Describing local and non-local electron heating by the Fokker–Planck equation

Abstract

The common description of kinetic effects in low-pressure plasmas is based on the Boltzmann equation. This applies especially to the description of Ohmic (collisional) and non-local (stochastic/collisionless) electron heating, where the Boltzmann equation is the starting point for the derivation of the corresponding heating operator. Here, it is shown, that an alternative and fully equivalent approach for describing the interaction between electrons and electric fields can be based on the Fokker–Planck equation in combination with the corresponding Langevin equation. Although, ultimately the final expressions are the same in both cases, the procedures are entirely different. While the Fokker–Planck/Langevin approach provides physical insights in a very natural way, the linearized Boltzmann equation allows straightforward calculation but requires some effort to interpret the mathematical structure in terms of physics. The Fokker–Planck equation for the present problem is derived, with particular emphasis on the consistent treatment of velocity-dependent elastic collision frequencies. The concept is tested for a simple case by comparing it with results from an ergodic Monte-Carlo simulation. Finally, the concept is applied to the problem of combined Ohmic and stochastic heating in inductively coupled plasmas. The heating operator is first analyzed for an exponential model field profile. Self-consistent field profiles are determined subsequently. In this context, a generalization of the plasma dispersion function is introduced, which allows for arbitrary forms of the distribution function and velocity dependence of the elastic collision frequency. Combined with the Fokker–Planck heating operator, a fully self-consistent description of the plasma and the fields is realized. Finally, a concept for integrating the operator in a standard local Boltzmann solver and using the local solver for determination of the global electron velocity distribution function in a low-pressure plasma is provided.

Editorial : Reviews of Modern Plasma Physics: Volume 6

“…There are number of invited papers from AAPPS-DPP2020 talks, (Shen et al 2022;Rodriguez-Fernandez et al 2022;Bhattacharjee et al 2022;Shin et al 2022;Czarnetzki and Alves 2022;Lee 2022) and from AAPPS-DPP2021 talk ).…”

Charged-particle transport models for global models

“…As in 4.2.1, the electrons are assumed in Boltzmann equilibrium with the space-charge potential, but here the ion momentum-balance equation further includes the corresponding inertia term. Czarnetzki and Alves [29] solved the resulting continuity and momentum-balance equations in Cartesian coordinates, obtaining the following implicit expression for the ion lossfrequency ν h−fact transp + , due to transport across any x, y, z direction of a cubic discharge with edge length L…”

“…Explicit inverted solutions can only be obtained in the LP (ν ≫ 1) and the HP (ν ≪ 1) limiting cases, which motivates an heuristic formula spanning the whole pressure range. For the one-dimensional situation the bridging formula is [29]…”

“…• Equation ( 38) is more accurate at very LPs (L/λ + ≪ 1), since it was deduced to bridge the EP and the EN regimes, following (24a) and (29). Indeed, in the LP limit the results shown in [36] are notably different, even if the solution of (31a) does not diverge at low L/λ + , contrary to what is claimed in [36]; • The fluid model used in [36] assumes Boltzmann negative ions, which is valid only at LP, when the negative ions are not rapidly destroyed (either by recombination with positive ions or by detachment on neutral excited states), at the location where they have been created (by attachment).…”

Synthesis of Acetylene and Benzene in Controlled Methane-Plasma System

“…Non-thermal plasma is created either at low or atmospheric pressure [6]. While low-pressure plasma needs cost-intensive process chambers to maintain a vacuum, higherpressure plasma does not require such arrangements and hence is more advantageous [7,8]. For the last 30 years, household microwave ovens have been converted into plasma reactors and used for many scientific applications as a laboratory investigation equipment [9].…”