Kinetic two-dimensional modeling of inductively coupled plasmasbased on a hybrid kinetic approach

In this paper, we present a two-dimensional (2-D) kinetic model for low-pressure inductively coupled discharges. The kinetic treatment of the plasma electrons is based on a hybrid kinetic scheme in which the range of electron energies is divided into two subdomains. In the low energy range the electron distribution function is determined from the traditional nonlocal approximation. In the high energy part the complete spatially dependent Boltzmann equation is solved. The scheme provides computational efficiency and enables inclusion of electron-electron collisions which are important in low-pressure high-density plasmas. The self-consistent scheme is complemented by a 2-D fluid model for the ions and the solution of the complex wave equation for the RF electric field. Results of this model are compared to experimental results. Good agreement in terms of plasma density and potential profiles is observed. In particular, the model is capable of reproducing the transition from on-axis to off-axis peaked density profiles as observed in experiments which underlines the significant improvements compared to models purely based on the traditional nonlocal approximation     

Self-consistent multidimensional electron kinetic analysis forinductively coupled plasma sources

Based upon the kinetic equations coupled with electromagnetic analysis for the recently developed inductively coupled plasma sources (ICPS), a self-consistent electron kinetic model is presented for 2-D (r, z) in a cylindrically symmetric configuration space and 2-D (ν _∂, ν_z) in the velocity space, The EM model is based on the mode analysis, while the kinetic analysis gives the perturbed Maxwellian distribution of electrons by solving the Boltzmann-Vlasov equation. The kinetic analysis shows that the RF energy in an ICPS is extracted by a collisionless dissipation mechanism, once the electron thermovelocity is close to the RF phase velocities determined by the reactor height and mode indexes. In this context, the effect of varying the reactor geometry is reported in terms of the electron energy distribution function. The analytical results are compared to the experimental data of Barnes et al. (see Appl. Phys. Lett., vol.62, no.21, p.2622-4 (1993)), which shows qualitative agreements in many aspects     

A unified gas-kinetic scheme for continuum and rarefied flows

With discretized particle velocity space, a multiscale unified gas-kinetic scheme for entire Knudsen number flows is constructed based on the BGK model. The current scheme couples closely the update of macroscopic conservative variables with the update of microscopic gas distribution function within a time step. In comparison with many existing kinetic schemes for the Boltzmann equation, the current method has no difficulty to get accurate Navier–Stokes (NS) solutions in the continuum flow regime with a time step being much larger than the particle collision time. At the same time, the rarefied flow solution, even in the free molecule limit, can be captured accurately. The unified scheme is an extension of the gas-kinetic BGK-NS scheme from the continuum flow to the rarefied regime with the discretization of particle velocity space. The success of the method is due to the un-splitting treatment of the particle transport and collision in the evaluation of local solution of the gas distribution function. For these methods which use operator splitting technique to solve the transport and collision separately, it is usually required that the time step is less than the particle collision time. This constraint basically makes these methods useless in the continuum flow regime, especially in the high Reynolds number flow simulations. Theoretically, once the physical process of particle transport and collision is modeled statistically by the kinetic Boltzmann equation, the transport and collision become continuous operators in space and time, and their numerical discretization should be done consistently. Due to its multiscale nature of the unified scheme, in the update of macroscopic flow variables, the corresponding heat flux can be modified according to any realistic Prandtl number. Subsequently, this modification effects the equilibrium state in the next time level and the update of microscopic distribution function. Therefore, instead of modifying the collision term of the BGK model, such as ES-BGK and BGK–Shakhov, the unified scheme can achieve the same goal on the numerical level directly. Many numerical tests will be used to validate the unified method.     

Kinetic modeling of positive ions in a low-pressure RF discharge

The evolution of the ion-velocity distribution function in a planar RF discharge is computed by numerically solving the Boltzmann equation in phase space. The electric field in this equation is computed with regard to the ion density, assuming Maxwellian electrons with a given uniform temperature. The collision term in the Boltzmann equation contains creation of ions by electron-impact ionization of the background gas and the effect of charge-exchange collisions. Examples are given of the behavior of discharges in argon at RF frequencies of 50 kHz, 300 kHz, and 15 MHz at a very low pressure and at a pressure of approximately 40 Pa. A good agreement is found with published experimental observations of the time-dependent behavior of the electric field profile in the RF sheath     

Two-dimensional, self-consistent, three-moment simulation of RFglow discharge

A two-dimensional self-consistent nonequilibrium fluid model is used to simulate radio frequency (RF) glow discharges to evaluate the quantitative effects of the radial and axial flow dynamics inside a cylindrically symmetric parallel-plate geometry. This model is based on the three moments of the Boltzmann equation and on Poisson's equation. Radial/axial flow dynamics of plasma in low-pressure parallel plate RF glow discharges are investigated. Instead of uniform profiles along the radial direction, which are assumed in one-dimensional models, nonplate profiles are obtained from the two-dimensional simulations. Ionization rate and three moment distributions of plasma density, average velocity, and mean energy are presented in a two-dimensional configuration. The maximum ionization rate occurs in the radial sheath region and agrees with experimental results. Variations in ion density distributions at different positions, various gas pressures frequencies, and applied fields are discussed     

Response of the Electron Kinetics on Spatial Disturbances of the Electric Field in Nonisothermal Plasmas

An efficient method for solving the inhomogeneous electron Boltzmann equation for a weakly ionized collision dominated plasma is represented. As a first application this method is used to investigate in a helium plasma the response of the electron velocity distribution function and of the relevant macroscopic quantities to the impact of spatially limited disturbances in the electric field. In addition to the field action elastic and (conservative) inelastic collisions of electrons with gas atoms are taken into account in the kinetic treatment. In this way the spatial relaxation behaviour of the electrons and their establishment into homogeneous plasma states could be studied on a strict kinetic basis. Unexpectedly large relaxation lengths in electron acceleration direction have been found at medium electric fields.     

Inelastic collisional effect on a dilute granular shock layer with a heated wall

The inelastic collisional effect on a shock layer of a dilute granular gas with a heated wall is numerically studied. To investigate the inelastic collisional effect via the gain term in the inelastic Boltzmann equation on the shock layer, an inelastic Bhatnagar-Gross-Krook (BGK) type equation, whose loss term is equivalent to that in the inelastic Boltzmann equation, is formulated on the basis of the kinetic theory of the granular gas. The inelastic BGK-type equation formulated for a hard-sphere particle is generalized to that for an inverse power law (IPL) molecule. Numerical results in a weakly inelastic regime confirm the nonequilirium contribution to the cooling rate, when the collision frequency depends on the particle velocity. The profile of the negative high-velocity tail of the distribution function in the generation regime of the shock wave obtained by the Direct Simulation Monte Carlo method is higher than that obtained by the proposed BGK-type equation when the collision frequency depends on the particle velocity because of the inelastic collisional effect via the gain term in the inelastic Boltzmann equation, which is not included in the proposed BGK-type equation.     

RF electric field penetration and power deposition into nonequilibrium planar-type inductively coupled plasmas

The RF electric field penetration and the power deposition into planar-type inductively coupled plasmas in low-pressure discharges have been studied by means of a self-consistent model which consists of Maxwell equations combined with the kinetic equation of electrons. The Maxwell equations are solved based on the expansion of the Fourier--Bessel series for determining the RF electric field. Numerical results show the influence of a non-Maxwellian electron energy distribution on the RF electric field penetration and the power deposition for different coil currents. Moreover, the two-dimensional spatial profiles of RF electric field and power density are also shown for different numbers of RF coil turns.     

Nonlocal electron kinetics in collisional gas discharge plasmas

Nonlocal phenomena in electron kinetics of collisional gas discharge plasmas, their kinetic treatment by a nonlocal approach, and relevant experimental results are reviewed in this paper. Using the traditional two-term approximation for the electron distribution function, a general method to analyze electron kinetics in nonuniform plasmas in DC and RF fields for atomic gases is presented for the nonlocal case, when the electron energy relaxation length exceeds the characteristic spatial scale of bounded plasmas. The nonlocal method, which is based on the great difference between the electron mean free path for the momentum transfer and the electron energy relaxation length, considerably simplifies the solution of the kinetic equation and, in a number of cases, allows one to obtain analytical and semi-analytical solutions. The main simplification is achieved for trapped electrons by averaging the Boltzmann equation over space and fast electron motion. Numerous examples of spatial nonlocality are considered in the positive column and near the electrodes of DC glow discharges, in spatial relaxation of the electron distribution and in striations, and in capacitively and inductively coupled low-pressure RF discharges. The modeling of fast beam-like electrons is based on a continuous-energy-loss approximation with the assumption of forward scattering. Simple analytic expressions for the fast electron spectrum are obtained in cathode regions of DC discharges with planar and hollow cathodes     

On the three-term approximation of the solution of the Boltzmann equation for weakly ionized gases

In this paper we discuss an improvement of the current technique of solution of the Boltzmann equation for weakly ionized gases in an electric field. The method is based on the usual expansion of the velocity distribution in spherical harmonics, but three terms of the expansion are retained instead of two. In the light of the results obtained for a particular interaction law between charged and neutral particles, a procedure is established which is consistent in the orders of approximation. This procedure requires an improvement of the degree of accuracy commonly used for the terms deriving from the Boltzmann collision operator. For this reason, the general expression of the isotropic collision operator correct to second order in the ion-neutral mass ratio is calculated. Finally, a new steady-state solution of the Boltzmann equation is obtained which is valid both for electrons and light ions in heavy gases.     

Radial Distributions of Dusty Plasma Parameters in a DC Glow Discharge

A non‐stationary non‐local kinetic model for radial distributions of dusty plasma parameters based on the solution of Boltzmann equation for electron energy distribution function is presented. Electrons and ions production in ionizing collisions and their recombination on dust particle surface were taken into account. The drift‐diffusion approximation for ions was used. To obtain the self‐consistent radial distribution of electric potential the Poisson equation was used. It is shown that at high dust particle density the recombination of electrons and ions can exceed their production in ionization collisions in the region of dusty cloud. In this case the non‐monotonous radial distribution of the electric field is formed, the radial electric field becomes reversed and the radial electron and ion fluxes change their direction toward the center of the tube (© 2012 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling electrokinetic flows in microchannels using coupled lattice Boltzmann methods

We present a numerical framework to solve the dynamic model for electrokinetic flows in microchannels using coupled lattice Boltzmann methods. The governing equation for each transport process is solved by a lattice Boltzmann model and the entire process is simulated through an iteration procedure. After validation, the present method is used to study the applicability of the Poisson–Boltzmann model for electrokinetic flows in microchannels. Our results show that for homogeneously charged long channels, the Poisson–Boltzmann model is applicable for a wide range of electric double layer thickness. For the electric potential distribution, the Poisson–Boltzmann model can provide good predictions until the electric double layers fully overlap, meaning that the thickness of the double layer equals the channel width. For the electroosmotic velocity, the Poisson–Boltzmann model is valid even when the thickness of the double layer is 10 times of the channel width. For heterogeneously charged microchannels, a higher zeta potential and an enhanced velocity field may cause the Poisson–Boltzmann model to fail to provide accurate predictions. The ionic diffusion coefficients have little effect on the steady flows for either homogeneously or heterogeneously charged channels. However the ionic valence of solvent has remarkable influences on both the electric potential distribution and the flow velocity even in homogeneously charged microchannels. Both theoretical analyses and numerical results indicate that the valence and the concentration of the counter-ions dominate the Debye length, the electrical potential distribution, and the ions transport. The present results may improve the understanding of the electrokinetic transport characteristics in microchannels.     

Temporal relaxation of plasma electrons acted upon by directcurrent electric and magnetic fields

On the basis of the time-dependent electron Boltzmann equation the temporal relaxation of the electrons in the presence of electric and magnetic fields in weakly ionized, collision dominated plasmas has been studied. The relaxation process is treated by using a strict time-dependent two-term approximation of the velocity distribution function expansion in spherical harmonics. A new technique for solving the time-dependent electron kinetic equation in this two-term approximation for arbitrary angles between the electric and magnetic fields has been developed and the main aspects of the efficient solution method are presented. Using this new approach and starting from steady-state plasmas under the action of time-independent electric fields only, the impact of superimposed DC magnetic fields on the electron relaxation is analyzed with regard to the control of a neon plasma. The investigations reveal an important effect of the magnetic field on the temporal relaxation process. In particular, it has been found that the relaxation time of the electron component with respect to the establishment of steady-state can be enlarged by some orders of magnitude when increasing the magnetic field strength     

Glow discharge in rare-gas and metal vapour mixture

Applying the Boltzmann equation to a He-Cd mixture discharge the electron energy distribution functions, kinetic coefficients and collision frequencies are numerically calculated. Calculations are made for a homogeneous and stationary discharge plasma subjected to an externally applied electric field. The collision processes which have been taken into account are elastic and inelastic collisions of electrons with He and Cd atoms as well as mutual encounters of electrons. In this case the electron energy distribution and all the quantities calculated from it are dependent on the reduced electric field, the ionization degree and the relative cadmium concentration.     

Monte Carlo solution of the Boltzmann equation via a discrete velocity model

A new discrete velocity scheme for solving the Boltzmann equation is described. Directly solving the Boltzmann equation is computationally expensive because, in addition to working in physical space, the nonlinear collision integral must also be evaluated in a velocity space. Collisions between each point in velocity space with all other points in velocity space must be considered in order to compute the collision integral most accurately, but this is expensive. The computational costs in the present method are reduced by randomly sampling a set of collision partners for each point in velocity space analogous to the Direct Simulation Monte Carlo (DSMC) method. The present method has been applied to a traveling 1D shock wave. The jump conditions across the shock wave match the Rankine–Hugoniot jump conditions. The internal shock wave structure was compared to DSMC solutions, and good agreement was found for Mach numbers ranging from 1.2 to 10. Since a coarse velocity discretization is required for efficient calculation, the effects of different velocity grid resolutions are examined. Additionally, the new scheme’s performance is compared to DSMC and it was found that upstream of the shock wave the new scheme performed nearly an order of magnitude faster than DSMC for the same upstream noise. The noise levels are comparable for the same computational effort downstream of the shock wave.     

Convergence of a Distributional Monte Carlo Method for the Boltzmann Equation

Direct Simulation Monte Carlo (DSMC) methods for the Boltzmann equation employ a point measure approximation to the distribution function, as simulated particles may possess only a single velocity. This representation limits the method to converge only weakly to the solution of the Boltzmann equation. Utilizing kernel density estimation we have developed a stochastic Boltzmann solver which possesses strong convergence for bounded and $L^\infty$ solutions of the Boltzmann equation. This is facilitated by distributing the velocity of each simulated particle instead of using the point measure approximation inherent to DSMC. We propose that the development of a distributional method which incorporates distributed velocities in collision selection and modeling should improve convergence and potentially result in a substantial reduction of the variance in comparison to DSMC methods. Toward this end, we also report initial findings of modeling collisions distributionally using the Bhatnagar-Gross-Krook collision operator.     

A hybrid method for hydrodynamic-kinetic flow – Part II – Coupling of hydrodynamic and kinetic models

In this work we present a non stationary domain decomposition algorithm for multiscale hydrodynamic-kinetic problems, in which the Knudsen number may span from equilibrium to highly rarefied regimes. Our approach is characterized by using the full Boltzmann equation for the kinetic regime, the Compressible Euler equations for equilibrium, with a buffer zone in which the BGK-ES equation is used to represent the transition between fully kinetic to equilibrium flows.In this fashion, the Boltzmann solver is used only when the collision integral is non-stiff, and the mean free path is of the same order as the mesh size needed to capture variations in macroscopic quantities. Thus, in principle, the same mesh size and time steps can be used in the whole computation. Moreover, the time step is limited only by convective terms.Since the Boltzmann solver is applied only in wholly kinetic regimes, we use the reduced noise DSMC scheme we have proposed in Part I of the present work. This ensures a smooth exchange of information across the different domains, with a natural way to construct interface numerical fluxes. Several tests comparing our hybrid scheme with full Boltzmann DSMC computations show the good agreement between the two solutions, on a wide range of Knudsen numbers.     

On the solution of the kinetic equation for a heat flux between concentric spheres

The problem of calculating a heat flux in a spherical layer is considered. The results are obtained in terms of the Bhatnagar-Gross-Krook model and the Boltzmann collision integral. A general (independent of the form of the kinetic equation and the solution method) expression for the heat flux as a function of the energy accommodation coefficient is derived. The results are compared with the experiment and the analytical results obtained previously.     

Strict Kinetic Treatment of the Electron Response to Spatial Field Disturbances in Nonequilibrium Plasmas

In generalization of former approaches for the simplified solution of the inhomogeneous electron Boltzmann equation a higher order solution technique has been developed. This technique is based on a multi-term expansion of the electron velocity distribution function and allows a strict study of the electron kinetics in plasmas acted upon by space-dependent electric fields. This solution technique is used to investigate the response of the plasma electrons to spatially limited disturbances of the electric field in weakly ionized plasmas of helium and mercury. By solving the kinetic equation with increasing order of the multi-term expansion the convergent solution of the kinetic problem and thus the strict spatial behaviour of the velocity distribution and of significant macroscopic properties of the electrons has been determined and analysed. Furthermore, the impact of higher order terms of the expansion has been revealed and the falsification of the velocity distribution and of related macroscopic properties has been evaluated when instead of the multi-term solution the simpler two-term solution of the kinetic equation is used.     

Convective scheme solution of the Boltzmann transport equation for nanoscale semiconductor devices

A model for the simulation of the electron energy distribution in nanoscale metal–oxide–semiconductor field-effect transistor (MOSFET) devices, using a kinetic simulation technique, is implemented. The convective scheme (CS), a method of characteristics, is an accurate method of solving the Boltzmann transport equation, a nonlinear integrodifferential equation, for the distribution of electrons in a MOSFET device. The method is used to find probabilities for use in an iterative scheme which iterates to find collision rates in cells. The CS is also a novel approach to 2D semiconductor device simulation. The CS has been extended to handle boundary conditions in 2D as well as to calculation of polygon overlap for polygons of more than three sides. Electron energy distributions in the channel of a MOSFET are presented.