Plasma Science and Technology, Vol.14, No.9, Sep. 2012 Multidimensional Numerical Simulation of Glow Discharge by Using the N-BEE-Time Splitting Method Benyssaad KRALOUA, Ali HENNAD Electrical Engineering Faculty, University of Sciences and Technology (USTO-MB), Oran, Algeria Abstract In this work, a new numerical technique is proposed for the resolution of a fluid model based on three Boltzmann moments. The main purpose of this technique is to calculate electric and physical properties in the non-equilibrium electric discharge at low pressure. The transport and Poisson s equations form a self-consistent model. This equation system is written in cylindrical coordinates following the geometric shape of a plasma reactor. Our transport equation system is discretized using the finite volume approach and resolved by the N-BEE explicit scheme coupled to the time splitting method. This programming structure reduces computation time considerably. The 2D code is carried out and tested by comparing our results with those found in literature. Keywords: 2D cylindrical fluid, glow discharge, N-BEE scheme, time splitting method PACS: 51.50.+v, 52.30.Ex, 52.80.Hc DOI: 10.1088/1009-0630/14/9/06 1 Introduction During the past few years, glow discharge been used for a large number of important applications, such as water treatment, sterilization medical material, low energy consumption source light, screen plasma for micro and nanoelectronic industries [1 7]. In this contribution, a recent numerical scheme is proposed for the two-dimensional resolution of the transport equations of the second-order fluid model in non-equilibrium electric discharges at low pressure. Our model is based on the first three moments of Boltzmann s equations for ions and electrons, coupled to Poisson s equation. By considering the geometry of a plasma reactor, the transport equations are written in cylindrical coordinates. In this work, we consider that glow discharge arises between two metal electrodes circular, plane and parallel and radius R. These two electrodes are spaced a distance d and the wall of our reactor is a dielectric (see Fig. 1). The bidimensional transport equations are transformed into one-dimensional equations (driftdiffusion equation), while a time splitting approach [8,9] is adopted. The transport equations are then solved by using the N-BEE explicit scheme [10,11]. The numerical solution of the transport equations depends essentially on the choice of the boundary and initial conditions. On the discharge symmetrical axis, Neumann boundary conditions are mainly used for the electron and ion densities, electric potential and electron temperature. Dirichlet boundary conditions are used at the electrodes and the dielectric walls. Schematic representation of the glow discharge re- Fig.1 actor 2 Fluid model presentation In the present model, the transport equations derived from the first three moments of Boltzmann s equation are written only for electrons and positive ions. The electron and ion densities are described by continuity equations expressed in cylindrical geometry [12 14] : e t + J ez + 1 (r J er ) = S e, (1) + + J +z + 1 (r J +r ) = S +. (2) t In the model, the momentum conservation equation is replaced by the drift-diffusion approximation; hence, the transport equation is represented by two separate
Benyssaad KRALOUA et al.: Multidimensional Numerical Simulation of Glow Discharge terms, i.e., drift and diffusion terms. The electron and ion density fluxes are given by: J ez = µ e n e E z D e e, (3) J er = µ e n e E r D e e r, (4) J +z = +µ + n + E z D + +, (5) J +r = +µ + n + E r D + + r. (6) Here J e and J + are electron and ion density fluxes, respectively. Poisson s Eq. (7) is resolved for the determination of the electric field and the discharge potential. In this equation, V is the electric potential, e is the elementary charge, and ε 0 is the permittivity of free space. 2 V 2 + 1 r V r + 2 V r 2 = e (n + n e ). (7) ε 0 The axial E z and radial E r components of the electric field strength are then found from the partial derivatives of the potential function, and are given by E z = V and E r = V r. (8) In our model, the ionization coefficient of neutral particles is a function of electron temperature. The third moment of Boltzmann s equation (electron energy Eq. (9)) is used to calculate the electron temperature profile). ε t + J εz + 1 (r J εr ) = S ε, (9) J εz = 5 3 µ en ε E z 5 3 D ε e, (10) J εr = 5 3 µ en ε E r 5 3 D ε e r, (11) where n ε is the electron density energy and J ε is the electronic energy density flux. The right-hand sides of Eqs. (1), (2) and (9) include all source terms, S e, S + and S ε are calculated with the corresponding coefficients rates tabulated in Table 1. Table 1. Transport parameters used in the present simulation [4,10,11] Transport parameters Values N 2.83 10 +16 (cm 3 ) µ e 2 10 5 (cm 2 V 1 s 1 ) D + 10 2 (cm 2 s 1 ) µ + 2 10 3 (cm 2 V 1 s 1 ) E i 24 (ev) H i 15.578 (ev) γ 0.046 V -77.4 (V) T c 0.5 (ev) K i 2.5 10 6 (cm 3 s 1 ) d 3.525 (cm) R 5.08 (cm) ( S e = S + = K i Nn e exp E i k b T e ( S ε = ej + E H i K i Nn e exp E i k b T e ), (12) ). (13) The pre-exponential factor, energy of process collisional activation, energy lost by ionization and density of the neutral gas are K i, E i, H i, and N, respectively. Our equation system is solved by means of two electrical and physical approximations: local electric field approximation for ions and local energy approximation for electrons [12,15]. 3 Boundary conditions The use of these boundary conditions for the above mentioned transport equations is essential for the description of our problem. As a matter of fact, we used mainly Neumann boundary conditions on the symmetrical axis: for the gradient particle densities: = 0, for the electric potential: T e r = 0. V r + r = 0 and e r = 0, for the electron temperature: The Dirichelet boundary conditions for the electric potential at electrodes are: V =0 V at the grounded electrode (anode) and V = 77.4 V at the powered electrode (cathode). At the anode; the gradient ion density + = 0, and for the electron density n 0 = 0, for the gradient of the electron temperature is T e = 2e V 5k b. At the dielectric wall; the gradient potential is supposed to be a floating potential. This potential is determined in an auto-coherent approach based on the flow of the charged particles which affects the dielectric according to the following relation: V r = e ε 0 t 0 (J + J e )dt. (14) We assume that the recombination of the charged particles on the wall is immediately completed and the conductivity of dielectric is perfect. Consequently, we suppose that the values of the electron and ion densities on the dielectric wall are nulls (n + =0 and n e =0). The gradient of the electron temperature is supposed to be floating: T e r = 2e V 5k b r. (15) At the cathode, the electron temperature is T c = 0.5 ev, the gradient ion density is + /=0 and the electron density is n e = 0. In this study, the secondary electron emission coefficient, γ 0, is equal to 0.046. Fig. 2 shows the schematic representation of the different boundary conditions used in our model. 803
Plasma Science and Technology, Vol.14, No.9, Sep. 2012 Here the flux J m i+1/2,j is: J m i+1/2,j = nm i,j + 1 2 (1 ν i) ( n m i+1,j n m i,j) ϕ (ri,j, ν i ). (19) The condition of Current Frederic Lewy (CFL) is: ν i = t z ν e (20) The flux limiter ϕ(r i,j, ν i ) of Eq. (17) is: [ ( ϕ(r i,j, ν i ) = Max 0, Min 1, 2r ) ( i,j, Min r i,j, ν i By using the density ratio: 2 1 ν i (21) )]. Fig.2 Schematic representation of the boundary conditions used in our model 4 N-BEE-time splitting method The numerical resolution of the transport equations of our 2D auto-coherent fluid model is laborious. In this section, we use the N-BEE algorithm coupled to the time splitting method for the numerical solution of the transport equations [8 11]. The splitting methods are well-known to solve this kind of multidimensional and multi-physical problems. In effect, the equations written initially in 2D are transformed into one-dimensional equations in each spatial direction. So by using this time-splitting method, it is possible to find n m i,j firstly for time step m with no knowledge of the numerical approximation, then for the new time step (m +1) and finally calculate Ji,j m using the solution that has already been computed for n m+1 i,j. In this section, we present in detail the N-BEE algorithm used in this work for the resolution of the transport equation cited above [16,17]. t + J + 1 (r J) = S. (16) By using the time splitting method, we have seen that the transport equations written initially in 2D can be transformed into 1D equations in each spatial direction: t + J = 0, (17.a) t + 1 (r J) = 0, (17.b) t = S. n m+1 i,j n m i,j t + J m i+1/2,j J m i 1/2, j z + r j+1/2j m i,j+1/2 r j 1/2J m i,j 1/2 r j r (17.c) = S m i,j. (18) r i,j = nm i,j nm i 1,j n m i+1,j. (22) nm i,j The solution of Eq. (17.a) is as follows: n m+1 i,j = n m i,j ν i (Ji+1/2,j m J i 1/2,j m ). (23) From Fig. 3, we have to compute the first step and then get the result of n m+1 i,j that will be considered as a further initial condition for the second step, where the n m+1 i,j solution is obtained. In the third step, we have put Si,j m as a further initial condition and get finally the result of n m+1 i,j. Fig.3 Time splitting method In this figure, the scheme represents the different steps of the transport equations resolution of the fluid model using the time splitting technique. With t, being the time step n m i,j and the discretization in time of n i,j at time m t, z and r are space steps and Ji,j m is the discretization in space following the nodes of the grid named i and j. 5 Results and interpretations A glow discharge reactor has a cylindrical symmetry and therefore the 2D geometry can be reduced to 1D for the fluid model [18]. This reduction enables a comparison with the 1D numerical simulation results obtained by LIN and ADOMAITIS [19,20] in which a good agreement can be noticed. Fig. 4 shows one-dimensional distributions of the steady-state electron and ion densities where the three different regions characterizing glow discharge (cathode 804
Benyssaad KRALOUA et al.: Multidimensional Numerical Simulation of Glow Discharge and anode sheaths and positive column) appear clearly. The anode region is characterized by a relatively important ion density compared to the electron density. The positive column associated with the plasma may be defined by electron and ion densities which are quasi equal characterising the plasma neutrality. Consequently, in the positive column the net space charge is negligible. The density of the species charged in this region has a maximum value. In the cathode region, the electron density is very low compared to the density of the ions. This gradient of density in this region is due to the fact that the electrons move away much more quickly from the cathode region than the ions in the presence of a potential gradient. This causes the depopulation of the electrons in this region. Fig.4 One-dimensional distributions of electron and ion densities at the At the level of the cathode sheath, the values of electron density resulting from our model and also from the model of LIN and ADOMAITIS [19 20] are different from zero, because this zone represents a source of electron production by the secondary emission process. Figs. 5, 6 and 7 represent the one-dimensional distributions of the potential, the electric field and the electron temperature in the steady state of the glow discharge. We notice in Fig. 5 an important gradient of potential in the region of the cathode sheath. This potential gradient is one of the characteristics of the glow discharge. In the positive column and the anode region, the potential is almost constant. The value of the potential in these two regions is nearly equal to the potential applied to the anode. This behaviour is normal, because of the value of the net space particle density that offers toward zero in the positive column. In Fig. 6, the variation of the field in the cathode sheath is linear because of the potential gradient. The field is quasi null in the positive column because the potential is almost unvaried. In the anode region, we can observe an inversion of the electric field because the gradient of plasma density is sufficiently important in this region. Fig. 7 shows the spatio-temporal distribution of the steady-state electron temperature over the whole distance. From this distribution, one can deduce the electron energy. We can easily see that the electron energy in the cathode region is relatively important because of presence of the relatively intense electric field. Figs. 8 and 9 represent contour lines of the electron and ion densities at a steady state where the behaviour of the luminescent discharge is correctly represented by its three regions. The drift velocity of electrons is 100 times larger than that of ions. We note that the contours level lines show the rapid movement of electrons of the cathode region. Fig.7 One-dimensional distributions of electron temperature at the Fig.5 One-dimensional distributions of electric potential at the Fig.6 One-dimensional distributions of electric field at the Fig.8 Bi-dimensional contour of electron density at the 805
Plasma Science and Technology, Vol.14, No.9, Sep. 2012 due to the increase of electronic temperature. We also notice that the variation of the electron temperature in the positive column and the anodic region is almost independent of the value of the applied voltage. This is reasonable because of the nature of the electric field in both two regions. Fig.9 Bi-dimensional contour of ion density at the steady state (color online) The contour lines of the steady-state electric potential are illustrated in Fig. 10. One can see a potential gradient relatively important at the cathode sheath level. In the positive column region where the plasma is formed, the potential is practically constant. Fig.10 Bi-dimensional contour of electric potential at the Fig. 11 represents the contour level lines of the steady-state axial electric field, which confirms the relatively high value of the electric potential at the cathode sheath. In the plasma column and near the anode region, the axial field is almost negligible or null because of the net space charge. We can notice on contour level lines of the steadystate radial electric field in Fig. 12 that this component of the field presents an asymmetric distribution with respect to the symmetry axis. Due to this fact, the resulting radial field is null in the inter-electrodes gap. The main role of this component is to maintain the glow discharge confined around the symmetry axis. Fig. 13 represents contour lines of the steady-state electron temperature. We can see the profile of expected electron temperature because of the high electric field region near the cathode. Generally, the increase of electron density is produced by secondary emission 806 Fig.11 Bi-dimensional contour of axial electric field at the Fig.12 Bi-dimensional contour of transversal radial field at the Fig.13 Bi-dimensional contour of the steady-state electron temperature (color online)
Benyssaad KRALOUA et al.: Multidimensional Numerical Simulation of Glow Discharge 6 Conclusion In this paper, we have developed a bidimensional model of the glow discharge being maintained by a term source of electrons and ions uniformly produced by a secondary electron emission at the cathode. The N-BEE algorithm based on the finite difference method coupled to the time splitting technique is a good approach to resolve the bi-dimensional transport equation of the glow discharge model at low pressure. Our simulation of glow discharge with a symmetry axis are in good agreement with the 1D work of LIN and ADOMATIS [19,20]. References 1 Sugama C, Tochikubo F, Uchida S. 2006, J. Appl. Phys., 45: 8858 2 Arif Malik M. 2010, Plasma Chem. Plasma Process., 30: 21 3 Moisan M, Barbeau J, Crevier M C, et al. 2002, Pure Appl. Chem., 74: 349 4 Lee H W, Park G Y, Seo Y S, et al. 2011, J. Phys. D: Appl. Phys., 44: 053001 5 Arkhipenko V I, Kirillov A A, Callegari T, et al. 2009, IEEE Transactions Plasma Science, 37: 740 6 Duan Y, Huang C, Yu Q. 2005, IEEE Transactions Plasma Science, 33: 328 7 Samanta M, Srivastava A K, Sharma S, et al. 2010, Electrical Discharge as an Inspection Method for Imperfect Plasma Display Cells. J. Phys. Conf. Ser., 208: 012113 8 Bott A. 2010, Improving the time-splitting errors of one-dimensional advection schemes in multidimensional applications. Atmospheric Research, 97: 619; Boris J P, Book D L. 1973, Flux-corrected transport. I. SHASTA, A fluid transport algorithm that works, J. Comput. Phys., 11: 38 9 Tahmouch G. 1995, A numerical multidimensionalmodelization and simulation of the interaction dynamics between charged particles and neutral particles during the transition to a point-plane transitory discharge in gas [Ph.D]. University Louis Pasteur, France 10 Bokanowski O, Zidani H. 2007, Journal of Scientific Computing, 30: 1 11 Bokanowski O, Martin S, Munos R, et al. 2006, Applied Numerical Mathematics, 56: 1147 12 Meyyappan M, Kreskovsky J P. 1990, J. Appl. Phys., 68: 1506 13 Lee C, Graves D B, Lieberman M A, et al. 1994, J. Electrochem, Soc., 141: 1546 14 Lee I, Graves D B, Lieberman M A. 2008, Plasma Sources Sci. Technol., 17: 015018 15 Kraloua B. 2010, Two dimensional modeling of the electrical discharges in low pressure: Second order fluid model [Ph.D]. University Sciences and Technology Mohamed Boudiaf Oran, Algeria 16 Kraloua B, Hennad A. 2008, Transport equations resolution by N-BEE anti-dissipative scheme in 2D model low pressure glow discharge. AIP Conf. Proc., 1047: 240 17 Kraloua B, Hennad A, Yousfi M. 2009, Application anti-dissipative scheme N-BEE for 2D characterization glow discharge. IREE, 4: 509 18 Kraloua B, Hennad A. 2010, Bidimensional Modelling Non-Equilibrium Fluid Model of Glow Discharge at Low Pressure. IREE, 5: 2653 19 Lin Y H, Adomatis R A. 1998, Physics Letters A, 243: 142 20 Lin Y H. 1999, From Detailed Simulation to Model Reduction: Development of Numerical Tools for a Plasma Processing Application [Ph.D]. Faculty graduate school of the University of Maryland, USA (Manuscript received 13 January 2011) (Manuscript accepted 5 January 2012) E-mail address of corresponding anthor Ali HENNAD: ali hennad@yahoo.fr 807