Technical collection One dimensional hybrid Maxwell-Boltzmann model of shearth evolution 27 - Conferences publications P. Sarrailh L. Garrigues G. J. M. Hagelaar J. P. Boeuf G. Sandolache S. Rowe B. Jusselin
Schneider Electric 27 - Conferences publications 28 th ICPIG, July 15, 27, Prague, Czech Republic One Dimensional Hybrid Maxwell-Boltzmann Model of Sheath Evolution Comparison with Simulations P. Sarrailh 1, L. Garrigues 1, G.J.M. Hagelaar 1, J.P. Boeuf 1, G. Sandolache 2, S. Rowe 2, and B. Jusselin 2 1 LAboratoire PLAsma et Conversion de l Energie (LAPLACE), UMR5213 CNRS Université Paul Sabatier, bât. 3R2, 118 route de Narbonne, 3162 Toulouse cedex 4, France 2 Schneider Electric, Centre de Recherche 38 TEC, 385 Grenoble Cedex 9, France The purpose of this paper is to analyse the accuracy of a Hybrid model (particle ions, Maxwell- Boltzmann electrons) in describing the transient sheath and plasma erosion when a voltage ramp is applied across an initially neutral plasma. Results from the hybrid Maxwell-Boltzmann simulation are compared with those of a Particle-In-Cell () simulation for non-collisional as well as collisional plasmas. The plasma density is not supposed to be constant as in most ion implantation simulations, but its decay due to the charged particles collection by the electrodes is taken into account. The results show that the cathode sheath evolution is well described by the hybrid model. Some differences between the two approaches appear on the anode side, and are mainly due to the assumption on the electron energy in the Hybrid Maxwell-Boltzmann model. 1. Introduction The transient evolution of the space charge sheath has been studied in different models reported in the literature in the context of ion implantation [1], post-arc phase of vacuum circuit breakers [2], electro-negative plasma sheaths [3]. In the sheath evolution model, the electron density is given by a Maxwell-Boltzmann (MB) relation assuming that electrons follow instantaneously the electric potential variation. The ion transport is treated with a kinetic description, or is described by fluid equations. The self-consistent electric potential is calculated from Poisson s equation. This type of model is often referred to as model [4]. The timestep scales as the ion CFL timestep. A more accurate study of the problem requires a Particle-In-Cell () model, where a kinetic description of both ions and electrons is coupled with the Poisson equation. The approach implies severe constraints on the time step t ( t<.2/ω pe, ω pe is the electron plasma frequency) and on the grid spacing x ( x~λ D, λ D is the Debye length). In this paper we compare the two approaches in a 1D case, in the context of the post-arc phase of vacuum circuit breakers (VCB). At the end of the arc phase of a VCB, a quasineutral plasma is present between the two-electrodes. After the current goes through zero, the fast increase of the electrode voltage (up to 5 kv in a few tens of µs) in the external circuit leads to the formation of a cathode sheath that expands and expels the plasma from the gap. The study of the post-arc phase is important for a better understanding and control of possible failures of the circuit breaker. The paper is organized as follows. Section 2 presents the and models, section 3 describes the results and the conclusions are presented in section 4. 2. Description of and models The model and the model are briefly described in sub-section 2.1 in 2.2. 2.1. The hybrid-mb model The ion density n i (x, is deduced from a kinetic description by integrating the trajectories of a large number of ions (Cu + ). Each particle is pushed at each time step, according to Newton s law: & x t = e. E x, t m (1), ( ) ( ) i where e is the electric charge, m i (63.5 amu) the ion mass and E(x, is the electric field. Electrons are in equilibrium with the electric potential V(x, and the electron density n e (x, is written as (MB relation) ne ( x, = n ( exp ev ( x, ( kbte ) (2), where kb is the Boltzmann constant, Te is the electron temperature, and n ( is the reference density, taken at the location where the potential is zero. In the post-arc problem, n decreases as the charges are collected by the electrodes. The n ( variation is deduced from current conservation [5]:. J T = (3) where J T is the total current. The electric potential is obtained by solving Poisson s equation:
Schneider Electric 27 - Conferences publications 28 th ICPIG, July 15, 27, Prague, Czech Republic V [ i e, t ] (4) ε ( x, = e n ( x, n ( x ) where ε is the vacuum permittivity. This equation is strongly non-linear because of the Maxwell- Boltzman electron density (2), and must be linearized [5]. The tail of the distribution is depleted due to the loss of electrons with energy larger than the anode potential drop. The model has thus been completed with an electron energy equation, which writes (neglecting ): ( x, nε + Γ ( x, = Γ ( x V( x t e,., ε (5), where n ε is the electron energy density (in ev.m -3 ), Γ e is the electron flux, Γ ε is the electron energy flux, V is the electric potential. Integration of (5) between anode and cathode gives the variations of the electron temperature ( n = 3 ε 2 ne T e). The effect of the variation of the electron temperature on the results is examined in section 3. 2.2. The model Electrons (as well as ions) are now treated with a kinetic description, by integrating the Newton s law q x ( t ) = E( x, t ) (6) ms where q=-e and m s = m e for the electrons, and q=e and m s = m i for the ions. The self consistent electric field is calculated from Poisson s equation (4). A review of techniques can be found in Ref. [6]. The influence of the electron-neutral elastic on the sheath evolution will be considered in section 3 for a pressure of 2 torr. Only elastic are included here, with a null-collision technique [7]. 3. Results and discussion The results obtained with the and models are compared in the following conditions. The geometry is one-dimensional. At t =, a uniform plasma with density equal to n p = 1 18 m -3 is bounded by two electrodes (cathode at x= and anode, at x=d). When elastic are taken into account in the model, the mean free path is d/1. The inter-electrode distance is d = 1 cm. The initial electron temperature T e = 1 ev. Before t =.1 µs, cathode and anode potentials are fixed to zero, ambipolar diffusion is observed, the plasma potential reaches a few V. After this period, a negative voltage with a time-rise of 1 kv/µs is applied to the cathode. 3.1. Sheath thickness and currents at the cathode Figure 1 represents the evolution of the cathode sheath thickness and of the total current density as a function of time, for a ramp voltage and for a decaying plasma density. We can see two phases in the sheath growth. In the first 5 µs, the sheath grows slowly (Fig. 1a) and a high ion current is collected at the cathode (Fig. 1b). This phase can be explained by the classical Child-Langmuir sheath evolution. Theoretically, in the case of a plasma of given density n s and for a constant applied voltage V, the sheath evolution s( is obtained solving this differential equation [8]: 1 2 3 2 ds( 2 9 4 V m e ens + en u dt s B = ε (7), 2 i s( k where u B T e B = is the Bohm velocity and the rhs mi of (7) represents the Child Langmuir flux. When the sheath thickness is small, a large current is extracted from the plasma. During the sheath expansion, the current decreases, as can be observed in Fig. 1 and as predicted by eq. (7). Sheath thickness (cm) Total current density (A.m ) 1..8.6.4.2 with Energy Eq. with. -1 2 4 6 8 1-5 with Energy Eq. -1-15 5 with -3-1 2 4 6 8 1 Figure 1 : sheath thickness and total current density evolution for an initial plasma density n p = 1 18 m -3 and a voltage ramp (dashed line) of 1 kv in 1 µs. Different models are presented : with and without energy equation, with and without electronatom elastic.
Schneider Electric 27 - Conferences publications 28 th ICPIG, July 15, 27, Prague, Czech Republic Equation (7) shows that in the case of constant voltage and plasma density, a steady state can be reached where the current extracted from the plasma is equal to the Child-Langmuir current and the sheath length no longer increases (the Bohm current is exactly equal to the Child-Langmuir curren. In our conditions (Fig. 1) of increasing voltage and decaying plasma density, there is no steady state. The second period (after t=5 µs) of the sheath evolution presents a rapid increase of the sheath thickness and a decrease of the ion current collected at the cathode. This phenomenon is due to the global decrease of the plasma density in the plasma bulk especially because the charged particles are collected by the electrodes. The two different models give very similar results, even though small differences appear between and MB methods for large sheath thickness. The effect of electron-atom in the method, as well as the description of the electron temperature evolution in the model, do not influence significantly the sheath and current density evolution under the conditions of Fig. 1. 3.2. Currents at the anode 2 Current density (A.m ) Current density (A.m ) 1-1 Ion current Displacement current Total current -3-1 Electron current 2 4 6 8 1 2 Ion current Displacement 1 current -1 Total current -3-1 Electron current 2 4 6 8 1 Figure 2 : Comparison of the anode currents with model with energy equation, with electron. The voltage time-rise (dashed) of 1 kv in 1 µs is also reported. The conditions are the same as in Fig. 1. The anode space charge and the displacement currents are shown in Figure 2. Fig. 2 shows that the displacement current is negligible during the sheath expansion and that the anode collects both electrons and ions, the ion current at the anode being non negligible with respect to the cathode current. This is because the anode pre-sheath occupies a larger region of the gap than the cathode presheath. Both models (Fig. 2a and Fig. 2b) give similar results, but the anodic ion current is larger in the Hybrid MB model than in the model. We also note that the fluctuations (statistical or numerical noise) in the Hybrid MB model are larger than in the model. The differences in the Hybrid and model predictions of the ion current at the anode can be better understood by looking at the time evolution of the plasma potential. This is described below. 3.3. Plasma bulk evolution The plasma potential and the electron temperature are plotted in Figs. 3a and 3b respectively. Plasma potential (V) Electron temperature (ev) 1 8 6 4 2 with with Energy Eq. -1 2 4 6 8 1 2. 1.8 1.6 1.4 1.2 1..8.6.4 with.2 with Energy Eq.. -1 2 4 6 8 1 Figure 3 : Plasma potential and electron temperature. The voltage time-rise (dashed) of 1 kv in 1 µs is also reported. The conditions are the same as in Fig. 1. At t=, the applied voltage between the electrodes is zero. Some electrons are quickly
Schneider Electric 27 - Conferences publications 28 th ICPIG, July 15, 27, Prague, Czech Republic collected at the cathode and the cathode and anode sheaths form. The potential drop between the plasma and the electrodes V p is established according to the classical relationship [8] T V = e i p ln m (8). 2 2πme As expected, V p ~ 5 V (for Cu + ions, m i =63.5 amu). The plasma potential then evolves as the voltage applied across the electrodes increases. Figure 3 shows the evolution of the plasma potential and electron temperature (defined as T e =23ε ) as predicted by the different models (Hybrid MB with constant electron temperature, Hybrid MB with electron energy equation, without electron, and with electron ). The Hybrid MB model gives a plasma potential higher than the other models because of the assumption of a constant electron temperature. When the decrease of the electron temperature due to the collection of the higher energy electrons at the anode is taken into account in the Hybrid model (with the fluid equation (5)), the plasma potential is smaller and decreases with time, in better agreement with the models. The best agreement is obtained when we compare the Hybrid MB model with energy equation, to the model with electron (see Figs. 3a and 3b). This is because when are included in the model, the electron velocity distribution function becomes more isotropic (which is consistent with the approximations of the electron energy equation), while in the collisionless case, only the electron temperature in the axial (x) direction decreases due to the collection of higher energy electrons at the anode. 4. Conclusions The plasma erosion by an expanding sheath when a linearly increasing voltage is applied to an initially neutral plasma has been analysed under conditions corresponding to the post-arc phase of a vacuum circuit breaker. The decay of the plasma density due to charged particle collections by the electrodes and to the finiteness of the plasma region has been taken into account. Results from a Hybrid Maxwell-Boltzmann model and a model have been compared. The results show very good agreement between the two models for the time evolution of the sheath length and the cathode current. Some discrepancies between the models exist for the ion (and electron) current collected on the anode side. These discrepancies are larger for lager initial plasma densities (not discussed here), and are due to the different treatments of the electron temperature evolution in the bulk plasma, in the different models. Adding an energy equation to the Hybrid MB model in order to take into account the electron temperature decay due to the loss of electrons with energy higher than the anode sheath voltage leads to an improvement of the agreement between Hybrid and model, especially when electron-neutral are present. Nevertheless, this discrepancy has a limited influence on the plasma erosion, which is controlled by the growth of the cathode sheath. The Hybrid- MB model description is thus sufficient to simulate the erosion of the bounded plasma under the conditions considered in this paper. The gain in computation time between the Hybrid and model is very substantial (typically 2 CPU minutes for the Hybrid model, compared with 6 minutes for the model on a 2 GHz PC, in the conditions considered in this paper). Acknowledgement This work is supported by Schneider Electric. References [1] P. Vitello, C. Cerjan, and D. Braun, Phys. Fluids B 4, 1447 (1992). [2] L. Garrigues, G.J.M. Hagelaar, T.W. Kim, J.P. Boeuf, and S.W. Rowe, 27 th International Conference on Phenomena in Ionized Gases, Eindhoven, the Netherlands (25). [3] P. Chabert and T. E. Sheridan, J. Phys. D: Appl. Phys. 33, 1854 (2). [4] K.L. Cartwright, J.P. Verboncoeur, and C.K. Birdsall, Phys. Plasmas 7, 3252 (2). [5] G.J.M. Hagelaar, How to normalize Maxwell-Boltzmann electrons in transient plasma models, submitted for publication. [6] J.P. Verboncoeur, Plasma Phys. Control. Fusion 47, A231 (25). [7] J.P. Boeuf and E. Marode, J. Phys. D 15, 2169 (1982). [8] M.A. Lieberman and A.J. Lichtenberg, Principles of Plasma Discharges and Materials Processing, John Wiley & Sons, Inc. (1994).