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A two-dimensional particle-in-cell code for simulation of low-frequency electromagnetic processes in laboratory plasmas has been developed. The code uses the Darwin method omitting the electromagnetic wave propagation. The Darwin method separates the electric field into solenoidal and irrotational parts. The irrotational electric field is the electrostatic field calculated with the direct implicit algorithm. The solenoidal electric field is calculated with a new algorithm based on the equation for the electric field vorticity. The new algorithm is faster and more reliable than the Streamlined Darwin Field Formulation introduced decades ago. The system of linear equations in the new algorithm is solved using a standard iterative method. The code is applied to simulate an inductively coupled plasma with the driving current flowing around the plasma region.

In this study, we conduct 2D3V Particle-In-Cell simulations of hollow cathodes, encompassing both the channel and plume region, with an emphasis on plasma switch applications. The plasma in the hollow cathode channel can exhibit kinetic effects depending on how fast electrons emitted from the insert are thermalized via Coulomb collisions. The criterion that determines whether the plasma operates in a Duid or kinetic regime is given as follows. When Coulomb collisions occur at a much greater rate than ionization or excitation events, the Electron Energy Distribution Function relaxes to aMaxwellian distribution and the plasmawithin the channel can be describedwith aDuidmodel. In contrast, if inelastic processes are much faster, then the Electron Energy Distribution Function in the channel exhibits a notable high-energy tail, and a kinetic treatment is required. This criterion is applied to other kinds of hollow cathodes from the literature, revealing that a Duid approach is suitable for most electric propulsion applications, whereas a kinetic treatment might be necessary for plasma switches. Additionally, amomentumbalance reveals that a diffusion equation is sufAcient to predict the plasma plume expansion, a crucial input in the design of hollow cathodes for plasma switch applications.

A physical and numerical model of the Townsend discharge in molecular hydrogen and deuterium has been developed to meet the needs of designing a plasma-based switching device for power grid application. The model allows to predict the low-pressure branch of the Paschen curve for applied voltage in the range of several hundred kiloVolts. In the regime of interest, electrons are in a runaway state and ionization by ions and fast neutrals sustains the discharge. It was essential to correctly account for both gas-phase and surface interactions (electron emission and electron back-scattering), especially in terms of their dependence on particle energy. The model yields results consistent with prior data obtained for lower voltage. The three-species (electrons, ions and fast neutrals) model successfully captures the essential physics of the process., Comment: 21 pages, 13 figures

The process of numerical thermalization in particle-in-cell (PIC) simulations has been studied extensively. It is analogous to Coulomb collisions in real plasmas, causing particle velocity distributions (VDFs) to evolve towards a Maxwellian as macroparticles experience polarization drag and resonantly interact with the fluctuation spectrum. This paper presents a practical tutorial on the effects of numerical thermalization in 2D PIC applications. Scenarios of interest include simulations which must be run for many thousands of plasma periods and contain a population of cold electrons that leave the simulation space very slowly. This is particularly relevant to many low temperature plasma discharges and materials processing applications. We present numerical drag and diffusion coefficients and their associated timescales for a variety of grid resolutions, discussing the circumstances under which the electron VDF is modified by numerical thermalization. Though the effects described here have been known for many decades, direct comparison of analytically derived, velocity-dependent numerical relaxation timescales to those of other relevant processes has not often been applied in practice due to complications that arise in calculating thermalization rates in 1D simulations. Using these comparisons, we estimate the impact of numerical thermalization in several example low temperature plasma applications including capacitively coupled plasma (CCP) discharges, inductively coupled plasma (ICP) discharges, beam plasmas, and hollow cathode discharges. Finally, we discuss possible strategies for mitigating numerical relaxation effects in 2D PIC simulations.