Numerical Simulation of Atmospheric-pressure Non-equilibrium Plasmas: Status and Prospects

Transcription

1 104 International Journal of Plasma Environmental Science & Technology, Vol.7, No.2, JULY 2013 Numerical Simulation of Atmospheric-pressure Non-equilibrium Plasmas: Status and Prospects W. S. Kang 1, M. Hur 1, Y.-H. Song 1, and S. H. Hong 2 1 Department of Plasma Engineering, Korea Institute of Machinery & Materials, Republic of Korea 2 Department of Nuclear Engineering, Seoul National University, Republic of Korea Abstract We present a brief review of recent work on the numerical simulation of atmospheric-pressure nonequilibrium plasma. In order to simulate plasma at atmospheric pressure, a theoretical model and numerical scheme are presented for solving Poisson s equation and a set of continuity equations using plasma properties estimated from an electron energy distribution function. Numerical methods, including determination of the computational domain and adoption of correction methods to minimize numerical divergence, are proposed and discussed, and recent work on simulating corona and dielectric barrier discharge is presented. Finally, we assess the prospects of using numerical simulation to develop new insights into the fundamental mechanisms behind atmospheric-pressure non-equilibrium plasmas for improved practical use. Keywords Atmospheric-pressure non-equilibrium plasmas, simulation, corona discharge, dielectric barrier discharge I. INTRODUCTION Over the past few decades, atmospheric-pressure non-equilibrium plasmas have become a topic of interest in the plasma research field [1,2]. The use of plasmas beyond the limits of low-pressure operation can enable the development of new applications for traditional ozone generators [1], flue gas and greenhouse gas decomposition systems [3], lighting and display panels [4], combustion enhancement devices [5], high-power lasers [6], as well as in the fields of functional surface treatment [7] and biomedicine [8]. For the intensive use of the plasma, the deeper insight into fundamental issues is required to understand physical and chemical phenomena. However, the highly collisional nature of atmospheric-pressure conditions often restricts the use of diagnostic approaches. Plasma simulation can provide a robust, nondiagnostic method for better understanding detailed discharge mechanisms in atmospheric-pressure plasmas. As these have characteristics differing from those of low pressures, simulation of atmospheric-pressure plasmas requires the consideration of theoretical and technical aspects of geometrical effects and the application of numerical schemes that reflect the unique characteristics of atmospheric-pressure conditions and the phenomena induced under such conditions by high electric fields. In this paper, the current state of the art in simulation modeling of atmospheric-pressure non-equilibrium plasmas using numerical methods is reviewed, and challenges and outlooks in this field are prospectively examined. Corresponding author: Woo Seok Kang address: kang@kimm.re.kr Presented at the Joint Symposium on Plasma and Electrostatic Technologies for Environmental Applications, in May 2013 II. ATMOSPHERIC-PRESSURE NON-EQUILIBRIUM PLASMAS Low-pressure plasmas are uniform and stable, exhibiting low breakdown voltages, high-energy electron energies, low neutral gas temperatures, high ion/radical concentrations, and wide operational windows. For these reasons, low-pressure plasmas have been synonymous with non-equilibrium plasmas in which there are differing energy level states between the electrons and ions; i.e., only the electrons are energetic, while the neutral/ion species remain at low or room-temperature [1, 6, 9]. However, use of plasmas operated at low pressure has often been restricted owing to the need for complex and expensive auxiliary vacuum systems. If plasma can be generated in a non-equilibrium state at atmospheric pressure, many of space/process time constraints of arising from the need for a vacuum chamber can be avoided; however, higher pressure operation tends to cause transitioning to an equilibrium state (i.e., a thermal plasma) owing to frequent energy transfer between particles. To achieve atmospheric-pressure plasmas in a nonequilibrium (non-thermal) state, some care must be taken in planning plasma reactor geometry and employing operational methods to sustain low energy neutral/ion species by reducing their energy exchange with energetic electrons. Typical configurations to achieve this include pulsed corona discharges (PCDs) and alternating-current or high-frequency dielectric barrier discharges (DBDs), as shown in Fig. 1. Pulsed corona discharge can be generated in a reactor with wire-plate or wire-cylinder geometry consisting of a sharp or curved high-voltage electrode and a plate or cylindrical ground electrode. In DBD also referred to as silent discharge a plasma is generated within a short discharge gap between electrodes that are covered with thin dielectrics. Pulse operation and the use of dielectric materials are the keys to sustaining the non-equilibrium state at atmospheric

2 Kang et al. 105 Fig. 1. Schematic of corona discharge in (a) pin-plate and (b) wirecylinder structure and dielectric barrier discharge in (c) parallel-plate double barrier and (d) pellet-packed reactor geometry. pressure in PCD and DBD, respectively [3, 10]. To simulate such plasmas, a model should contain criteria for generating highly collisional conditions given reactorelectrode structural considerations. III. THEORETICAL MODEL Depending on operational pressure, plasma simulation models can be classified as kinetic, fluid, or hybrid. Low-pressure plasma simulation requires the use of a kinetic model that solves the Boltzmann equation for particle distribution or a hybrid model that solves for heavy particle dynamics using a fluid model and for electron dynamics using a kinetic model. The fluid model has proven effective in atmospheric-pressure plasma simulation, although some attempts have been made to adopt kinetic models [10, 11]. The computational procedure used in a plasma simulation can be summarized by the following steps as shown in Fig. 2: 1) determine the simulation target and simplify the model; 2) set up a computational dimension/domain and initialize the simulation; 3) solve for the electric field; 4) solve the continuity equation for particle generation and decay, and; 5) increase the time iteration counter, t n in Fig. 2, and repeat steps 3 and 4 above [10]. After a calculation target has been determined, the governing equations and related physical phenomena should be carefully selected in accordance with the objectives of the simulation, and an adequate, modelbased computational domain should be designed in which the electric potential and field within the domain can be calculated by solving Poisson s equation: ( ) ( ) (1) where ε r and ε o are the permittivity of the dielectric material and of free space, respectively; n e, n p, and n n are the electron, positive, and negative ion number densities, (2) Fig. 2. Simulation proceduce. respectively; ϕ is the electric scalar potential; and E is the electric field. The boundary conditions at the plasma-electrode or plasma-barrier surface must be determined before the electric potential calculation. In the case of a bare electrode, it is necessary to assume secondary emission from the metal electrodes. For plasma-barrier interfaces, the positive ion or electron fluxes moving toward the dielectric barriers can be assumed to accumulate on the barrier as surface charges: (3), (4) where σ e and σ i are the accumulated surface charges of electrons and ions, respectively; Γ e and Γ i are the electron and ion fluxes, respectively; and n is the normal vector directed from the plasma to the dielectric barrier. At plasma-dielectric barrier interfaces, the divergence theorem and Stokes theorem can be applied to Gauss and Faraday s laws, respectively, to evaluate the electric potential boundary conditions [12]: ( ) (5) ( ) (6) where ϕ D is the electric potential on the barrier side. On the basis of the drift-diffusion approximation, a set of continuity equations can be developed for various species in which physical and chemical phenomena occurring during discharge are considered as either source or sink terms [10]. After calculating the electric potential and field distribution, the set of continuity equations can be solved based on the source and sink terms, S k : ( ) (7) (8)

3 106 International Journal of Plasma Environmental Science & Technology, Vol.7, No.2, JULY 2013 where n k, v k, and μ k are the number density, drift velocity, and mobility, respectively, of a species k. The term, S k, includes the reactions of electrons and photons with neutral particles, such as electron impact ionization, electron attachment, and photon impact ionization, and interactions of excited molecules and neutral particles [10, 13]. IV. TRANSPORT PROPERTIES AND NUMERICAL METHODS Plasma transport properties depend on the electron energy distribution function (EEDF) that is obtained by solving the Boltzmann equation: ( ) (9) where f is the distribution function of electron energy ε. Because collisions occur between accelerated electrons and other species with high frequency at high pressure, the relaxation time for achieving a steady-state EEDF is shorter than the time needed for discharge development; therefore, the well-known local field approximation (LFA) can be satisfied allowing for the estimation of plasma properties such as drift velocity, mobility, diffusion coefficients, and some electronrelated reaction coefficients as functions of the reduced electric field (E/N) or electron energy (ε) after the EEDF has been calculated: ( ) ( ) (10) ( ) ( ) (11) where k k and σ k are the rate coefficient and cross section of reaction k, respectively; and μ e and m e are electron mobility and mass, respectively. Software to solve the Boltzmann equation ELENDIF [13] or BOLSIG [14] can be used to obtain the EEDF. Photoionization is an important phenomenon as well in the streamer head, and the relevant parameter is usually assumed from experimental results or calculation from ionization radiation emitted by excited species [10, 15, 16]. In order to enable efficient simulation by a model, the computational dimension should be carefully selected. Although one-dimensional (1-D) models can be used in simple approaches, they are too crude to reflect streamer dynamics; instead, a 1.5-D model the so-called disk method, in which a streamer channel is assumed to be a collection of infinitesimally thin disks of finite radius can be used to complement the shortcomings of 1-D modeling [10, 17]. Having a short calculation time like that of an ordinary 1-D model, the 1.5-D model can be easily validated by 2-D or 3-D simulation [3]. A more sophisticated approach involves describing the localized nature of individual streamers by means of axisymmetric 2- or 3-D models, which can be successfully used for simulating pin- or wire-plate corona discharges. However, such multi-dimensional models can consume huge amounts of debugging/computational time and CPU/memory resources. Calculations of electric field and continuity equations are carried out sequentially. Because solving Poisson s equation in order to determine the electric field distribution is the most time-consuming step, the matrix solver for partial differential equations should be carefully chosen in order to allow for efficient calculation. As the simulation progresses, calculated values will often diverge owing to the creation of abrupt electric field changes by streamer propagation. To reduce numerical diffusion, the grid size and calculation time should obey the Courant Friedrichs Lewy condition [18] with a typical simulation time scale of less than tens of picoseconds and a sub-micrometer grid size under atmospheric-pressure conditions. Depending on specific geometry and dimensions, either the finite difference method (FDM), in which the calculation domain is subdivided with a rectangular mesh, or the finite element method (FEM), in which the domain is divided by means of a triangular mesh into an unstructured grid, can be adopted. A flux-corrected transport (FCT) algorithm [19] combining highly stable, low-accuracy low-order parts and highly accurate, low-stability high-order parts by means of stability enhancing correction methods is often applied to enable stable convection calculation [20]. At each step, transport parameters are re-calculated as functions of the changed electric fields; these in turn have an effect on the electric field, and vice versa. The lifetime of a streamer is only a few to tens of nanoseconds, which is too short for an interval to assess either the reactions of excited and ground state species or flow-induced phenomena. To save computational resources, the simulation code is often broken into three modules the plasma, reaction, and flow modules combined via time-slicing methods of differing time scales [21]. V. PREVIOUS WORK In this section, we present highlights of previous research in simulating atmospheric-pressure nonequilibrium plasmas, corona discharges, and dielectric barrier discharges. A. Corona Discharge Kim et al. described streamer characteristics using two-dimensional simulations of pulsed corona discharge in wire-plate or wire-cylinder geometry [3, 22]. In such cases, a build-up of high electric potential near an electrode from external voltage application leads to the initiation of discharge. When a positive streamer propagates from the high-voltage electrode to the ground electrode, a high electric field from space charges and thus most of the field-dependent reaction occurs at the streamer head. As the streamer head moves toward the ground electrode, ions and excited species are generated

4 Kang et al. 107 and remain in the plasma state along the streamer body. While the initial electric potential distribution and shape of the applied voltage affect the course and dynamics of a streamer, its propagation is also affected by neighboring streamers [22]. Three-dimensional simulation has been used to develop more realistic descriptions of the streamer dynamics in a wire-cylindrical geometry [3]. B. Dielectric Barrier Discharge The discharge of DBD can either be classified as streamer or homogeneous mode, and any simulation used should include the physical characteristics of each model. The effects of the DBD barrier arrangement on 1.5- and 2-D single streamer models are described by [3, 10], and [23]. Discharge propagation in single streamer mode shows unique characteristics in terms of the avalanche, streamer, and decay phases. By allowing an accumulation over the barrier of surface charges that reduce the discharge gap voltage, the dielectric material prevents streamer-to-spark transition. The barrier also affects the loss of surface charges that lead to discharge recurrence. Streamers in DBD can exhibit propagation characteristics similar to those of corona discharges when the electric field is high enough. By introducing the decay of accumulated surface charges in 1.5-D simulation, recurring streamers will appear, but only during the rising phase of applied voltage (active period), and that is the unique characteristics of the DBD [3]. If ferroelectric pellets are inserted as a ferroelectric packed discharge (FPD) into the plasma region, the 2-D simulation shows that the pellet geometry will distort the electric field distributions, causing streamer propagation to mimic high electric field-induced corona discharges with high electron density and temperature [10]. To model homogeneous (i.e., glow and Townsend) modes, it is reasonable to use a 1-D model in which the plasma is distributed uniformly along the direction parallel to the electrode position. In such cases, parameters such as applied voltage, frequency, gas mixture, and gas temperature will govern the generation and decay of metastable species that determine the discharge modes [24]. C. Spray-type Dielectric Barrier Discharge Because of the narrow discharge gap in a typical DBD, a large area surface treatment of the inside of the DBD reactor is often impractical. For this reason, the concept of spray-type DBD has been suggested as a means of treating the material outside of the DBD reactor using flow-driven plasmas. For linear treatment, an electrode structure parallel to the gas flow direction is adequate [25]; for large-area treatment, a reactor with planar electrodes covered with dielectric material and with a ground electrode perforated by an array of spray holes is suitable [21]. As the gas flow-rate increases, both types of reactor will eject the radicals generated within toward a substrate at a distance. High advective flow velocity operation of the discharge gas is effective because it allows species to reach the substrate with less loss owing to the shortened transport time [21, 25]. Kim et al. developed a 2-D simulation of a spraytype DBD using nitrogen sulfur hexafluoride (SF6) discharge in which the electric field distortion caused by the spray-hole structure contributed to the generation of initial seed electrons in the region [21]. In this geometry, gas flow affects both the electric field and the species distribution in the discharge, spray-hole, and spray regions, resulting in an altered radical distribution over the substrate. VI. PROSPECTS AND OUTLOOK Although plasma equipments are widely used in the modern electronics manufacturing industry, plasma simulation codes and the concept of computer-aided engineering (CAE) have been still behind comparing to the commercial computational fluid dynamics (CFD) simulation used in the fields of fluid, thermal, structural, and electrical analysis. Facing some challenges of plasma application, the plasma simulation is getting more interests validating the two approaches to understand physical phenomena behind plasmas experiment and simulation. Especially in the atmospheric pressure plasmas, the exploration of emerging topics such as plasma bullets, plasmas in liquid, and plasma medicine requires that there be a role for simulation in the development of new insights into the fundamentals of plasmas. Further simulation of atmospheric-pressure nonequilibrium plasmas should focus on taking on challenging topics based on multi-physics phenomena such as gas flow, heat and mass transfer, and plasmasurface (tissue or liquid) interactions with efficient computational techniques. Although obstacles such as computational instability, lack of information on reactions and related parameters, and the need for large amounts of computational time remain, we expect these will be solved in the near future owing to advances in hardware development (CPU and memory) and to the development of new computational techniques, including parallel programing [3], adaptive meshing [26], and modern numerical schemes and solvers. Finally, and most importantly, we expect that the major breakthroughs in atmospheric-pressure nonequilibrium plasma research will occur as a result of the convergence of theoretical, numerical, and experimental work. ACKNOWLEDGMENT The work was supported in part by the National Research Foundation of Korea (NRF) and in part by the

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