Local Mean Energy Approximation Research Articles

Synchronization of chaos in semiconductor gas discharge model with local mean energy approximation

The delta synchronization is a useful method to analyze appearance of chaotic synchronization in gas discharge systems. In recent studies, the generalized synchronization method has been implemented in various gas discharge systems. However, synchronization is not detected with this conventional method. In our previous study, we introduced the delta synchronization method and applied it to the gas discharge-semiconductor system (GDSS) via the one-dimensional ‘simple’ fluid model approach. In the present study, we implement this method in the more detailed (in terms of plasma chemical reactions and treatment of the electron transport) fluid model, namely the ‘extended’ fluid model or ‘local mean energy approximation’ model. The description of the GDSS model is given, and a bifurcation diagram demonstrates the system’s transition to the chaotic regime. The unpredictable motion, which proves the existence of Poincaré chaos, and the delta synchronized motion are confirmed by the numerical simulations, and corresponding algorithms are given. The time sequences corresponding to the unpredictability and delta synchronization are presented in tables. For consistency, the absence of generalized synchronization is also shown via the auxiliary system approach. The numerical characteristics indicating the degrees of chaos and synchronization are described and implemented in the analysis. These features are also used to compare our results to the simpler model.

Nanosecond dielectric barrier discharge on a curved surface in atmospheric air: streamer evolution and aerodynamic perturbations

A curved nanosecond-pulsed dielectric barrier discharge actuator is proposed to fit the flexibly complex surface shape of air vehicles. Streamer evolution and aerodynamic perturbations driven by a single-pulse voltage applied on the anode, where the pulse width is 35 ns and the peak voltage is 14 kV, are numerically studied. Continuity equations that consider 15 species and 34 reactions are solved based on the drift–diffusion approximation. The electron temperature is obtained using the local mean energy approximation method. Discharge characteristics during the voltage-rise, plateau and decay stages are discussed, respectively. The streamer is initiated at 2.2 ns. The maximum electric field is progressively located at the head of the streamer, between the electrodes and in the dielectric layer during the three voltage stages, while the maximum value of electron density is settled at the downstream tip of the driven anode. The electron number density at the streamer head increases at the voltage-rise stage, keeps constant during the voltage-plateau stage, decreases at the beginning of the voltage-decay stage and then increases due to the quenching effect of the excited species. Compared with the discharge on a flat surface, the initial discharge propagation velocity is smaller, while it decreases more slowly during the entire applied voltage. The simulated deposited energy matches the analytical solution well. Aerodynamic perturbations are investigated by solving the compressible Navier–Stokes equations. The ambient air is rapidly heated to the maximum temperature of 1883 K within 0.04 μs. The temperature rise remains greater than 1000 K for 0.32 μs and greater than 500 K for 21.6 μs. A ‘semiring-like’ compression wave is formed from the discharge region; the propagation speed decreases as it propagates away from the wall and then converges to 345 m s−1 at 13 μs. High-speed flows are rapidly induced at the beginning, and two vortexes are formed successively due to the interaction between the flow induced by the actuator and the backflow induced by the compression wave.

An uncoupled implementation of the local mean energy plasma model

The study of cold plasma represents a very active field of applied physics with technical applications ranging from medical treatments to estimation of aging of electrical components due to internal partial discharges and treeing. In particular, the simulation of this category of plasma plays an increasingly important role since more and more complex, and technically relevant, configurations can be represented. Various kinds of models have been considered, one possible classification is relative to the way the electronic energy is computed. In the local electric field approximation a simple algebraic relationship is used which directly links the electric field strength to the electron energy. On the contrary, in the local mean energy approximation a proper differential equation is solved. In most cases this equation is coupled with a conservation equation which predicts the electron concentration. We will tackle this latter case and we will introduce a formulation capable of decoupling the electron density equation from the electron energy one. We will study the properties of the new formulation and we will build a proper numerical scheme capable of preserving, at a discrete level, these properties. Moreover, we will also discuss the existence of the discrete solution and test the performances of the scheme both in simple test cases, where an exact solution is known, and in a technically relevant configuration such as the formation of a treeing structure.

Modeling the thermalization of electrons in conditions relevant to atmospheric pressure He-O2 nanosecond pulsed discharges

The electron thermalization process is significant in nanosecond pulsed discharges due to the applied voltage pulse's short duration and rapid rise and fall times. In this contribution, a comparison was made between two approaches to modeling the electron kinetics of electron thermalization in atmospheric pressure helium plasma with an oxygen admixture. Modeling based on the direct solution of the local time-dependent electron Boltzmann equation was compared with modeling based on the commonly used but less general local mean energy approximation. For modeling based on the local time-dependent electron Boltzmann equation, a temporary faster decay in the population of electrons in the high energy tail and a slower decay in the population of intermediate energy electrons were observed while the electron swarm cooled from an average energy of above 8 eV, without an electric field present. During that period, the electron impact reaction rate coefficients of helium direct ionization and electronic excitation decreased by more than three orders of magnitude as compared to the modeling based on the local mean energy approximation. Global modeling of the evolution of plasma species densities in response to an electric field typical of atmospheric pressure pulsed discharges was performed with the two approaches to electron kinetics. Differences in the species densities were observed between the two approaches, with a 100% increase in the maximum density of electrons found with the modeling based on the local mean energy approximation.

Modeling of fast ionization waves in pure nitrogen at moderate pressure

The fast ionization wave (FIW) discharges in pure nitrogen in a capillary tube at 27 mbar, initiated by positive polarity, high-voltage nanosecond pulses, are numerically studied by coupled two-dimensional plasma fluid modeling. The 2D fluid code based on local mean energy approximation is validated and used, an extended three-exponential Helmholtz photoionization model is proposed for pure nitrogen. The development and structure of the nitrogen FIW is analyzed, the electric field and current are calculated compared with experimental measurements. The evolution of radial distribution of electrons and N2(C 3Π u ) during the FIW development stage and in the afterglow are analyzed, the radial profiles of electron density show a ‘hollow’ structure in the FIW development stage, the temporal-spatial evolution of N2(C 3Π u ) is dominated by the competition between the pooling reaction of and the quenching by electrons. The role of photoionization on the nitrogen FIW radial morphology is discussed, the equivalent background electron density of photoionization in nitrogen discharge is suggested to be (4–6) × 1013 m−3. Spatial distribution of specific energy deposition (SED), fast gas heating (FGH) energy and temperature rise are obtained, heating efficiency varies with electric field E/N and SED and tends to be 10% at high SED. The dominating reactions responsible for FGH and their fractional contribution in space and time are analyzed, in the near axis region, pooling reactions of and N2(B 3Π g ) contribute up to 80% FGH energy, electron impact dissociation of molecular nitrogen contributes about 10%, while e- recombination and quenching of N(2 D) atoms by N2 molecules contribute rest.

Development of a Massively Parallelized Fluid-Based Plasma Simulation Code With a Finite-Volume Method on an Unstructured Grid

A massively parallelized unstructured-grid plasma simulation code based on a multifluid plasma model has been developed and validated. The code uses a collocated cell-centered finite-volume method and is designed to simulate intermediate low-pressure and atmospheric-pressure (AP) plasma sources with complex machine geometries. One of the novel features of this code is that it is implemented in a highly flexible computational platform named ultrafast massively parallel processing (ultraMPP), which allows straightforward addition and integration of different partial differential equation (PDE) solvers in a self-consistent manner. As to the numerical methods, the Scharfetter–Gummel scheme is adopted to handle the drift-diffusion flux of electrons, whereas the Harten–Lax-van Leer (HLL)-type approximate Riemann solver is used to handle the convection terms of ion momentum equations. For discretization of the diffusion terms in an unstructured grid, the Taylor expansion is used to deal with the effects of nonorthogonality of the cells, and the cell-center gradient is calculated using a least-squares method. The simulation code with a local-field approximation (LFA) was validated for two cases of AP plasmas as well as a case of an argon capacitively coupled plasma generated in a Gaseous Electronic Conference (GEC) reference cell with local mean energy approximation (LMEA) at intermediate low pressure. The simulation results were found to be in good agreement with the previously published experimental and simulation data.

Investigation of surface boundary conditions for continuum modeling of RF plasmas

This work was motivated by a lacking general consensus in the exact form of the boundary conditions (BCs) required on the solid surfaces for the continuum modeling of Radiofrequency (RF) plasmas. Various kinds of number and energy density BCs on solid surfaces were surveyed, and how they interacted with the electric potential BC to affect the plasma was examined in two fundamental RF plasma reactor configurations. A second-order local mean energy approximation with equations governing the electron and ion number densities and the electron energy density was used to model the plasmas. Zero densities and various combinations of drift, diffusion, and thermal fluxes were considered to set up BCs. It was shown that the choice of BC can have a significant impact on the sheath and bulk plasma. The thermal and diffusion fluxes to the surface were found to be important. A pure drift BC for dielectric walls failed to produce a sheath.

Plasma fluid modeling of microwave streamers: Approximations and accuracy

Fluid models of microwave streamers at 110 GHz in atmospheric pressure air predict the formation of filamentary plasma patterns that show a good qualitative agreement with experiments. In order to perform more quantitative comparisons with experiments, in this paper, we study the consequences of different types of approximations that are generally used in the fluid models. We consider here the streamer dynamics before gas heating effects become important, i.e., the first few tens of ns after breakdown at atmospheric pressure. The influence on the results of the local effective field approximation vs. the local mean energy approximation is analyzed in detail. Other approximations that are related to the choice and method of calculation of electron transport parameters are also discussed. It is shown that the local effective field approximation is rather good for a large range of conditions of high frequency breakdown at atmospheric pressure in air while the results may be very sensitive to the choice of transport coefficients.

Rarefied gas electro jet (RGEJ) micro-thruster for space propulsion

This article numerically investigates a micro-thruster for small satellites which utilizes plasma actuators to heat and accelerate the flow in a micro-channel with rarefied gas in the slip flow regime. The inlet plenum condition is considered at 1 Torr with flow discharging to near vacuum conditions (<0.05 Torr). The Knudsen numbers at the inlet and exit planes are ~0.01 and ~0.1, respectively. Although several studies have been performed in micro-hallow cathode discharges at constant pressure, to our knowledge, an integrated study of the glow discharge physics and resulting fluid flow of a plasma thruster under these low pressure and low Knudsen number conditions is yet to be reported. Numerical simulations of the charge distribution due to gas ionization processes and the resulting rarefied gas flow are performed using an in-house code. The mass flow rate, thrust, specific impulse, power consumption and the thrust effectiveness of the thruster are predicted based on these results. The ionized gas is modelled using local mean energy approximation. An electrically induced body force and a thermal heating source are calculated based on the space separated charge distribution and the ion Joule heating, respectively. The rarefied gas flow with these electric force and heating source is modelled using density-based compressible flow equations with slip flow boundary conditions. The results show that a significant improvement of specific impulse can be achieved over highly optimized cold gas thrusters using the same propellant.

Unified fluid model analysis and benchmark study for electron transport in gas and liquid analogs

The interaction of plasmas with liquids requires an understanding of charged particle transport in both the gaseous and liquid phases. In this study we present a generalized fluid-equation framework to describe bulk electron transport in both gaseous and non-polar liquid environments under non-hydrodynamic non-equilibrium conditions. The framework includes liquid structural effects through appropriate inclusion of coherent scattering effects and adaption of swarm data to account for the modification to the scattering environment present in such systems. In the limit of low-densities it reduces to the traditional gas-phase fluid-equation model. Using a higher-order fluid model (four moments), it is shown that by applying steady state electron swarm data in both the gaseous and liquid phases, to close the system of equations and evaluate collisional rates, an improvement in macroscopic electron transport results over popular existing assumptions used. The failure of the local mean energy approximation in fluid models to accurately describe complex spatial oscillatory structures in both the gaseous and liquid phases is discussed in terms of the spatial variation of the electron distribution function itself.

Numerical study of low pressure air plasma in an actuated channel

A model for air plasma discharge based on drift-diffusion with local mean energy approximation is described. The model consists of 7 species and 18 reactions. The code is benchmarked with experimental and numerical results for low pressure glow discharge in a cylindrical tube. The code is used to simulate the discharge produced by a wire placed in a rectangular channel with grounded electrodes at the top and bottom walls. The discharge is concentrated near the wire. The actuator acts on the neutral gas through a body force and Joule heating. Around 80%–90% of the electrical power is converted to Joule heating of the neutral gas and the wall. The actuator produces a body force on the order of 0.1 mN/m. The effectiveness of the actuator increases from 100 to 300 V, and plateaus from 300 to 600 V. The results of the study suggest a further exploration of the channel concept.

High-order fluid model for streamer discharges: I. Derivation of model and transport data

Streamer discharges pose basic problems in plasma physics, as they are very transient, far from equilibrium and have high ionization density gradients; they appear in diverse areas of science and technology. This paper focuses on the derivation of a high-order fluid model for streamers. Using momentum transfer theory, the fluid equations are obtained as velocity moments of the Boltzmann equation; they are closed in the local mean energy approximation and coupled to the Poisson equation for the space charge generated electric field. The high-order tensor in the energy flux equation is approximated by the product of two lower order moments to close the system. The average collision frequencies for momentum and energy transfer in elastic and inelastic collisions for electrons in molecular nitrogen are calculated from a multi-term Boltzmann equation solution. We then discuss, in particular, (1) the correct implementation of transport data in streamer models; (2) the accuracy of the two-term approximation for solving Boltzmann's equation in the context of streamer studies; and (3) the evaluation of the mean-energy-dependent collision rates for electrons required as an input in the high-order fluid model. In the second paper in this sequence, we will discuss the solutions of the high-order fluid model for streamers, based on model and input data derived in this paper.

Capacitively coupled radio-frequency discharges in nitrogen at low pressures

This paper uses experiments and modelling to study capacitively coupled radio-frequency (rf) discharges in pure nitrogen, at 13.56 MHz frequency, 0.1–1 mbar pressures and 2–30 W coupled powers. Experiments performed on two similar (not twin) setups, existing in the LATMOS and the GREMI laboratories, include electrical and optical emission spectroscopy (OES) measurements. Electrical measurements give the rf-applied and the direct-current-self-bias voltages, the effective power coupled to the plasma and the average electron density. OES diagnostics measure the intensities of radiative transitions with the nitrogen second-positive and first-negative systems, and with the 811.5 nm atomic line of argon (present as an actinometer). Simulations use a hybrid code that couples a two-dimensional time-dependent fluid module, describing the dynamics of the charged particles (electrons and positive ions and , and a zero-dimensional kinetic module, describing the production and destruction of nitrogen (atomic and molecular) neutral species. The coupling between these modules adopts the local mean energy approximation to define space–time-dependent electron parameters for the fluid module and to work out space–time-averaged rates for the kinetic module. The model gives general good predictions for the self-bias voltage and for the intensities of radiative transitions (both average and spatially resolved), underestimating the electron density by a factor of 3–4.

Simulation of Excimer Ultraviolet (EUV) Emission from a Coaxial Xenon Excimer Ultraviolet Lamp Driven by Distorted Bipolar Square Voltages

Simulation of excimer ultraviolet (EUV) emission from a coaxial xenon excimer ultraviolet lamp driven by distorted bipolar square voltages is presented in this study. A self-consistent radial one-dimensional fluid model, considering local mean energy approximation (LMEA), along with a set of simplified xenon plasma chemistry was employed to simulate the discharge physics and chemistry. Emitted powers of EUV light and deposited powers to the charged species were simulated by varying the values of four key parameters, which include the driving frequency, gas pressure, gap distance and number of dielectric layers. Results show that there are three distinct periods that include pre-discharge, discharge and post-discharge ones. It is found that intensive EUV (172 nm) emission occurs during the early part of the discharge period, which correlates very well in time with the power deposition through electrons. In addition, power deposition through $$ {\text{Xe}}^{ + } $$ and $$ {\text{Xe}}_{2}^{ + } $$ occurs mainly in the discharge period and later part of discharge period, respectively. Surprisingly, the emission efficiency of 172 nm increases slightly with increasing driving frequency of power source, while it increases dramatically with increasing gap distance. In addition, the maximal emission efficiency is found to take place at gas pressure of 600 torr. The emission efficiency of one-dielectric case is found to be better than that of two-dielectric one. The underlying mechanisms in the above observations are discussed in detail in the paper.

Why the local-mean-energy approximation should be used in hydrodynamic plasma descriptions instead of the local-field approximation

The local-mean-energy approximation (LMEA) and the local-field approximation (LFA) are commonly applied to include the electron properties like transport and rate coefficients into a hydrodynamic description of gas discharge plasmas. Both the approaches base on the solution of the stationary spatially homogeneous Boltzmann equation for the electron component, but the consequences of these approaches differ drastically. These consequences of using both the approaches are studied and discussed on a kinetic level and by comparison of results of hydrodynamic investigations of low-pressure glow discharge plasmas. It is found that the LMEA is to be strongly recommended for the application to a hydrodynamic description of dc as well as rf discharge plasmas, while the LFA is conditionally suitable to describe dc glow discharges with rough reaction kinetics only and its application to rf discharge plasmas is inappropriate.

Fluid modelling of the positive column of direct-current glow discharges**Paper originally presented at the European Summer School ‘Low Temperature Plasma Physics: Basics and Applications’ and ‘Master Class: Biotechnical and Medical Applications’ (Bad Honnef, Germany, 26 September–8 October 2004).

This paper presents a tutorial on the fluid modelling of the positive column of direct-current cylindrical glow discharges. The model writes the continuity and momentum-transfer equations for electrons and ions, coupled with Poisson's equation and the electron mean energy transport equations. The model is solved for helium at pressures p ≃ 0.1–10 Torr, gas temperature Tg = 500 K, discharge currents Idc = 5–200 mA, and tube radius R = 1 cm. The electron transport parameters and rate coefficients are calculated using the local mean energy approximation in articulation with a two-term Boltzmann solver. Model equations are integrated from the discharge axis to the wall, passing through the space-charge sheath and monitoring the charge separation within it. The results obtained allow a systematic characterization of discharge energy deposition and space-charge sheath formation, as a function of pressure and axial current. A general good agreement is found between model results and experimental measurements available in the literature.

Nonequilibrium positive column revisited

The paper presents a fluid modeling of the nonequilibrium, dc-positive column of cylindrical gas discharges, by using a robust numerical code that describes the transport of electrons and positive ions, over a broad domain of pressures p/spl sime/100 mtorr-10 torr and discharge currents I/sub dc//spl sime/5-200 mA. The model writes the continuity and momentum transfer equations for electrons and ions, coupled with Poisson's equation and the electron-mean energy transport equations, and adopts adequate boundary conditions for the particle fluxes at the wall. The electron transport parameters and rate coefficients are calculated using the local mean energy approximation in articulation with a two-term Boltzmann solver. For coherence, the electron transport equations and boundary conditions are deduced in the same two-term approximation framework, which induces some specific features when writing the mean energy transport equations. The model equations are integrated from the discharge axis to the wall, passing through the space-charge sheath and monitoring the charge separation within it. An eigenvalue formulation is adopted to find a univocal solution to the particle and energy balance equations. The model is solved for pure helium dc discharges, as an application example. The results obtained allow a systematic study of the discharge energy deposition and sheath formation, leading to a general criterion defining the transition region in pressure between nonlocal and local regimes.

Two-dimensional fluid modelling of charged particle transport in radio-frequency capacitively coupled discharges

This paper reviews the formulation and updates some numerical procedures usually adopted in two-dimensional, time-dependent fluid models to study the transport of charged particles in radio-frequency capacitively coupled discharges. The description of charged particle transport is made by solving the continuity and momentum transfer equations for electrons and ions, coupled with Poisson's equation and the electron mean energy transport equations. Inertia terms are considered in the ion momentum transfer equations, by generalizing the earlier definition of effective electric field. The electron mean energy equations are written using specific energy transport parameters, deduced from integration over the electron energy distribution function (EEDF). The model adopts the local mean energy approximation, i.e. it computes the electron transport parameters as a function of the electron mean energy, using either a homogeneous, two-term Boltzmann equation solver or a Maxwellian EEDF. More appropriate boundary conditions for the electron and ion fluxes are used successfully. The model is solved for a GEC Cell reactor type (with 6.4 cm radius and 3.2 cm interelectrode distance) operating at frequency 13.56 MHz, pressures between 10 mTorr and 10 Torr and applied voltages from 100 to 500 V, in electropositive (helium) and electronegative (silane–hydrogen) gases or gas mixtures. The ion kinetics in silane and hydrogen is updated with respect to previous works, by further considering SiH2+, H+ and H3+ ions. In general, simulation results for some typical electrical parameters are closer to experimental measurements available than calculations reported in previous works.