Electron Energy Balance Equation Research Articles

On fundamental inconsistencies in a commonly used modification of a fluid model for glow discharge

This work considers the fundamental contradictions in the concept of one of the most well-known and widely used modifications of the fluid model for simulation of a glow discharge (GD), the ‘local mean energy approximation’ (LMEA). In this model, it is proposed to determine the kinetic coefficients in the electron particle and energy balance equations as functions of the electron mean energy (temperature) rather than local electric field, using a one-to-one correspondence between these parameters through the electron Boltzmann equation. It is shown that the scope of applicability of this model, like any other modification of the fluid model, is limited by the local mode of formation of the electron energy distribution function (EEDF). Therefore, as demonstrated by the examples of typical 1D and 2D problems for a GD in argon, its extension to the region of nonlocal EEDF is in no way justified and leads not only to serious errors in the results, but also to a logically intractable situation in attempts to apply the main postulate of the LMEA model to the region of a weak (or even reverse) electric field in a negative glow plasma. At the same time, the apparent reliability of calculations within the framework of the LMEA model for a number of parameters, in our opinion, only slows down progress in modeling of gas discharge plasma.

Influence of radio frequency wave driving frequency on capacitively coupled plasma discharge

A two-dimensional symmetric fluid model is established to study the influence of radio frequency (RF) wave driving frequency on the capacitively coupled plasma discharge. The relationship between the driving frequency and electron density is obtained by solving the electron energy balance equation. The calculation results show that the average electron density first increases rapidly with the increase in driving frequency and then gradually tends to saturation at a threshold frequency. A fluid simulation is also carried out, which provides similar results. Physical studies on this phenomenon are conducted, revealing that the essence of this phenomenon is due to the inability of electrons to quickly respond to potential changes within the boundary sheath when the driving frequency of RF exceeds the plasma frequency. In addition, it is also found that increasing gas pressure can enhance the electron density and the type of gas can also affect the electron density.

Validation of a plasma burn-through simulation with an ECH power absorption model in KSTAR

An electron cyclotron heating (ECH) power absorption model was integrated with a plasma burn-through simulator, DYON, and the new version, DYON-EC, was validated against KSTAR ECH-assist start-up experiments. The absorbed ECH power was calculated with an analytic formula which is a function of the electron density and electron temperature and ECH hardware settings, namely, the injected power, wave frequency, harmonic number, mode fraction, and beam injection angles. Wave parameter changes by wall reflection was also included to simulate multiple reflections. The absorbed ECH power was self-consistently included in the electron energy balance equation. The simulation settings of the plasma-wall interaction model and the electromagnetic scenario including the eddy current model were optimized to reproduce the plasma parameter evolution and line emission data in a pure ohmic start-up discharge. The study revealed that assumption of double-path EC beam absorption is required to reproduce a KSTAR ECH-assisted start-up discharge. Using the same optimized settings, DYON-EC modelling successfully reproduced multiple KSTAR EC-assisted discharges in a large range of operation parameters. The good statistical reproduction of measured plasma parameter evolution confirms the validity of the DYON burn-through modelling with ECH.

2D simulations of inductive RF heating in the drivers of the SPIDER device

This paper presents the basic physical and numerical principles of a fluid model of the negative ions source. The model gives self-consistent two-dimensional description of the source, including the neutral gas flow, plasma chemistry, RF coupling in the source driver and plasma transport through the magnetic filter. The different particle species (electrons, the three types of the positive ions: H+, H2+, H3+, negative ions H− and the neutral species: hydrogen atoms H and molecules H2) are described by separate continuity equations and the electron temperature is governed by the electron energy balance equation. The particle fluxes are found from momentum equations neglecting the inertia terms (drift-diffusion approximation). The electrostatic coupling between electrons and ions is described by the Poisson equation. The paper focuses on details of the RF model and on the initial code developments to simulate the currents in the source induced by the RF currents flowing in the feeding coils. In our approach, the RF electrical field is split in a plasma part plus another one in vacuum, E = Ep + EV, which simplifies the boundary conditions for Ep while EV is obtained from a theoretical formulation for the field generated by a current loop, which results in a fast and flexible numerical algorithm providing converged solution for the RF field after a few iterations, even for the cases with high conductivity of the plasma. The numerical method is based on finite volume approximation and 9-point discretization is used to account on anisotropy due to magnetic field. The semi implicit numerical solver allows for large time steps (>1000× explicit time step) producing steady-state solution in a reasonable time (few hours for a typical mesh). The RF model has been successfully coupled to the fluid equations of the source and self-consistent solutions are analyzed in order to see the influence of the driver parameters and RF model assumptions (e.g. stochastic conductivity, induced magnetic field) on the heating efficiency and the plasma dynamics. It has been found that without magnetic field the plasma dynamics in the SPIDER source is almost independent on the details of the RF coupling model.

Estimation of the scrape-off-layer width in tokamak using the electron balance equations

The width of the scrape-off-layer has been a critical issue in the design of fusion reactors. This width has been derived using Braginskii electron energy balance and electron momentum balance equations. The most significant influence on the edge plasma profiles, especially on the divertor heat flux, was the drift element. Therefore, the drift mechanisms are used to find the width of scrape-off-layer. The derivation of these formulas with some presumptions leads to the width of scrape-off-layer ∼ qρci, where q is a safety factor, and ρci is an ion Larmor radius. The plasma parameters for the small-size divertor tokamak are solved using the B2SOLPS5.02D code at different locations. Then, they are applied in the scrape-off-layer width relation to correlate the width with these variables. In these locations, the scrape-off-layer width was found to be uneven. The results show that the diffusion coefficient, heat flux density, impurities, and the creation of edge transport barriers significantly affect scrape-off-layer width.

Role of excimer formation and induced photoemission on the Ar metastable kinetics in atmospheric pressure Ar–NH3 dielectric barrier discharges

Tunable diode laser absorption spectroscopy was used to record the space-and time-resolved number density of argon metastable atoms, Ar(1s3) (Paschen notation), in plane-to-plane dielectric barrier discharges (DBDs) operated in a Penning Ar–NH3 mixture at atmospheric pressure. In both low-frequency (LF 650 V, 50 kHz) discharges and dual LF–radiofrequency (RF 190 V, 5 MHz) discharges operated in α–γ mode, the density of Ar(1s3) revealed a single peak per half-period of the LF voltage, with rise and decay times in the sub-microsecond time scale. These results were compared to the predictions of a 1D fluid model based on continuity and momentum equations for electrons, argon ions (Ar+ and Ar2 +) and excited argon 1s atoms as well electron energy balance equation. Using the scheme commonly reported for Ar-based DBDs in the homogeneous regime, the Ar metastable kinetics exhibited much slower rise and decay times than the ones seen in the experiments. The model was improved by considering the fast creation of Ar2 * excimers through three-body reactions involving Ar(1s) atoms and the rapid loss of Ar2 * by vacuum ultraviolet light emission. In optically thin media for such photons, they can readily reach the dielectric barriers of the DBD electrodes and induce secondary electron emission. It is shown that Ar2 * and photoemission play a significant role not only on the Ar metastable kinetics, but also on the dominant ionization pathways and possible α–γ transition in dual frequency RF–LF discharges.

Two-temperature balance equations implementation, numerical validation and application to H2O–He microwave induced plasmas

Global Models are widely used to study reaction kinetics in low-temperature plasma discharges. The governing conservative equations are simplified into a system of ordinary differential equations in order to provide computationally feasible conditions to study complex chemistries with hundreds of species and thousands of reactions. This paper presents a detailed two-temperature global model for a H2O–He mixture. The model developed in this work uses a statistical thermodynamics approach to solve the heavy particles energy equation self-consistently together with the electron energy and particles balance equations in order to improve the description of reactive plasma environments. Three analytical test cases are presented to validate and demonstrate the capability of this newly developed functionality embedded in PLASIMO software’s [] global model module. The developed H2O–He models are compared with the reported results for a radio-frequency plasma [] and then with experimental measured electron densities and gas temperature for a microwave induced plasma. In addition, conversion and energy efficiencies of hydrogen and hydrogen peroxide productions are compared with experimental values (only for hydrogen) for a pure H2O microwave induced plasma and with available literature results. This comparison underlines the challenges toward finding an optimal plasma configuration and conditions for production of hydrogen from water. The three analytical test cases for validation of the gas-temperature balance implementation in the PLASIMO global model and the detailed developed H2O–He model can be used as benchmarks for other global plasma models. The PLASIMO input files for the presented H2O–He model are available as supplementary materials (https://stacks.iop.org/PSST/30/075007/mmedia); for any future update, please consult the PLASIMO website, https://plasimo.phys.tue.nl.

On the Validity of Two-Chamber Configuration for the Generation of Electromotive Force in Photoplasma

This study investigates different geometry configurations for obtaining photoplasma in a gas cell of Na–Ar mixture. For this purpose, 2-D simulations with the fluid model of plasma have been conducted for single-chamber and two-chamber cells. For the studied range of pressure, the drift-diffusion approximation has been used. The simulation model is based on solving the continuity, momentum balance for charged particles, coupled with electron energy balance equation and the Poisson equation. Detailed plasma chemistry has been considered, and plasma parameters have been obtained, such as electron density and temperature, and the electric potential of the plasma. The obtained results show the effectiveness of the two-chamber cell configuration over the single-chamber one in generating a notable electromotive force (EMF).

Benchmark of the KGMf with a coupled Boltzmann equation solver

The Kinetic Global Model framework (KGMf) is an open-source general-purpose global model (spatially averaged) simulation code developed to explore the reaction kinetics and pathways in plasma discharge systems. It contains species continuity and electron energy balance equations, with a time-dependent evaluated electron energy distribution function (EEDF) for electron impact reactions. The EEDF is utilized to determine the rate coefficients for electron impact reactions, which can have profound impact on the temporal evolutions of plasma parameters. Previously, the EEDF was commonly assumed as Maxwellian or an analytical function of the effective electron temperature. In this work, the KGMf is coupled with a Boltzmann equation (BE) solver to self-consistently compute the EEDF. The EEDF evolution frequency is determined based on relative changes of the reduced electric field. The KGMf is benchmarked with the ZDPlasKin code based on high-pressure low-temperature argon plasma discharge cases. The temporal evolutions of reduced electric field, electron temperature, EEDF, reaction rates, and species densities, are obtained and compared under different discharge conditions, showing good agreement between the KGMf and the ZDPlasKin simulations. The application of the KGMf for predicting breakdown times in high power microwave discharges is also presented, which shows qualitative agreement with particle-in-cell simulations. The KGMf can be further applied for more complicated plasma discharge systems, where the reaction kinetics are more intricate (e.g., plasma-assisted combustion systems).

Plasma-Enhanced Chemical Vapor Deposition of Silicon Films at Low Pressure in GEC Reference Cell

A self-consistent fluid simulation of inductively coupled plasma-enhanced chemical vapour deposition (ICP-CVD) using silane, argon, and hydrogen mixture is presented. The model solves the continuity equations for charged species, the drift-diffusion equations to describe their transport and the electron energy balance equation, coupled with Maxwell’s and Poisson’s equations, using COMSOL Multiphysics software. In order to obtain a better comprehending of the discharge parameters, a discharge in a Gaseous Electronics Conference cell (GEC) reactor is simulated. In this study, the GEC reactor with five-turn inductive coils is driven at a radio frequency of 13.56 MHz in order to sustain the plasma discharge at low temperature for a mixture of SiH4/Ar/H2, with an intial gas pressure fixed at 20 mTorr. The simulations yield the profile of plasma components such as electron density and temperature as well as the electrical potential in the centre of the discharge during silicon films deposition. The effects of external settings, such as chamber pressure, applied power and hydrogen dilution, on the growth rate of the silicon films deposited by ICP-CVD are then investigated. It is shown that the increase of the applied power from 1000 to 3000 W and the gas pressure from 10 to 40 mTorr results in a moderate increase in the deposition rate of the silicon films, whereas the increase in the dilution of the hydrogen in the mixture produces its decrease.

Reduced kinetics model for X-ray-generated atmospheric air plasmas fitted by microwave transmission measurements

We elaborate a reduced kinetics model to study humid air plasmas at atmospheric pressure generated by X-ray irradiation. The originality of the present approach is to use the experimental results of the transmission measurements, in the case of a microwave signal by the X-ray-induced plasma filled waveguide, to fit the calculated time evolutions of some plasma parameters such as average electron energies and an effective loss coefficient. The reduced kinetics model used to restitute the transmission measurements is based on the solution of a one-dimensional transport of a guided microwave signal coupled to the calculation of the complex electric conductivity of the plasma. The conductivity is calculated using a simplified kinetics scheme based on three species (electrons, positive ions, and negative ions) and coupled to the electron energy balance equation. The input parameters of the model are the collision cross sections of the electrons impact with air molecules (N2, O2, and H2O) and the electron energy distribution functions pre-tabulated for a large set of average electron energies. The latter takes into account the main processes leading to the decrease of average electron energies. This model is more generally usable for the modelling of weakly ionized atmospheric air plasmas during, for instance, the streamer development in corona or dielectric barrier discharges.

Fluid-model analysis on discharge structuring in the RF-driven prototype ion-source for ITER NBI

A 2D self-consistent fluid plasma model is employed for studying the plasma in the radio-frequency (RF)-driven negative hydrogen ion source prototype developed to equip the neutral beam injector systems for ITER. The source is considered in its usual configuration with a cylindrical driver and magnetic filter field (MF) produced by permanent magnets arranged in a magnet frame movable along the expansion chamber. The model accounts for the RF injection, the bias potential applied at the plasma grid (PG) and the losses of particles and electron energy in the third dimension, i.e. along the magnetic field lines, are evaluated. The study is focused on the role of the processes which govern the spatial structuring of the plasma parameters caused by the topology of the MF, the PG bias and the losses along the MF. The obtained numerical results are analyzed based on the contribution of the local and nonlocal processes in the electron- and negative hydrogen ion-balance and electron energy balance equations. It is shown how the fluxes associated with the diamagnetic drift caused by the gradient of the electron temperature, and the E × B-drift as well as the electron heating and the negative ion drift in the dc electric field are involved in the formation of the pattern of the plasma parameters. Effects due to the partial penetration of the MF in the driver are also investigated. A comparison with the experimental data shows that, accounting for the losses along the MF, the present model is a reliable tool for study and optimization the ITER relevant sources.

Effects of the plasma-facing materials on the negative ion H− density in an ECR (2.45 GHz) plasma

Within the framework of fundamental research, the present work focuses on the role of surface material in the production of H− negative ion, with a potential application of designing cesium-free H− negative ion sources oriented to fusion application. It is widely accepted that the main reaction leading to H− production, in the plasma volume, is the dissociative attachment of low-energy electrons (Te ≤ 1 eV) on highly ro-vibrationally excited hydrogen molecules. In parallel with other mechanisms, the density of these excited molecules may be enhanced by means of the recombinative desorption, i.e. the interaction between surface absorbed atoms with other atoms (surface adsorbed or not) through the path . Accordingly, a systematic study on the role played by the surface in this reaction, with respect to the production of H− ion in the plasma volume, is here performed. Thus, tantalum and tungsten (already known as H− enhancers) and quartz (inert surface) materials are employed as inner surfaces of a test bench chamber. The plasma inside the chamber is produced by electron cyclotron resonance (ECR) driving and it is characterized with conventional electrostatic probes, laser photodetachment, and emission and absorption spectroscopy. Two different positions (close to and away from the ECR driving zone) are investigated under various conditions of pressure and power. The experimental results are supported by numerical data generated by a 1D model. The latter couples continuity and electron energy balance equations in the presence of magnetic field, and incorporates vibrational kinetics, H2 molecular reactions, H electronically excited states and ground-state species kinetics. In the light of this study, recombinative desorption has been evidenced as the most probable mechanism, among others, responsible for an enhancement by a factor of about 3.4, at 1.6 Pa and 175 W of microwave power, in the case of tantalum.

English

Problems of using proton-Lithium-6 (p6Li) fuel are energy losses that occur in a fusion reactor. Investigating the energy balance equation in this fuel is significant. The p6Li reaction is termed aneutronic, as it produces relatively few neutrons and requires none for breeding. The energy from the charged reaction products can be directly converted to electrical power at a much higher efficiency than Deuterium-tritium (DT). In this paper, the approach of optimum performance of p6Li fuel in fusion reactors was presented investigating the energy balance equations for ions and electrons. The optimum fuel mixture is almost . The performance was determined to be p6Li and is favorable for Ti=800 keV. Key words: Fuel, reactor, energy, radiation.

Radial structure of the constricted positive column: Modeling and experiment.

We present a detailed self-consistent model of a positive column in argon glow discharge at moderate pressures and currents. This model describes the discharge transition between diffuse and constricted states. The model includes an extensive set of plasma chemical reactions and equation for inhomogeneous gas heating. The nonequilibrium behavior of an electron distribution function is also considered. One of the main features of the model is an accurate treatment of radiation trapping by solving the Holstein-Biberman equation directly. Influence of the radiation trapping on macroscopic parameters of the constricted positive column is studied. We propose a method for solving a boundary-value problem, including particle and energy balance equations for electrons, ground state atoms, atomic and molecular ions, and excited species. Unlike traditional solution approaches for similar systems, the method provides continuous Z- and S-shaped characteristics of discharge parameters, describing hysteresis in transition between diffuse and constricted discharge regimes. Performed experiments include measurements of volt-ampere characteristics and spectroscopic study of radial density profiles of excited atoms by measuring line emission and absorption, and electrons by measuring bremsstrahlung intensity. The role of resonance radiation trapping in spatial redistribution of 1s and 2pstates of argon is demonstrated. Results of modeling are compared to the experimental data.

Global Energy Transfer Model of a Microwave Electrothermal Thruster Operating With Helium Propellant at 2.45-GHz Frequency

Microwave electrothermal thruster (MET) is a type of space propulsion system that uses free floating plasma to heat the propellant gas. This paper presents a 0-D model that investigates the energy transfer from the electromagnetic wave to the propellant gas via atmospheric pressure plasma, and evaluates the steady-state plasma properties of an MET thruster. In the model, governing equations for electron particle balance, electron energy, balance and heavy particle energy balance equations are solved to obtain the plasma parameters. Electron temperatures of around 1 eV, electron number densities of $10^{19}~\#/\text {m}^{3}$ , and heavy particle temperatures of about 3000 K are evaluated for the free floating plasma inside the MET resonant cavity for 1200-W delivered power at 2.45-GHz frequency. Obtained solutions are in good agreement with the experimental results from the literature. Along with the plasma parameters, rocket performance parameters, and specific impulse and thrust are computed. The developed model is also used to predict the plasma and performance parameters of a prototype MET system developed at the Bogazici University Space Technologies Laboratory.

Low-pressure low-frequency inductive discharge with ferrite cores for large-scale plasma processing

Electrophysical characteristics of a low–frequency (100 kHz) low–pressure (10–100 Pa) argon inductive discharge with ferrite cores have been investigated for discharge currents of 1–50 A, discharge chamber diameter of 230 mm. The dependencies of electric field strength on the argon pressure and discharge current were measured. Radial profiles of electron density and temperature were determined with double electric probes. Experimental results were compared with numerical results obtained by a self-consistent kinetic model of the low-frequency ICP, based on the simultaneous solution of a non-local Boltzmann equation for electron energy distribution function and balance equations for the metastable argon atoms density and gas temperature. Comparison of numerical and experimental results is presented.

3-D Numerical Characterization of a Microwave Argon PECVD Plasma Reactor at Low Pressure

A 3-D model of a microwave plasma (mwp)-enhanced chemical vapor deposition (PECVD) reactor at 2.45 GHz in argon at low pressure describing self-consistently, the coupling of the microwave energy into the plasma is presented. The characteristics of the discharge are simulated using a fluid plasma model which solves the electron and ion continuity equations, electron energy balance equation, and the Poisson’s equation by finite element method, using COMSOL Multiphysics software. The physical behavior of the microwave PECVD discharge, such as plasma density, electron temperature, electric field and plasma potential, are simulated and analyzed. The chemical reactions considered in this paper are: elastic, superelastic, excitation, ionization, penning ionization and metastable quenching processes, involving electrons, ions (Ar+), neutral atoms (Ar), and excited metastable argon atoms (Ar*). The plasma characterization results are studied for a gas temperature of 300 K, a gas pressure of 100 mtorr and a microwave power of 600 W. The effect of varying gas pressure from 50 to 200 mTorr has been studied. The obtained results turn out to be in agreement with previous measurements and show that this kind of model can lead to a better understanding of the physical processes occurring in this kind of microwave reactor and thus allow optimization of this device.

CO2 conversion in a gliding arc plasma: 1D cylindrical discharge model

CO2 conversion by a gliding arc plasma is gaining increasing interest, but the underlying mechanisms for an energy-efficient process are still far from understood. Indeed, the chemical complexity of the non-equilibrium plasma poses a challenge for plasma modeling due to the huge computational load. In this paper, a one-dimensional (1D) gliding arc model is developed in a cylindrical frame, with a detailed non-equilibrium CO2 plasma chemistry set, including the CO2 vibrational kinetics up to the dissociation limit. The model solves a set of time-dependent continuity equations based on the chemical reactions, as well as the electron energy balance equation, and it assumes quasi-neutrality in the plasma. The loss of plasma species and heat due to convection by the transverse gas flow is accounted for by using a characteristic frequency of convective cooling, which depends on the gliding arc radius, the relative velocity of the gas flow with respect to the arc and on the arc elongation rate. The calculated values for plasma density and plasma temperature within this work are comparable with experimental data on gliding arc plasma reactors in the literature. Our calculation results indicate that excitation to the vibrational levels promotes efficient dissociation in the gliding arc, and this is consistent with experimental investigations of the gliding arc based CO2 conversion in the literature. Additionally, the dissociation of CO2 through collisions with O atoms has the largest contribution to CO2 splitting under the conditions studied. In addition to the above results, we also demonstrate that lumping the CO2 vibrational states can bring a significant reduction of the computational load. The latter opens up the way for 2D or 3D models with an accurate description of the CO2 vibrational kinetics.

Numerical Analysis of Substrate‐Incident Carbon Flux in Low‐Pressure Radio‐Frequency CH4 Plasmas for Deposition of Diamond‐Like Carbon Films

SUMMARYCapacitively coupled hydrocarbon (CH4) plasmas used in the deposition of diamond‐like carbon films have been simulated using a self‐consistent one‐dimensional fluid model, incorporating the mass balance equations for electrons, ions, radicals, and nonradicals, as well as the electron energy balance equation, coupled with the Poisson equation. Despite the conditions in the low‐pressure CH4 gas, many positive‐ion species, such as C2H, CH, C2H, and CH have been found in the plasmas. Nonradical neutrals such as C2H4, C3H8, C2H2, and C2H6, have also been found at higher densities comparable to the source gas density. This result indicates that this complexity in the background gas of CH4 plasmas has a substantial effect on the electron energy distribution function, which is important when making determinations about the plasmas properties.