A regularized high-order moment model to capture non-Maxwellian electron energy distribution function effects in partially ionized plasmas

Abstract

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.

The need for accurate measurements of thermal velocity distribution functions in the solar wind

“…These observations suggest the proton core shares the same multitude of populations as the electrons below ∼1 keV in the solar wind. What's perhaps more confusing is how to reconcile the implications for the self-similar distribution, given that the deviation from a bi-Maxwellian is evidence of inelastic collisions (Wilkinson and Edwards, 1982;Martin and Piasecki, 1999;Alvarez Laguna et al, 2022). This realization was made when comparing to studies of flow through disordered porous media and shocks in coarse grain media (Villani, 2006;Siena et al, 2014;Matyka et al, 2016).…”

Discussion on the transport processes in electrons with non-Maxwellian energy distribution function in partially-ionized plasmas

“…We refer the interested reader to Ref. [1] where the 9-M set of moment equations is presented. The Grad's non-equilibrium distribution function for this choice of moments can be decomposed into an isotropic part of the distribution function and a anisotropic part along the first spherical harmonic, as follows:…”

“…By following the procedure described in Ref. [1], we find that the moment equations can be regularized. Whereas the density, energy, and fourth moment are described by a non-linear mixed hyperbolic-parabolic equations, the electron particle and heatfluxes are described as transport fluxes that are function of the gradients of the density, pressure, and fourth moment and the electric field.…”

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In a previous work [1], we have developed a non-linear moment model for electrons that self-consistently captures non-Maxwellian electron energy distribution function (EEDF) effects. The model does not rely in the local approximation and the transport coefficients are calculated by expanding the distribution function into Hermite polynomials and by taking moments of the Boltzmann equation, including the collision operator for elastic and inelastic collisions with arbitrary cross sections.

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“…Collision frequenciesThe collisional frequencies, which are proportional to the heavy particle density, read ν (u,1) n α (16 + 30∆ e )Ω(1,1) eα − 40∆ e Ω(1,2) eα + 8∆ e Ω(1,3) n α (64 + 120∆ e )Ω (1,2) eα − 160∆ e Ω(1,3) eα + 32∆ e Ω(1,4) eα (A.4)ν (∆,2) eα = 16n α Ω (1,2) eα − (1,1) eα , ν (∆,4) eα = 64n α Ω (1,3) eα − (1,2)eα (A.5) where α ∈ {i, g} and Ω (l,r) eα (T e ) 2r+3 e −ξ 2 Q (l) eα dξ with ξ = β e g and β e = m e 2eT e . (A.6)…”