. Technol.
26
, 075003
(2017
).26.
H.
Grad
, “ On the kinetic theory of rarefied gases
,” Commun. Pure Appl. Math.
2
(4
), 331
–407
(1949
).27.
H.
Struchtrup
, Macroscopic Transport Equations for Rarefied Gas Flows–Approximation Methods in Kinetic Theory
(Springer
, Berlin/Heidelberg
, 2005
).28.
G.
Kremer
, An Introduction to the Boltzmann Equation and Transport Processes in Gases. Interaction of Mechanics and Mathematics
(Springer
, Berlin/Heidelberg
, 2010
).29.
M.
Torrilhon
and H.
Struchtrup
, “ Regularized 13-moment equations: Shock structure calculations and comparison to Burnett models
,” J. Fluid Mech.
513
, 171
–198
(2004
).30.
H.
Struchtrup
and M.
Torrilhon
, “ Regularization of Grad's 13 moment equations: Derivation and linear analysis
,” Phys. Fluids
15
, 2668
(2003
).31.
G. M.
Kremer
and W.
Marques
, Jr. , “ Fourteen moment theory for granular gases
,” Kinet. Relat. Models
4
(1
), 317
–331
(2011
).32.
H.
Struchtrup
, “ Extended moment method for electrons in semiconductors
,” Physica A
275
, 229
–255
(2000
).33.
S.-F.
Liotta
and H.
Struchtrup
, “ Moment equations for electrons in semiconductors: Comparison of spherical harmonics and full moments
,” Solid-State Electron.
44
, 95
–103
(2000
).34.
S. I.
Braginskii
, “ Transport processes in a plasma
,” Rev. Plasma Phys.
1
, 205
(1965
).35.
R.
Balescu
, “ The classical transport theory
,” in Transport Processes in Plasmas
, Classical Transport, edited by R.
Balescu
(North-Holland, Amsterdam
, 1988
), pp. 211
–276
.36.
P.
Hunana
, T.
Passot
, E.
Khomenko
, D.
Martínez-Gómez
, M.
Collados
, A.
Tenerani
, G.
Zank
, Y.
Maneva
, M.
Goldstein
, and G.
Webb
, “ Generalized fluid models of the Braginskii-type
,” Astrophys. J. Suppl. Ser.
260
(2
), 26
(2022
).37.
V. M.
Zhdanov
, Transport Processes in Multicomponent Plasma
(CRC Press
, 2002
), Vol. 44
.38.
R. E.
Robson
and K. F.
Ness
, “ Velocity distribution function and transport coefficients of electron swarms in gases: Spherical-harmonics decomposition of Boltzmann's equation
,” Phys. Rev. A
33
, 2068
–2077
(1986
).39.
K.
Kumar
, H. R.
Skullerud
, and R. E.
Robson
, “ Kinetic theory of charged particle swarms in neutral gases
,” Aust. J. Phys.
33
, 343
(1980
).40.
S. L.
Lin
, R. E.
Robson
, and E. A.
Mason
, “ Moment theory of electron drift and diffusion in neutral gases in an electrostatic field
,” J. Chem. Phys.
71
(8
), 3483
–3498
(1979
).41.
K. F.
Ness
and R. E.
Robson
, “ Velocity distribution function and transport coefficients of electron swarms in gases. II. Moment equations and applications
,” Phys. Rev. A
34
, 2185
–2209
(1986
).42.
Z.
Cai
, Y.
Fan
, and R.
Li
, “ Globally hyperbolic regularization of Grad's moment system
,” arXiv:1111.3409 (2012
).43.
I.
Müller
, D.
Reitebuch
, and W.
Weiss
, “ Extended thermodynamics—Consistent in order of magnitude
,” Continuum Mech. Thermodyn.
15
, 113
–146
(2003
).44.
S.
Chapman
and T. G.
Cowling
, “ An account of the kinetic theory of viscosity, thermal conduction and diffusion in gases
,” in The Mathematical Theory of Non-Uniform Gases
(Cambridge University Press
, 1970
).45.
J.
Ferziger
and H.
Kaper
, Mathematical Theory of Transport Processes in Gases
(North-Holland Publishing Company
, 1972
).46.
A.
Aanesland
, J.
Bredin
, P.
Chabert
, and V.
Godyak
, “ Electron energy distribution function and plasma parameters across magnetic filters
,” Appl. Phys. Lett.
100
(4
), 044102
(2012
).47.
V.
Godyak
, “ Rf discharge diagnostics: Some problems and their resolution
,” J. Appl. Phys.
129
(4
), 041101
(2021
).48.
A.
Meurer
, C. P.
Smith
, M.
Paprocki
, O.
Čertík
, S. B.
Kirpichev
, M.
Rocklin
, A.
Kumar
, S.
Ivanov
, J. K.
Moore
, S.
Singh
, T.
Rathnayake
, S.
Vig
, B. E.
Granger
, R. P.
Muller
, F.
Bonazzi
, H.
Gupta
, S.
Vats
, F.
Johansson
, F.
Pedregosa
, M. J.
Curry
, A. R.
Terrel
, v
Roučka
, A.
Saboo
, I.
Fernando
, S.
Kulal
, R.
Cimrman
, and A.
Scopatz
, “ SymPy: Symbolic computing in Python
,” PeerJ Comput. Sci.
3
, e103
(2017
).49.
See www.lxcat.net for “
Phelps database
;” accessed on 14 April 2021.50.
T. E.
Magin
, G.
Martins
, and M.
Torrilhon
, “ Regularized grad equations for multicomponent plasmas
,” AIP Conf. Proc.
1333
(1
), 99
–104
(2011
).51.
G.
Martins
, M.
Kapper
, M.
Torrilhon
, and T.
Magin
, “ Hypersonics simulations based on the regularized grad equations for multicomponent plasmas
,” AIAA Paper No. 2011-3305.52.
See www.lxcat.net for “
Siglo database 2013
;” accessed 14 April 2021.53.
G.
Dimarco
, R.
Caflisch
, and L.
Pareschi
, “ Direct simulation Monte Carlo schemes for coulomb interactions in plasmas
,” Commun. Appl. Ind. Math.
1
(1
), 72
–91
(2010
).54.
V.
Vahedi
and M.
Surendra
, “ A Monte Carlo collision model for the particle-in-cell method: Applications to argon and oxygen discharges
,” Comput. Phys. Commun.
87
(1
), 179
–198
(1995
).55.
G. J. M.
Hagelaar
, “ Coulomb collisions in the Boltzmann equation for electrons in low-temperature gas discharge plasmas
,” Plasma Sources Sci. Technol.
25
, 015015
(2015
).56.
M.
Lieberman
and A.
Lichtenberg
, Principles of Plasma Discharges and Materials Processing
(Wiley
, 2005
).© 2022 Author(s). Published under an exclusive license by AIP Publishing.
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