Nonlinear hybrid Boltzmann–particle-in-cell acceleration algorithm

Abstract

Kinetic simulation of plasmas in which equilibrium occurs over ion time scales poses a computational challenge due to disparity with electron time scales. Hybrid electrostatic particle-in-cell (PIC) algorithms are presented in which most of the electrons reach thermodynamic equilibrium [Maxwell–Boltzmann (MB) distribution function] each time step. Conservation of charge enables convergence of the nonlinear Poisson equation. Energy conservation is used to determine the temperature of the Boltzmann species. This article first develops an algorithm where all the electrons have a MB energy distribution, either with a full MB distribution or with a truncation of the high energy tail. Second, high energy PIC electrons are added to the truncated distribution so that high energy electrons are modeled kinetically by PIC and low energy electrons (the majority) are modeled by the MB distribution. Collisions for PIC electrons are included via a Monte Carlo model, while for the MB electrons, the distributions are integrated with energy dependent cross sections. The MB model is not constrained by the electron time scale which decreases the required computer time by about the square root of the mass ratio of ion to electron. However, the hybrid boundary conditions are more complex and the simulation is not quite self-consistent. Comparison between full PIC and the PIC–MB hybrid is made for simulations of photo-ionized sustained discharges and current-driven dc discharges.

How to normalize Maxwell–Boltzmann electrons in transient plasma models

“…In the second test, case we used a particle description of the ions as in [3][4][5][6][7], where we set the ion mass equal to 6.64 · 10 À26 kg (argon), and we turned off the ionisation source term. As an initial condition, we distributed 10 5 macro-ions of weight 10 9 m À2 uniformly over the domain at a Maxwellian temperature of 300 K and set the electron density equal to the ion density of 10 16 m À3 .…”

“…The problem with the above model scheme is that the electron reference density n 0 is not known a priori but has to be calculated self-consistently from electron conservation [7] and that this calculation leads to numerical difficulties. The global electron conservation equation is…”

A hybrid Boltzmann electrons and PIC ions model for simulating transient state of partially ionized plasma

“…However, in a lot of experimental setups, one of the electrodes can be radio-frequency (RF) powered up, and therefore, U(r) is not necessary self-consistent and determined. Cartwright et al proposed a different method [12]. In their method, the reference potential was set to zero and the electron density n 0 , where / = 0, will be determined.…”

“…In their method, the reference potential was set to zero and the electron density n 0 , where / = 0, will be determined. The Poisson's equation with the Boltzmann distribution of electrons was solved by Newton-Raphson iteration [12]. Poisson's equation was rewritten as…”

“…It was shown that proper choice of n 0 determined whether this method was going to converge and converge to the correct solution [12]. Hagelaar et al independently developed a similar method [13].…”