Finite element modeling of a two-fluid RF plasma discharge

Journal of Applied Physics, 2004

We present the development and application of a versatile finite-element method to discretize direct current and radio frequency (rf) induced plasma-sheath dynamics, using multifluid equations. For the former, argon gas is assumed, and the solution is verified by comparison with a theoretical model obtained from the literature. For rf discharges, partially ionized helium gas is considered between two electrodes coated in a dielectric material. The computed solutions for charge densities, the ion velocity and the neutral gas density and crossflow distributions show expected trends. Specifically, ion and electron number densities at the peak discharge current are compared with published numerical results. The derived electric field is utilized with a simple phenomenological model applicable to the transverse velocity in a one-dimensional situation to predict the anticipated hump in the near wall profile. The next step of extending the model, through future work, to two dimensions and for polyphase supply as implemented in realistic configurations is greatly facilitated by the generality of the chosen finite-element method.

IEEE Conference Record - Abstracts. 1996 IEEE International Conference on Plasma Science

For radio-frequency discharges of electronegative gases, one-dimensional equilibrium equations for plasma variables are formulated and the scaling formulae of the plasma variables are derived in terms of the control parameters. The control parameters consist of three parameters: p (pressure), lp (halfsystem length), and P (power) or ne (electron density). The classifications of the operating regions are performed according to the prevailing particle-loss mechanism (recombination-loss-dominated or ion-flux-loss-dominated) and according to the main heating mechanism (ohmic-heating-dominated or stochastic-heating-dominated). The variations of the charged particle densities with pressure and absorbed power are estimated and compared with the results of a particle-in-cell simulation.

Journal of Computational Physics, 1999

This paper describes a numerical method for the solution of a system of plasma fluid equations. The fluid model is similar to those employed in the simulation of high-density, low-pressure plasmas used in semiconductor processing. The governing equations consist of a drift-diffusion model of the electrons, together with an internal energy equation, coupled via Poisson's equation to a system of Euler equations for each ion species augmented with electrostatic force, collisional, and source/sink terms. The time integration of the full system is performed using an operator splitting that conserves space charge and avoids dielectric relaxation timestep restrictions. The integration of the individual ion species and electrons within the time-split advancement is achieved using a second-order Godunov discretization of the hyperbolic terms, modified to account for the significant role of the electric field in the propagation of acoustic waves, combined with a backward Euler discretization of the parabolic terms. Discrete boundary conditions are employed to accommodate the plasma sheath boundary layer on underresolved grids. The algorithm is described for the case of a single Cartesian grid as the first step toward an implementation on a locally refined grid hierarchy in which the method presented here may be applied on each refinement level.

Lecture Notes in Computer Science, 2006

In the frame of the internal project PUMA (Plasma Used to Master Aerodynamics), ONERA is conducting fundamental studies of plasma-flow interactions. In this paper, the ionic wind created by corona discharges is studied in the case of a subsonic flow over a flat plate. The proposed mechanism of the ionic wind proposed is the addition of momentum by collisions between charged and neutral particles. In order to evaluate the effect of plasma on aerodynamics, a kinetic modeling of the discharge is coupled with a Fluid Dynamics code.

Journal of Computational Physics

To date, fluid models of plasma sheaths have consisted of the coupling of the electric field potential equation obtained through Gauss’s law to the charged species conservation equations obtained through the drift–diffusion approximation. When discretized using finite-difference stencils, such a set of equations has been observed to be particularly stiff and to often require more than hundreds of thousands of iterations to reach convergence. A new approach at solving sheaths using a fluid model is here presented that reduces significantly the number of iterations to reach convergence while not sacrificing on the accuracy of the converged solution. The method proposed herein consists of rewriting the sheath governing equations such that the electric field is obtained from Ohm’s law rather than from Gauss’s law. To ensure that Gauss’s law is satisfied, some source terms are added to the ion conservation equation. Several time-accurate and steady-state cases of dielectric sheaths, anode sheaths, and cathode sheaths (including glow and dark discharges) are considered. The proposed method is seen to yield the same converged solution as the conventional approach while exhibiting a reduction in computational effort varying between one-hundred-fold and ten-thousand-fold whenever the plasma includes both quasi-neutral regions and non-neutral sheaths.

2000

Defining and finding the plasma-sheath boundary (also referred to as "plasma edge", "sheath edge" or "sheath entrance") is a problem of general relevance in plasma physics, and particu- larly so in the context of laboratory, space and fusion plasmas. In a most general approach this problem starts from the Poisson equation under the condition that (i) the electron density distri-

Plasma Sources Science and Technology, 2015

This work describes a new 1D hybrid approach for modeling atmospheric pressure discharges featuring complex chemistry. In this approach electrons are described fully kinetically using Particle-In-Cell/Monte-Carlo (PIC/MCC) scheme, whereas the heavy species are modeled within a fluid description. Validity of the popular drift-diffusion approximation is verified against a "full" fluid model accounting for the ion inertia and a fully kinetic PIC/MCC code for ions as well as electrons. The fluid models require knowledge of the momentum exchange frequency and dependence of the ion mobilities on the electric field when the ions are in equilibrium with the latter. To this end an auxiliary Monte-Carlo scheme is constructed. It is demonstrated that the drift-diffusion approximation can overestimate ion transport in simulations of RF-driven discharges with heavy ion species operated in the γ mode at the atmospheric pressure or in all discharge simulations for lower pressures. This can lead to exaggerated plasma densities and incorrect profiles provided by the drift-diffusion models. Therefore, the hybrid code version featuring the full ion fluid model should be favored against the more popular drfit-diffusion model, noting that the suggested numerical scheme for the former model implies only a small additional computational cost.

ANU, 1990

A one-dimensional, electrostatic particle-in-cell code with non-periodic boundary conditions is used to simulate a low pressure capacitive rf plasma created between two planar electrodes. Ion and electron motion is included and ionising collisions by energetic electrons allow a steady state to be reached and maintained. Realistic values of mi/me are used but there is no attempt to model a real gas and, except for ionisation, no binary collision processes are considered. The simulation plasma is generated by driving one boundary with a sinusoidal rf voltage at a frequency of 10 MHz. The effects of scaling on the steady state, the structure and the impedance of the resulting discharge are investigated. Changes resulting from varying the amplitude of the driving voltage are examined and scaling laws for the plasma potential, electron density and power loss obtained. Sheath heating is shown to be the main electron heating process and power balance is checked. The structure of the rf sheath obtained in the simulation is compared to theoretical models of both the current driven and the voltage driven sheath. Disagreement in the maximum sheath width between the simulation and the model is ascribed to neglect of the period of sheath collapse and the use of an idealised electron density profile in the model. Sheath scaling is shown to underlie the variation of electron density and temperature with rf voltage. The electron sheath interaction is examined and found to differ considerably from current theoretical models. In the range of parameters investigated, it is essential to consider the distortion of the electron velocity distribution in the sheath. A beam-like distribution is observed when the sheath velocity changes rapidly near the time of sheath collapse and an instability develops when electrons are accelerated into the plasma as the sheath expands.

A solution procedure for the Navier-Stokes equations coupled with the full Maxwell equations is described. The approach implements a strongly conservative fluid formulation in which the Lorentz force and Ohmic heating terms are recast as convective terms. This removes explicit sources from the Navier-Stokes equations, which have previously introduced severe sti ness and demanded a very delicate numerical treatment. The coupling with the full Maxwell equations enables the displacement current to be incorporated directly. To demonstrate the e ectiveness of this technique, a fully explicit finite volume approximate Riemann solver is used to obtain numerical solutions to the Brio and Wu electromagnetic shock problem. Comparisons with the analytical solution show good agreement, and the implementation requires less than an hour of computational time on a single processor machine. Simulations using large and small conductivities confirm that the formulation captures both wave and di usion limits of the magnetic field.

arXiv (Cornell University), 2020

We present a new multi-fluid, multi-temperature plasma solver with adaptive Cartesian mesh (ACM) based on a full-Newton (non-linear, implicit) scheme for collisional low-temperature plasma. The particle transport is described using the drift-diffusion approximation for electrons and ions coupled to Poisson equation for the electric field. Besides, the electron-energy transport equation is solved to account for electron thermal conductivity, Joule heating, and energy loss of electrons in collisions with neutral species. The rate of electron-induced ionization is a function of electron temperature and could also depend on electron density (important for plasma stratification). The ion and gas temperature are kept constant. The spatial discretization of the transport equations uses a nonisothermal Scharfetter-Gummel scheme from semiconductor physics adapted for multi-dimensional ACM framework. We demonstrate the new solver for simulations of direct current (DC) and radio frequency (RF) discharges. The implicit treatment of the coupled equations allows using large time steps, and the full-Newton method enables fast non-linear convergence at each time step, offering greatly improved efficiency of fluid plasma simulations. We discuss the selection of time steps for solving different plasma problems. The new solver enables us to solve several problems we could not solve before with existing software: two-and three-dimensional structures of the entire DC discharges including cathode and anode regions with electric field reversals, normal cathode spot and anode ring, plasma stratification in diffuse and constricted DC discharges, and standing striations in RF discharges.

IEEE Transactions on Plasma Science, 2013

A time-dependent analytical model for capacitive sheath is developed for describing collisional radio-frequency sheath driven by two sinusoidal sources. In this model, it has been assumed that ion motion and velocity are time varying and sheath is collisional. The time-dependent terms in ion-fluid equations are ignored. Based on the assumption of steplike electron density profile model, analytical expressions for instantaneous sheath motion and sheath potential have been developed. The plasma-sheath motion and sheath potential are compared with a time-dependent model for collisionless and a time-independent model for collisional capacitively coupled plasma.

1993

We present an overview of models of low pressure, non-thermal gas discharges as commonly used in plasma processing. Significant progress has been made in the past decade towards the goal of a self-consistent model of the electrical properties of discharges, whether d.c., r.f. or microwave discharges. These models are based on solutions of the charged particle transport equations coupled with Poisson's equation for the electric field, and provide the space and time distribution of charged particle densities, current densities and electric field or potential. Some of the most sophisticated models also provide the electron and ion velocity distribution functions in the discharge at any point in space or time. It is now possible to describe reasonably accurately the physical properties of a discharge (including the plasma, the electrode regions and the walls) for two-dimensional cylindrical geometries, even for complex electrode configurations involving e.g. a hollow cathode or anode. A survey of the available models is presented here and we illustrate the current state ofthe art by results from one-and two-dimensional models ofd.c., r.f. and transient discharges.

35th AIAA Plasmadynamics and Lasers Conference, 2004

A multi-fluid formulation is implemented to numerically model direct current (DC) and radio frequency (RF) induced plasma wall interaction. The model uses quadratic finite elements to discretize the computational space. Argon gas properties are utilized for collisionless DC glow discharge under fully ionized conditions, and the solution is compared with a theoretical model. The dielectric barrier RF discharge between two insulated electrodes uses partially ionized helium gas. The plasma, together with the charge separated sheath region, is considered collisional. The computed charge densities at the peak discharge current are compared with published numerical results. The solutions predict the ion velocity and the neutral gas density and crossflow velocity distributions. Based on the derived electric field, the transverse gas velocity solution shows the anticipated hump in the near wall profile.

Fluid modeling approaches encounter several shortcomings when used for simulation of capacitively coupled plasma discharges, especially under low-pressure and high-frequency conditions. For example, fluid models fail to accurately predict important features such as the collisionless electron heating and the electron temperature profiles in these discharges. We improve the classical fluid modeling approach to include the full electron momentum equation instead of the approximate drift-diffusion and a nonlocal collisionless electron heat flux terms instead of the Fourier heat flux form. A one-dimensional form of the fluid model is used in our studies. Improved predictions of the collisionless electron heating effect, charged species densities, and sheath electron temperature profiles are shown. Also accurate prediction of discharge impedance characteristics in the low-pressure, high-frequency regime are demonstrated.

Journal of Computational Physics, 2016

High-order Discontinuous Galerkin finite element method Continuous Galerkin finite element method Implicit-explicit (IMEX) scheme Multi-fluid plasma model The multi-fluid plasma model represents electrons, multiple ion species, and multiple neutral species as separate fluids that interact through short-range collisions and longrange electromagnetic fields. The model spans a large range of temporal and spatial scales, which renders the model stiff and presents numerical challenges. To address the large range of timescales, a blended continuous and discontinuous Galerkin method is proposed, where the massive ion and neutral species are modeled using an explicit discontinuous Galerkin method while the electrons and electromagnetic fields are modeled using an implicit continuous Galerkin method. This approach is able to capture large-gradient ion and neutral physics like shock formation, while resolving high-frequency electron dynamics in a computationally efficient manner. The details of the Blended Finite Element Method (BFEM) are presented. The numerical method is benchmarked for accuracy and tested using two-fluid one-dimensional soliton problem and electromagnetic shock problem. The results are compared to conventional finite volume and finite element methods, and demonstrate that the BFEM is particularly effective in resolving physics in stiff problems involving realistic physical parameters, including realistic electron mass and speed of light. The benefit is illustrated by computing a three-fluid plasma application that demonstrates species separation in multi-component plasmas.