Monte Carlo Collision Research Articles

Numerical study on wave attenuation via 1D fully kinetic electromagnetic particle-in-cell simulations

Abstract Electromagnetic wave-plasma interaction has drawn much attention recently due to numerous important technologies and applications, taking advantage of phenomena such as electromagnetic waves being reflected or absorbed in a plasma medium. The physics of wave-plasma interaction can be complicated, when non-uniform, non-equilibrium, or anisotropic plasmas are involved, in which numerical simulations can be used to fill the gaps between theoretical solutions and experimental measurements. Among many numerical methods, the particle-in-cell method, which can solve accurately both the electromagnetic fields and particle trajectories
self-consistently, would be the best choice to study wave-plasma interaction problems as long as the computational cost can be accepted. However, the applications of PIC on wave-plasma interaction remain rare, and the numerical effects of the PIC method on accurately evaluating the wave attenuation have not been studied in depth. In this paper, a number of numerical parameters and physical parameters are tested using a 1D electromagnetic PIC method plus Monte Carlo collision model. It is found that as long the as the basic PIC criterion is met, the PIC results can be trustable, and the numerical noise due to limited number of particles has a minor effect. The physical parameters of the EM wave frequency, amplitude, the plasma temperature, thickness, and collision type are studied, and their effects on the wave attenuation are presented. In addition, strategies on establishing simulation setup and evaluating the wave attenuation in terms of power or energy are discussed.

Stochastic weighted particle control for electrostatic particle-in-cell Monte Carlo collision simulations in an axisymmetric coordinate system

The non-uniform grids in the axisymmetric coordinate system pose a significant challenge for electrostatic particle-in-cell/Monte Carlo collision (PIC/MCC) simulations because they require numerous macroparticles to manage numerical heating around the mid-axis. To address this, we have developed a stochastic weighted particle control method that selectively samples small-weight particles, effectively controlling the particle number without inducing numerical heating. This method is based on a rejection-acceptance probability merging scheme, which is easy to implement and has a low time complexity. We have also made essential modifications, including a corrected density deposition scheme, an energy conservation scheme, and the introduction of target weights. By applying this particle control method, the number of macroparticles in the simulation can be reduced by more than one order of magnitude, significantly reducing the required computing time and storage. Furthermore, appropriately setting target weights also enables enhanced resolution of dilute regions with an acceptable increase in computational cost.

Ion velocity separation mechanism during vacuum spark stage

Abstract Supersonic ion jets produced in vacuum arc discharges have a wide range of applications, where precise control of ion kinetic energy is crucial. However, a comprehensive understanding of the ion acceleration mechanism remains elusive, particularly regarding whether there is ion velocity separation in the vacuum spark stage. In this paper, a 1D spherical implicit particle-in-cell (PIC) with Monte Carlo collision (MCC) model is employed to investigate the ion velocity separation in multi-charged vacuum arc plasma with varying electrode bias voltages and plasma ion densities. The results show that ion kinetic energy can reach hundreds of electron volts due to continuous acceleration by the formed potential valley, which leads to ion velocity separation at low electrode bias voltage or low plasma density. An increasing electrode bias voltage flattens the potential valley, reducing the electric field acceleration. While increasing the plasma density deepens the valley and intensifies Coulomb collisions, resulting in nearly-equal velocities across ions in different charge states. These findings can theoretically explain the discrepancies observed in previous experiments regarding the dependence of the ion velocity on its charge state during the vacuum spark stage.

Kinetic investigation of discharge performance for Xe, Kr, and Ar in a miniature ion thruster using a fast converging PIC-MCC-DSMC model

Abstract The present work develops a full particle-based model that couples the particle-in-cell plus Monte Carlo collision (PIC-MCC) simulation for plasma dynamics and the direct simulation Monte Carlo (DSMC) method for neutral dynamics in a synergistic iterative manner. This new model overcomes the slow convergence issue in the conventional direct coupling approach caused by the disparity of the time scales between the plasma and neutral dynamics. This model is applied to simulate the behavior of xenon (Xe) and its potential alternatives, krypton (Kr) and argon (Ar), in the discharge chamber of a miniature direct current (DC) ion thruster. The results show that a stable discharge is difficult to achieve for Kr and Ar under the operating conditions optimal for Xe. While increasing the discharge voltage can effectively improve the stability of discharge for Kr and Ar, other common strategies such as changing the magnetic field strength, propellant flow rate, and cathode current are not successful. The propellant utilization efficiency and discharge efficiency are affected by both discharge voltage and propellant flow rate. A maximum utilization efficiency and an optimal discharge efficiency are observed for all three propellants, with the values decreasing in the order of Xe, Kr, and Ar. Moreover, the discharge voltage corresponding to the optimal efficiency is inversely proportional to the square root of the propellant mass, indicating that the ion diffusional loss to the wall, rather than the ionization energy, is the dominant factor affecting the discharge performance for alternative propellants in a miniature DC thruster.

Multiscale Modeling of Plasma-Assisted Non-Premixed Microcombustion

This work explores microcombustion technologies enhanced by plasma-assisted combustion, focusing on a novel simulation model for a Y-shaped device with a non-premixed hydrogen-air mixture. The simulation integrates the ZDPlasKin toolbox to determine plasma-produced species concentrations to Particle-In-Cell with Monte Carlo Collision analysis for momentum and power density effects. The study details an FE-DBD plasma actuator operating under a sinusoidal voltage from 150 to 325 V peak-to-peak and a 162.5 V DC bias. At potentials below 250 V, no hydrogen dissociation occurs. The equivalence ratio fitting curve for radical species is incorporated into the plasma domain, ensuring local composition accuracy. Among the main radical species produced, H reaches a maximum mass fraction of 8% and OH reaches 1%. For an equivalence ratio of 0.5, the maximum temperature reached 2238 K due to kinetic and joule heating contributions. With plasma actuation with radicals in play, the temperature increased to 2832 K, and with complete plasma actuation, it further rose to 2918.45 K. Without plasma actuation, the temperature remained at 300 K, reflecting ambient conditions and no combustion phenomena. At lower equivalence ratios, temperatures in the plasma area consistently remained around 2900 K. With reduced thermal power, the flame region decreased, and at Φ = 0.1, the hot region was confined primarily to the plasma area, indicating a potential blow-off limit. The model aligns with experimental data and introduces relevant functionalities for modeling plasma interactions within microcombustors, providing a foundation for future validation and numerical models in plasma-assisted microcombustion applications.

Analysis of Monte-Carlo collision method with argon and helium background gases at different RF-powers and pressures in ELTRAP device

The impact of confined excited electron with argon and helium background gases is systematically studied using Monte-Carlo particle-in-cell simulations for various low powers (1.0–8.0 V) and pressures (1.50 × 10−8 Torr and 3.75 × 10−9 Torr) in ELTRAP device. The results of these numerical studies showed that the heating effect of the excited confined electron plasma density of 5 × 1014 m−3 within the Brillouin limit (ne≤ε0B22me=31.086×1019m-3) resulted a higher temperature (eV) and increased collision time in argon as a background gas as compared to helium gas. The axial temperature (eV) has a higher value than the radial temperature (eV) and increases with the increase in RF-Powers and pressures. The maximum kinetic energy of excited confined electrons occurred in the range of 0.03–0.04 m radially due to the maximum self-electric field intensity in simulation time up to 20 µs. The electric field decreases when the collision frequency is increased. The secondary electron production and ionization are higher than expected at a background pressure of 1.50 × 10−8 Torr as compared to 3.75 × 10−9 Torr. The remaining secondary electrons were always ejected from the symmetry axis of the device. It was observed that the production of secondary electrons is proportional to the ionization rate.

Analysis of application range of simplified models for field to thermo-field to thermionic emission processes from the cathode

Electron emission plays a dominant role in plasma–cathode interactions and is a key factor in many plasma phenomena and industrial applications. It is necessary to illustrate the various electron emission mechanisms and the corresponding applicable description models to evaluate their impacts on discharge properties. In this study, detailed expressions of the simplified formulas valid for field emission to thermo-field emission to thermionic emission typically used in the numerical simulation are proposed, and the corresponding application ranges are determined in the framework of the Murphy–Good theory, which is commonly regarded as the general model and to be accurate in the full range of conditions of the validity of the theory. Dimensionless parameterization was used to evaluate the emission current density of the Murphy–Good formula, and a deviation factor was defined to obtain the application ranges for different work functions (2.5‒5 eV), cathode temperatures (300‒6000 K), and emitted electric fields (105 to 1010 V·m−1). The deviation factor was shown to be a nonmonotonic function of the three parameters. A comparative study of particle number densities in atmospheric gas discharge with a tungsten cathode was performed based on the one-dimensional implicit particle-in-cell (PIC) with the Monte Carlo collision (MCC) method according to the aforementioned application ranges. It was found that small differences in emission current density can lead to variations in the distributions of particle number density due to changes in the collisional environment. This study provides a theoretical basis for selecting emission models for subsequent numerical simulations.

Interplay between the non-resonant streaming instability and self-generated pressure anisotropies

ABSTRACT The non-thermal particles escaping from collision-less shocks into the surrounding medium can trigger a non-resonant streaming instability that converts parts of their drift kinetic energy into large amplitude magnetic field perturbations, and promote the confinement and acceleration of high energy cosmic rays. We present simulations of the instability using an hybrid-Particle-in-Cell approach including Monte Carlo collisions, and demonstrate that the development of the non-resonant mode is associated with important ion pressure anisotropies in the background plasma. Depending on the initial conditions, the anisotropies may act on the instability by lowering its growth and trigger secondary micro-instabilities. Introducing collisions with neutrals yield a strong reduction of the magnetic field amplification as predicted by linear fluid theory. In contrast, Coulomb collisions in fully ionized plasmas are found to mitigate the self-generated pressure anisotropies and promote the growth of the magnetic field.

A discharge plasma regulation method with spike current for electrical discharge machining

Improving material removal rate (MRR) has been one of the most important issues in electrical discharge machining (EDM). A major factor which hinders the improvement of MRR in EDM is the relatively low expulsion ratio of molten material. During a discharge process, a large portion of molten material cannot be expelled sufficiently from the molten pool but re-solidifies, ultimately resulting in a low machining efficiency. In this study, aiming at enhancing the expulsion ratio of molten material in EDM, a discharge plasma regulation method with a spike current is proposed. A novel discharge pulse shape, with a spike current aligned at pulse turn-off stage, is adopted. The transient behaviors of the discharge plasma when a spike current engaged are captured using a high-speed camera, and the morphologies of the corresponding discharge craters are also analyzed. A particle-in-cell with Monte Carlo collision simulation method is used to analyze the changes in particle state inside the plasma. It is found that the spike current applied at the stage when the plasma contracts to a smaller diameter is more conducive to facilitating molten material expulsion. The results of consecutive-pulse machining experiments verify that the proposed discharge plasma regulation method can increase MRR by at least 25.89 % for stainless steel.

Numerical Investigation of Electrodeless Plasma Thruster with Rotating Electric Field

Two-dimensional simulations of various cross-sections under different magnetic field strengths in an electrodeless plasma thruster acceleration part using rotating electric field were conducted. The Particle-In-Cell and Monte Carlo Collision method was used to better understand plasma movement and find the ideal condition for the azimuthal current formation. The findings suggest that raising the voltage amplitude can improve the electric field’s penetration into the plasma, diminish the disparity between the azimuthal current’s maximum and minimum values, and consequently generate a more uniform size and distribution. Peak azimuthal current values are directly proportional to the ac frequency at frequencies above 100 MHz and inversely proportional to the frequency below 100 MHz. A rotating electric field that matches the background magnetic field could be produced by adjusting the phase difference among the two sets of antennas. And then the plasma is accelerated by the effect of E×B to achieve a more favorable azimuthal current. The results also demonstrate that an appropriately sized and distributed azimuthal current benefits more from a magnetic field with a strength between 200 and 1000 G and a value that declines from the cross-section center to its periphery.

Hysteresis between gas breakdown and plasma discharge

In direct-current (DC) discharge, it is well known that hysteresis is observed between the Townsend (gas breakdown) and glow regimes. Forward and backward voltage sweep is performed using a one-dimensional particle-in-cell Monte Carlo collision (PIC-MCC) model considering a ballast resistor. When increasing the applied voltage after reaching the breakdown voltage (Vb), transition from Townsend to glow discharges is observed. When decreasing the applied voltage from the glow regime, the discharge voltage (Vd) between the anode–cathode gap can be smaller than the breakdown voltage, resulting in a hysteresis, which is consistent with experimental observations. Next, the PIC-MCC model is used to investigate the self-sustaining voltage (Vs) in the presence of finite initial plasma densities between the anode and cathode gap. It is observed that the self-sustaining voltage coincides with the discharge voltage obtained from the backward voltage sweep. In addition, the self-sustaining voltage decreases with increased initial plasma density and saturates above a certain initial plasma density, which indicates a change in plasma resistivity. The decrease in self-sustaining voltage is associated with the electron heat loss at the anode for the low pd (rarefied) regime. In the high pd (collisional) regime, the ion energy loss toward the cathode due to the cathode fall and the inelastic collision loss of electrons in the bulk discharge balance out. Finally, it is demonstrated that the self-sustaining voltage collapses to a singular value, despite the presence of a initial plasma, for microgaps when field emission is dominant, which is also consistent with experimental observations.

Particle-in-cell simulations of EUV-induced hydrogen plasma in the vicinity of a reflective mirror

Particle-In-Cell (PIC) simulations were performed in this work to study the dynamics of the EUV-induced hydrogen plasma. The Monte-Carlo Collision (MCC) model was employed to deal with the collisions between charged particles and background gas molecules. The dynamic evolution of the plasma sheath, as well as the flux and energy distribution of ions impacting on the mirror surface, was discussed. It was found that the emission of secondary electrons under the EUV irradiation on the ruthenium mirror coating creates a positively charged wall and then prevents the ions from impacting on the mirror and therefore changes the flux and energy distribution of ions reaching the mirror. Furthermore, gas pressure has a notable effect on the plasma sheath and the characteristics of the ions impinging on the mirrors. With greater gas pressure, the sheath potential decreases more rapidly. The flux of ions received by the mirror grows approximately linearly and at the same time the energy corresponding to the peak flux decreases slightly. However, the EUV source intensity barely changes the sheath potential, and its influence on the ion impact is mainly limited to the approximate linear increase in ion flux.

Developing of a Monte Carlo calculations-based on simulation code for binary collision approximation in amorphous materials

In this work, we present a computer code developed and named MCIS (Monte Carlo Collisions In Solids) in his first version allowing the study and determination of the structure properties (arrangement, vacancies, energy losses...) during ion implantation in an amorphous solid matrix. This code is based on both the binary collisions approximation (BCA) theory and statistical Monte Carlo method within C++ Builder environment. The properties of this code can be summarized as: simulate 92 chemical elements of periodic table, the energy of projectile tuned from 1eV to few MeV, variable angle of incidence projectile, it can simulated 4 layers with a maximum of 4 elements per layer, it can use of the electronic losses of energies by choose of Firsov or Robinson approximation, various interactions potentials such as the ZBL, Moliers and C-Kr. Possibility of choose of scattering phenomenon among the Magic Formula and the Rutherford scattering. The results generated by our code are very promising when compared to those given by SRIM (Stopping and Range of Ions in Matter) code along with the advantage that our code takes into account the effect of the interaction potential which is neglected in SRIM.

Numerical Analysis and Quantification of Transfer Efficiency Coupled with Capillary and Quadrupole Ion Guide in an API-MS System.

Ion trajectory simulation is a significant and useful tool for understanding ion transfer mechanisms within the first vacuum region of the atmospheric pressure ionization mass spectrometer (API-MS). However, the complex dynamic gas field and wide pressure range lead to inaccurate simulation and huge computational costs. In this work, a novel electrohydrodynamic simulation called the statistical diffusion-hard-sphere (SDHS) mixed collision model was developed for characterizing the ion trajectories. For the first time, the influence of the dynamic pressure on the ion trajectory is considered for simulation, which helps to avoid an intolerable computational cost. Comparing with the conventional Monte Carlo collision model, the SDHS method helps to improve the calculation accuracy of ion trajectories under the first vacuum region and reduce the computational cost for at least 12-folds. Simulation results showed that the maximum ion loss came from the gap of the electrodes. The distance of the capillary-quadrupole ion guide was also a non-negligible factor. The trend of quantitative experimental results matches the SDHS simulation results. The maximum ion transfer efficiencies of quantitative experiment and simulation were 55% and 52%, respectively. Moreover, three ions, caffeine, reserpine, and Ultramark 1621, were measured for evaluating the applicability of SDHS in real API-MS. The trend of experimental results showed good agreement with that of computation. And the results of caffeine further illustrated the reason that the small mass ion transfer efficiency decreased with increasing radio frequency voltage. SDHS method is expected to be useful in the design of ion guides for further improvement of the sensitivity of API-MS.

Effects of low-frequency voltage on nonlinear standing wave excitation, plasma uniformity, and ion dynamics in dual-frequency asymmetric capacitive discharges

It is known that in very-high-frequency (VHF) capacitively coupled plasmas, the higher harmonics generated by nonlinear sheath motion can enhance the standing wave effect (SWE), which can lead to center-peaked plasma density profiles. In this work, an improved nonlinear electromagnetic model incorporating a transmission line model, an electron momentum balance model, a bulk plasma model, a collisionless nonlinear numerical sheath model, and an ion Monte-Carlo collision (MCC) model is developed to study the effects of low-frequency (LF) voltage VL on the nonlinear standing wave excitation, plasma uniformity, and ion energy and angular distribution functions (IEDFs and IADFs) in dual-frequency (DF) asymmetric capacitive argon discharges at relatively low pressure of 3 Pa. The plasma diffusion in the radial direction and ion dynamics within the LF oscillating sheath are self-consistently considered. The LF voltage VL at 2 MHz varies from 0 to 700 V while the HF voltage VH at 60 MHz is fixed at 100 V. Simulation results indicate that without the addition of an LF source (i.e. VL= 0 V), there are a considerable number of high-order harmonics with short wavelengths, leading to significant SWE and central peak in the radial plasma density profile. Nevertheless, the high-order harmonic excitations tend to be weakened and merely occur around the phase of the full LF sheath collapse due to a shorter characteristic damping time of the surface waves as VL increases. This, combined with increased surface wavelengths of both the driving frequency and the higher harmonics at a higher VL , leads to suppressed standing waves and improved plasma uniformity. Meanwhile, the simulations show that both the low and the high energy peaks of IEDF move towards higher energies, and the energy peak separation width ΔE becomes wider with the increase of VL . The IEDF at the radial center of the powered electrode exhibits a broader ΔE than that at the edge. For the IADF, an increased VL results in more ions incident on the electrode with a smaller deflection angle. Because of a thinner sheath and a higher sheath voltage at the electrode center, the peak value of IADF at the electrode center is greater than that at the edge.

Simulating vacuum arc initiation by coupling emission, heating and plasma processes

Vacuum arcing poses significant challenges for high-field vacuum devices, underscoring the importance of understanding it for their efficient design. A detailed description of the physical mechanisms involved in vacuum arcing is yet to be achieved, despite extensive research. In this work, we further develop the modeling of the physical processes involved in the initiation of vacuum arcing, starting from field emission and leading to plasma onset. Our model concurrently combines particle-in-cell with Monte Carlo collisions (PIC-MCC) simulations of plasma processes with finite element-based calculations of electron emission and the associated thermal effects. Including the processes of evaporation, impact ionization and direct field ionization allowed us to observe the dynamics of plasma buildup from an initially cold cathode surface. We simulated a static nanotip at various local fields to study the thresholds for thermal runaway and plasma initiation, identifying the significance of various interactions. We found that direct field ionization of neutrals has a significant effect at high fields on the order of 10 GV/m. Furthermore, we find that cathode surface interactions such as evaporation, sputtering and bombardment heating play a major role in the initiation of vacuum arcs. Consequently, the inclusion of these interactions in vacuum arc simulations is imperative.

Multi-hole grid thruster plume simulation model based on single grid aperture analytical model

This paper presents a plume divergence angle prediction model for ion thruster. Firstly, an analytical model using Gaussian distribution for single grid hole plume is proposed. The parameters of this single hole model are obtained from the particle simulation results of ion optics which utilizes the immersed finite element Particle-in-cell Monte Carlo collision (IFE-PIC-MCC) method. Then, by the extension of the single hole model, a three-dimensional plume model for the whole ion optics is proposed. This plume model can be used to predict the divergence angle of ion thruster plume. Finally, a numerical example with the simulation of a 217-hole ion optics is provided to demonstrate the features of the proposed method.

Particle injection methods in 3D-PIC MCC simulations applied to plasma grid biasing

In negative ion sources for the ITER Neutral Beam Injection system, the co-extraction of electrons is one of the main limiting factors. The current of co-extracted electrons can be decreased by applying a positive bias voltage to the Plasma Grid (PG) with respect to its source walls. Simulations using three-dimensional Particle-in-Cell Monte Carlo Collision (3D-PIC MCC) model are a powerful tool for studying the extraction region of such ion sources. However, the inclusion of both PG and source walls in the simulation domain is difficult due to numerical constraints. This study uses the 3D-PIC MCC code ONIX to explore the effects of particle injection models on plasma characteristics, using a flux injection model to regulate particle influx for a flat transition in potential from the bulk plasma to the simulation domain. Biasing of the PG above floating potential is possible using the flux injection scheme and results in a notable reduction in co-extracted electrons, corroborating with established experimental observations.

Particle-in-Cell simulation of radio-frequency ion thruster RIT-1.0

Ion propulsion systems with a small thrust can be used in space missions for highly accurate controllability while possessing low power and propellant consumption. At the University of Giessen, radio-frequency ion thrusters are miniaturized with thrusts in the order of micro-Newtons (μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu$$\\end{document}N-RIT). As the size of the thruster becomes smaller the plasma properties can not be measured inside a closed plasma chamber. The simulations are needed to investigate plasma properties like temperature, density profile which would aid further opimization. In this work, we discuss kinetic simulations of plasma inside the RIT-1.0 thruster using in-house developed PlasmaPIC code that incorporates Monte-Carlo collisions (MCC) and Direct Simulation Monte Carlo (DSMC) for neutral gas distribution. We briefly present the features of PlasmaPIC and simulation of stable plasma generation. We also show the influence of neutral gas and extraction properties for the RIT-1.0 thruster under consideration.

3D simulations of positive streamers in air in a strong external magnetic field

We study how external magnetic fields from 0 to 40 T influence positive streamers in atmospheric pressure air, using 3D PIC-MCC (particle-in-cell, Monte Carlo collision) simulations. When a magnetic field B is applied perpendicular to the background electric field E , the streamers deflect towards the +B and −B directions which results in a branching into two main channels. With a stronger magnetic field the angle between the branches increases, and for the 40 T case the branches grow almost parallel to the magnetic field. Due to the E×B drift of electrons we also observe a streamer deviation in the opposite −E×B direction, where the minus sign appears because positive streamers propagate opposite to the electron drift velocity. The deviation due to this E×B effect is smaller than the deviation parallel to B . In both cases of B perpendicular and parallel to E , the streamer radius decreases with the magnetic field strength. We relate our observations to the effects of electric and magnetic fields on electron transport and reaction coefficients.