Derivation of Moment Equations for the Theoretical Description of Electrons in Nonthermal Plasmas

Abstract

The derivation of moment equations for the theoretical description of electrons is of interest for modelling of gas discharge plasmas and semiconductor devices. Usually, certain artificial closure assumptions are applied in order to derive a closed system of moment equations from the electron Boltzmann equation. Here, a novel four-moment model for the description of electrons in nonthermal plasmas is derived by an expansion of the electron velocity distribution function in Legendre polynomials. The proposed system of partial differential equations is consistently closed by definition of transport coefficients that are determined by solving the electron Boltzmann equation and are then used in the fluid calculations as function of the mean electron energy. It is shown that the four-moment model can be simplified to a new drift-diffusion approximation for electrons without loss of accuracy, if the characteristic frequency of the electric field alteration in the discharge is small in comparison with the momentum dissipation frequency of the electrons. Results obtained by the proposed fluid models are compared to those of a conventional drift-diffusion approximation as well as to kinetic results using the example of low pressure argon plasmas. It is shown that the results provided by the new approaches are in good agreement with kinetic results and strongly improve the accuracy of fluid descriptions of gas discharges.

Impact of hexamethyldisiloxane admixtures on the discharge characteristics of a dielectric barrier discharge in argon for thin film deposition

“…The particle fluxes of the heavy particles Γ j in are determined in the common drift‐diffusion approximation according to where b j and D j denote the mobility and diffusion coefficient of the species, respectively, and the function sgn( Z j ) yields the sign of Z j . Furthermore, the electron particle flux Γ e in and the electron energy flux Q e in are expressed by a consistent drift‐diffusion approximation, which was deduced by an expansion of the electron velocity distribution function (EVDF) in Legendre polynomials and the derivation of the first four moment equations from the Boltzmann equation of the electrons in Becker and Loffhagen . They are given by …”

Comparing plasma fluid models of different order for 1D streamer ionization fronts

“…The average electron speed can be properly described by models that contain a momentum balance equation (i.e. no drift-diffusion approximation) [13,14,16,19]. In figure 7 we show the average electron speed and average electron energy flux for Ne.…”

“…Fluid models are typically constructed by taking the velocity moments of the Boltzmann equation. Depending on the number of moments considered and the closure assumptions, different fluid models have been derived over the last four decades [1,[12][13][14][15][16]. Fluid models require less computational resources than particle models, yet they can provide reasonably accurate results.…”

Analysis and simulation for a model of electron impact excitation/deexcitation and ionization/recombination

“…Here λ e is the de Broglie wavelength of an electron (see [19] for example). We have also defined the energy gap 1 We emphasize that only electron-impact collisions are examined in tho work. Therefore T e , the temperature of the Maxwellian distribution of electrons (2.5) is also the governing temperature of Boltzmann and Saha equilibria, when these are reached.…”

“…A variety of physical and numerical models have been used to solve for the Boltzmann equation (mostly designed for transport studies) using fluid approximations (e.g. [1]), spherical harmonics expansions (e.g. [2][3][4]), or Monte-Carlo (e.g.…”