Particle Balance Equations Research Articles

Theoretical study of particle and energy balance equations in locally bounded plasmas

In this study, new particle and energy balance equations have been developed to predict the electron temperature and density in locally bounded plasmas. Classical particle and energy balance equations assume that all plasma within a reactor is completely confined only by the reactor walls. However, in industrial plasma reactors for semiconductor manufacturing, the plasma is partially confined by internal reactor structures. We predict the effect of the open boundary area ( ) and ion escape velocity ( ) on electron temperature and density by developing new particle and energy balance equations. Theoretically, we found a low ion escape velocity ( / ) and high open boundary area ( ) to result in an approximately 38% increase in electron density and an 8% decrease in electron temperature compared to values in a fully bounded reactor. Additionally, we suggest that the velocity of ions passing through the open boundary should exceed under the condition .

Study of the Pure Deuterium Fuel Burning Rate in Z-Pinch Devices with Magneto-Inertial Confinement

The burning rate of pure deuterium (D-D) fuel in Z-pinch devices with magneto-inertial confinement was studied in this paper. The system of particle and energy balance equations for D-D fuel burning with a mixed D-T-3He fusion cycle (D-D, D-T, and D-3He reactions) was solved numerically, taking into account the densities of all reacted and produced ions (protons, deuterium, tritium, helium-3, and alpha-particles). The obtained results indicate that effective D-D fusion in Z-pinch devices can be successfully achieved under conditions of a hot, dense plasma with an initial temperature of 31 keV or higher. The initial ion density of deuterium and electron density were equal due to quasi-neutrality condition of the plasma, with both reaching 1024 m−3. Although the obtained results show that the burning rate of D-D fuel is approximately 2.3 times slower and its power density notably lower than that of D-T fuel, pure deuterium plasma can be considered as a promising alternative to well-studied deuterium–tritium plasma, with potential future applications in magneto-inertial fusion (MIF) facilities.

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.

Time-resolved analysis of Ar metastable and electron populations in low-pressure misty plasma processes using optical emission spectroscopy

Misty plasmas have recently emerged as a promising tool for nanocomposite thin films deposition. However, aerosol-plasma interactions remain poorly documented, especially at low working pressure. In this work, optical emission spectroscopy is used to probe the temporal evolution of three fundamental plasma parameters during pulsed liquid injection in an inductively coupled argon plasma at low-pressure. Time-resolved values of metastable argon density, electron temperature, and electron density are determined from radiation trapping analysis and particle balance equations of selected argon 1s and 2p levels. Pulsed liquid injection is found to induce a sudden drop in metastable density and electron temperature, and an increase in electron density. These results are attributed to the lower ionization thresholds of the injected molecular species compared to the one of argon. In addition, upstream liquid temperature is found to affect the transitory kinetics for non-volatile solvents more than volatile ones, in accordance with a previously reported flash boiling atomization mechanism.

Deep bed filtration and formation damage by particles with distributed properties

Current models for deep bed filtration describe particles with uniform properties. Yet, the sizes, densities, and mineral composition of particles vary significantly in the same injection well. The aim of this work is to provide an effective mathematical model for water injection of particles with distributed properties and formation damage prediction. We average the set of traditional population balance equations for single-property particles and obtain one upscaled equation. The upscaled equation for particle retention rate contains a non-linear function of suspended concentration, which we call the 'suspension function'. We derive analytical solutions for the upscaled equation for linear (coreflood) and radial (well injectivity) flows. Then we treat lab coreflood data to determine the model suspension function and provide a model for well injectivity prediction. The retention profile for the flow of uniform particles has an exponential form. Frequently reported in the literature, hyper-exponential forms have been hypothetically explained by multiple particle properties. The inverse solution allows revealing the individual filtration coefficients for binary mixtures from total breakthrough concentrations during coreflood. Treatment of the data from lab experiments reveals individual filtration coefficients that belong to common intervals. For the first time, deep bed filtration of particles with distributed properties is upscaled and presented using a single equation that reflects the particle property distribution. This equation provides an effective mathematical model for tuning lab coreflood data, determines the model function, and uses it for injectivity decline prediction.

Numerical Investigation of the Two-Component Suspension Filtration in a Porous Medium Taking into Account Changes in the Characteristics of the Porous Medium

The paper explores a mathematical model of the filtration of dual-component suspension within a porous medium characterized by two distinct zones. This model encompasses mass balance equations of suspended particles, kinetic equations of deposition formation for both reversible and irreversible deposition types for each suspension component, and incorporates Darcy’s law. In order to solve the problem, we formulate a numerical algorithm for computer-based experimentation on the basis of the finite difference method. Through the analysis of numerical findings, we establish key features of two component suspension filtration within a porous medium. Furthermore, we examine the effects of model parameters on the transport and deposition of suspended particles in a two-component suspension within porous media. The polydispersity of the suspension and the multi-stage nature of deposition kinetics can induce effects that differ from those typically observed in the transport of onecomponent suspensions with single-stage particle deposition kinetics.

Refined interpretation of electron temperature response to neutral beam injection at DIII-D

Accurate particle and power deposition profiles of neutral beam injection (NBI) are essential to transport studies, and that information is usually acquired through Monte Carlo simulations with a given collisional model. The deposition process of the energetic beam particles leads to the informative electron temperature (Te) evolution trajectory, which can be captured by electron cyclotron emission (ECE) system due to its good spatial and temporal resolution. Previously, some work has been done to interpret the Te responses to the pulsed NBI as a linear heating source with Fourier-based techniques, although that approach fell short when the fast ion slowing-down time becomes significant (∼100 ms). It has been observed in DIII-D that the modulated NBI pulses (10–50 Hz) reduce local core Te values ∼0.1 keV through cold electron dilution in high-Te (>2 keV) plasmas alongside accumulative heating. Here, a novel approach to interpret the Te response to NBI was developed by linearizing and modeling the detailed Te evolution trajectory using coherently averaged ECE data based on the different time scales of the terms in the local power and particle balance equations. The technique does not require absolute calibrations of ECE and is independent of collisional models. The resulting beam deposition profiles show good consistency and reasonable agreement with Monte Carlo calculations based on the atomic data from the Atomic Data and Analysis Structure (ADAS). Local electron density response measured by Thomson scattering (TS) also suggests the same features when the beam pulse is large enough for that diagnostic to resolve. The remaining discrepancies are also discussed.

Performance analysis of photon-enhanced thermionic emission systems mediated by quantum tunneling

Reducing the gap between the electrodes to the nanoscale and utilizing quantum effects are an effective way to enhance the performance of a thermionic energy device. In this work, we establish the model of a photon-enhanced thermionic emission system with a nanoscale vacuum gap, where the electron transport due to electron tunneling and the near-field radiation resulting from photon tunneling are introduced. Analytical expressions for the thermionic emission current, electron tunneling current, and heat flux due to the near-field radiation are provided. By using the energy and particle balance equations, the electron concentration and the temperature of the cathode are determined. The impacts of the voltage, electron affinity, and gap distance on the performance are further analyzed. Results show that the suggested system can achieve high efficiency at the low-temperature cathode. Up to 34.7% of solar-to-electricity efficiency is possible at a cathode temperature of 472.5 K. The proposed model provides a strategy for designing highly efficient thermionic emission devices operating at low temperatures.

Role of impurity and thermal noise on the radiation sources in ITER using DT fuel

Abstract In this paper, the time evolution of bremsstrahlung radiation loss, plasma frequency and electron particles density and the relationship between these parameters and black body radiation are investigated. The model used in this work is based on numerical solution of particle and energy balance equations in ITER with DT fuel. The fusion reaction takes places in a plasma of deuterium and tritium heated to millions of degrees. It is expected that at this temperature, the thermal noise could have a significant effect on plasma behavior. This effect is considered in the solution of equations for the first time in this work. In order to attain a proper set of particle and energy balance equations, an appropriate thermal noise term is considered in the set of coupled differential equations. These equations are solved simultaneously by numerical methods. The results of the calculations for bremsstrahlung radiation loss, plasma frequency, intensity of blackbody radiation, absorption coefficient and quality factor show that in the absence of thermal noise blackbody radiation doesn’t occur but in the presence of thermal noise blackbody radiation occurs in times of 55.7 s and 42.73 s for two cases of considering and ignoring impurity respectively. As it can be seen that with the addition of impurities to the system, bremsstrahlung radiation and intensity of blackbody radiation increase while absorption coefficient and quality factor decrease.

Introduction and verification of FEDM, an open-source FEniCS-based discharge modelling code

This paper introduces the finite element discharge modelling (FEDM) code, which was developed using the open-source computing platform FEniCS (https://fenicsproject.org). Building on FEniCS, the FEDM code utilises the finite element method to solve partial differential equations. It extends FEniCS with features that allow the automated implementation and numerical solution of fully coupled fluid-Poisson models including an arbitrary number of particle balance equations. The code is verified using the method of exact solutions and benchmarking. The physically based examples of a time-of-flight experiment, a positive streamer discharge in atmospheric-pressure air and a low-pressure glow discharge in argon are used as rigorous test cases for the developed modelling code and to illustrate its capabilities. The performance of the code is compared to the commercial software package COMSOL Multiphysics® and a comparable parallel speed-up is obtained. It is shown that the iterative solver implemented by FEDM performs particularly well on high-performance compute clusters.

The role of thermal noise on key parameters governing the improvement of tokamak fusion reactor efficiency using aneutronic fuel

This article examines the time evolution of energy, particle density, synchrotron radiation loss, plasma frequency, electron particle density, and the relationship between these parameters and blackbody radiation in the presence of thermal noise for the tokamak fusion reactor with aneutronic fuel. To achieve this goal, a suitable thermal noise term is added to the set of energy and particle balance equations. These nonlinear point kinetic coupled differential equations are simultaneously solved by numerical methods. The results of our calculations related to fusion power, particle density, fusion energy density, and quality factor in cases: (i) without and (ii) with noise show that in the conditions where thermal noise is considered, the mentioned parameters and quality factor are increased and in fact system performance improves. Also, our numerical results related to synchrotron radiation loss, plasma frequency, electron particle density, and energy intensity in cases (i) and (ii) show that in the conditions when blackbody radiation occurs (3.75 and 7.225 s respectively for (i) and (ii) cases), synchrotron radiation reaches the maximum values of and for (i) and (ii) cases, respectively.

A new global model with two electron groups for weakly ionized argon discharges at low pressure

A new global model is developed for weakly ionized argon discharges at low pressure with a non-Maxwellian electron energy distribution. This model adopts the ‘two-electron group approximation’, as done in reference (Vriens L 1973 J. Appl. Phys. 44 3980; Ligthart F A S and Keijser R A J 1980 J. Appl. Phys. 51 5295; Kudryavtsev A A and Tsendin L D 2002 Tech. Phys. Lett. 28 841), but the effective temperatures for both groups of electrons, the energy boundary between them in energy space and the total volume-averaged electron density are all obtained in a self-consistent manner. By treating low- and high-energy electrons as two different species and involving particle and energy balance equations for both of them, the model-predicted electron distributions agree well with the measured ones under a wide range of discharge conditions. A detailed comparison between the results from the model presented in this work and those from the previous models is also made, along with the discussion of the reasons that cause the differences.

Transmission conductance of a cylindrical tube with wall pumping

The concept of transmission conductance is required to describe phenomena occurring in a tube with wall pumping. This is because the gas flow is not constant along the length of the tube but decreases, which implies that the geometric conductance determined by dimensional parameters of the tube cannot be further used. In one of the previous studies, the transmission conductance of a cylindrical tube was analytically derived using the particle balance equation based on the continuity principle. However, the calculated results well agreed with those of the test particle Monte Carlo (TPMC) simulation only when the sticking coefficient was less than 0.1. This is due to the intrinsic limitation of the analytical method, in which the constant conductance of a tube of unit length with uniform wall pumping was used. In this study, we introduced another analytical method based on the cosine law and finite element analysis using the conductance of the tube of unit length as a function of the position in the tube, considering the beaming effect. Using both methods, the relative difference in transmission conductance with TPMC considerably decreased compared to the previous study when the sticking coefficient was greater than 0.1. The results can be used in the design of a non-evaporable getter coated tube or a cryogenic water pump using an adsorbing wall to determine characteristic dimensions. Furthermore, this can be employed in a windowless vacuum coupling or a distributed pumping system for various light source systems.

Study of alpha particle concentration effects as the ash of deuterium-tritium fusion reaction on ignition criteria

In this study, the ignition criteria in the International Thermonuclear Experimental Reactor (ITER) associated with helium concentration is investigated. The main fuels used in this study are deuterium-tritium. Helium as the ash of deuterium-tritium reaction cannot be avoided. In this work, the effects of alpha particle concentration on plasma operation are investigated. The calculations were performed by using zero-dimensional power and particle balance equations. The results of our calculations show that the temporal evolution of alpha particle concentration becomes constant at 0.14. Then, ignition criteria and saddle points have been studied in n e τ E −T, and −T planes for different alpha particle concentrations.

A Global Model Study of Plasma Chemistry and Propulsion Parameters of a Gridded Ion Thruster Using Argon as Propellant

This work reports on the (zero-dimensional) global model study of argon plasma chemistry for a cylindrical thruster based on inductively coupled plasma (ICP) whose output has a system of two grids polarized with each other with direct current potential. The global model developed is based on particle and energy balance equations, where the latter considers both charged and neutral species. Thus, the model allows the determination of the neutral gas temperature. Finally, this study also investigated the role of excited species in plasma chemistry especially in the ions production and its implications for propulsion parameters, such as thrust. For this, the study was carried out in two different scenarios: (1) one taking into account the metastable species Arr and Arp (multi-step ionization), and (2) the other without these species (single-step ionization). Results indicates a distinct behavior of electron temperature with radiofrequency (RF) power for the investigated cases. On the other hand, the gas temperature is almost the same for investigated power range of up to 900 W. Concern propulsion analysis, a thrust of 40 mN at 450 W was verified for case (1), which represents a remarkable thrust value for electric thrusters.

A study on the influence of external magnetic field on Nitrogen RF discharge using Langmuir probe and OES methods

This paper reports the study of the effects of an externally applied magnetic field (0–300 G), in the mode transition as well as in the radial and axial variation of different plasma parameters such as electron density, temperature, etc, in nitrogen RF discharge with the help of an RF compensated Langmuir probe (LP). Also, Optical Emission Spectroscopy (OES) study is performed in order to have a good understanding of the properties of plasma at different magnetic fields. Data collected from LP shows all three mode transitions (E, H, and W mode) in presence of magnetic fields whereas for no magnetic field only two modes (E and H) are visible. The measured value of electron density by using LP is further verified and compared theoretically using particle and power balance equations. However, the overall density profile attains a higher value for no magnetic field. This rise in overall density at 0 G field is further explained in terms of EEPF plot and OES analysis. The EEPF plot reveals that the number of high energy electrons is reduced with the application of magnetic fields. Also from OES analysis, it is found that the molecular excitations in N2 second positive system [C 3Πu (ν′) → B 3Πg (ν′′)] are increased in the presence of magnetic fields whereas with no magnetic field the ionization peak of N2 first negative system and the molecular dissociation peak at 746.8 nm attains the largest value at a certain power. Plasma density values calculated with the OES method at the different magnetic fields and RF power show a similar trend with respect to the density values obtained from the LP method.

Global model study of plasma parameter variation in helicon plasma source in oxygen discharge

In this work, different plasma parameters, such as electron density, temperature, and negative ion density are measured in both the source and expansion chamber of the Helicon Plasma Source (HeliPS) in a radio frequency (RF) oxygen discharge of 13.56 MHz and are compared with the values obtained numerically using the volume-averaged global model. For this purpose, the global model of RF oxygen discharge suitable for the HeliPS having two chambers (source and expansion) together with the electromagnet in the source chamber and the magnetic cage in the expansion chamber to reduce the radial loss of plasma is developed using the particle and power balance equations. In this model, the radial confinement by the axial field of the electromagnet in the source chamber and the confinement in the expansion chamber by the cusped field of the magnetic cage is incorporated. Studies on the charged particle dynamics by both the model and the experiment show comparable values of the variation of species density and electron temperature with RF power and working pressure as well as with the applied magnetic field of the electromagnet in the source chamber.

The Effect of Excited Species on the Collisional Energy of Argon Inductively Coupled Plasmas: A Global Model Study

Global modeling of inductively coupled plasma (ICP) reactors is a powerful tool to investigate plasma parameters. In this article, the argon ICP global model is revisited to explore the effect of excited species on collisional energy through the study of different approaches to particle and energy balance equations. The collisional energy loss is much more sensitive to modifications in the balance equations than the electron temperature. According to the simulations, the multistep ionization reduces the collisional energy loss in all investigated reaction sets and the inclusion of heavy species reactions has negligible influence. The plasma parameters obtained, such as total energy loss and electron temperature, were compared with experimental results from the literature. The simulated cases that have more excited species and reactions in the energy balance are in better agreement with the experimental measurements.

Global model of miniature electron cyclotron resonance ion source

Miniature electron cyclotron resonance (ECR) ion sources are widely used in compact ion implanters, miniature neutron tubes, and miniature ion thrusters. To understand the mechanism of miniature ECR ion source, a miniature deuterium ion source developed by Peking University is taken as the research object. In this work, a global model based on particle balance equations is developed for studying the hydrogen plasma and the deuterium plasma inside the miniature ECR source. The research results show that both the hydrogen discharge process and the deuterium discharge process of the ion source are strongly dependent on the gas pressure and microwave power. The calculated results show that high power is beneficial to increasing the proportion of H<sup>+</sup>(D<sup>+</sup>) ions, low pressure is helpful in augmenting the ratio of <inline-formula><tex-math id="M3">\begin{document}$ {\text{H}}_2^ + $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20212250_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20212250_M3.png"/></alternatives></inline-formula>(<inline-formula><tex-math id="M4">\begin{document}$ {\text{D}}_2^ + $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20212250_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20212250_M4.png"/></alternatives></inline-formula>) ions, high pressure and low power are beneficial to enhancing the proportion of <inline-formula><tex-math id="M5">\begin{document}$ {\text{H}}_3^ + $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20212250_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20212250_M5.png"/></alternatives></inline-formula>(<inline-formula><tex-math id="M6">\begin{document}$ {\text{D}}_3^ + $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20212250_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20212250_M6.png"/></alternatives></inline-formula>) ions. In addition, there is a large difference in ion proportion between hydrogen discharge and deuterium discharge. Under the same operating parameters, the proportion of D<sup>+</sup> ions is 10%–25% higher than the proportion of H<sup>+</sup> ions since the plasma density of deuterium discharge is higher than that of hydrogen plasma. Therefore, during the operation of miniature source, H<sub>2</sub> gas, instead of D<sub>2</sub> gas, can be used in experiment, and the proportion of D<sup>+</sup> ions under the corresponding operating parameters can be estimated based on the proportion of H<sup>+</sup> ions. Finally, the calculated results show that high microwave power is a prerequisite for achieving the high proportion of H<sup>+</sup> (D<sup>+</sup>) ions. However, owing to the limitation of microwave coupling efficiency, the miniature ECR ion source cannot work when the microwave power is greater than 150 W, so that the H<sup>+</sup> (D<sup>+</sup>) proportion cannot be further increased, thereby limiting its further applications in neutron sources, implanters, etc. Therefore, how to improve the microwave coupling efficiency has become one of the key research contents of the miniature ECR ion source. The global model proposed in this paper is helpful in understanding the physical process of the miniature ECR ion source, but there are also some shortcomings. Firstly, the effect of the secondary electron emission coefficient is not considered in the model, so it is impossible to study the influence of wall materials on ion proportion in detail. Secondly, the dissociation degree depends on the plasma measurements, and the error of plasma measurements in turn affect the accuracy of the model to a certain extent. In addition, only the hydrogen plasma model and deuterium plasma model are established in this work, based on which it is impossible to study the processes of other gas discharge plasmas. In the future, the above factors will be considered and the model will be further improved to establish a complete and self-consistent global model of the miniature ECR ion source.

Development of model predictive control of fluorine density in SF6/O2/Ar etch plasma by oxygen flow rate

A model predictive controller (MPC) that controls the fluorine density to a constant level in the etching process plasma was developed. In SF6/O2/Ar etch plasma, an algorithm was developed to maintain the F radical density at a constant level by controlling the amount of oxygen inflow. In the development of the controller, the gain of control model was designed from the particle balance equation, and the time constants were designed in consideration of the dynamic characteristics of equipment parts. The performance of the controller algorithm is evaluated through a set-point tracking test. It is expected that this will be an element technology of an automated process controller that actively controls disturbances during the etching process.