Modelling of non-uniform DC driven glow discharge in argon gas

Transcription

1 Physics Letters A 367 (2007) Modelling of non-uniform DC driven glow discharge in argon gas I.R. Rafatov,1, D. Akbar, S. Bilikmen Physics Deartment, Middle East Technical University, TR Ankara, Turkey Received 18 November 2006; received in revised form 18 January 2007; acceted 19 February 2007 Available online 23 February 2007 Communicated by F. Porcelli Abstract Physical roerties of non-uniform DC-driven glow discharge in argon at ressure 1 torr are analyzed numerically. Satially two-dimensional axial-symmetric model is based on the diffusion-drift theory of gas discharge. Results resented comare favorably with the classic theory of glow discharges and exhibit good agreement with the exerimental result. Comarison with the result of satially one-dimensional model is erformed Elsevier B.V. All rights reserved. PACS: s; Cb; b Keywords: Glow discharge; Plasma; Modelling; Positive column 1. Introduction Low-ressure glow discharges are of toical interest for lots of technological alications like lasma light sources, lasma rocessing for surface modification, and lasma chemistry. Therefore, a detailed understanding of the fundamental rocesses of glow discharges is required for the design, characterization, and otimization the discharge arameters in these lasma alications. Such understanding can be gained from modelling the rocesses occurring in the lasma. The modelling aroaches can be classified as fluid methods, kinetic (article) methods and their combinations, called as hybrid methods, which reresent a comromise between the comutationally very efficient fluid models and fully kinetic article models that require very extensive comutations. The glow discharge models are usually develoed in uniform geometry. The aim of this work is to study the effect of non-uniformity of the discharge tube on the discharge roerties. We used a simle fluid model based on the diffusion-drift theory of gas discharge [1 3] and consisted of continuity equations for electrons and ions, as well as Poisson equation for the * Corresonding author. address: rafatov@metu.edu.tr (I.R. Rafatov). 1 Also at American University Central Asia, 205 Abdumomunov Street, Bishkek , Kyrgyzstan. self-consistent electric field. We adoted a local field aroximation: the ionization source term and the article transort coefficients (drift and diffusion) are defined as functions of the local value of the reduced electric field E/ (where E is the magnitude of the field and is the gas ressure). The model is satially two-dimensional (axial-symmetric). Though such a diffusion-drift theory aroach has serious limitations, e.g. in the rediction of absolute values of charge densities (see, e.g. [4]), and in adequate describing of the sheath regions of a sace charge at the immediate vicinity to electrodes [1,5,6], it is still able to redict some general characteristics of the glow discharge [1 3,7 9]. For the testing of the model, we aly conditions and use exerimental results for the non-uniform DC-driven argon glow discharge described in [10]. Schematic diagram of this discharge system is shown in Fig. 1. The system was made of two Pyrex glasses, one of them is uniform (external) and the other is non-uniform (internal). The inner tube consists of two quasi-shere arts with diameters 4.1 cm, and each side of the sheres is connected with cylindrical arts with diameters 3 cm. The distance between the cylindrical electrodes was adjusted to be 30 cm. A floating double robe characteristic has been emloyed for this measurements. For the details we refer to [10]. This Letter is organized as follows. In Section 2, we define the model, erform dimensional analysis, describe hysical arameters and solution aroach. In Section 3, we resent and /$ see front matter 2007 Elsevier B.V. All rights reserved. doi: /j.hysleta

2 I.R. Rafatov et al. / Physics Letters A 367 (2007) Fig. 1. Exerimental setu [10]. discus calculation results. Finally, concluding remarks are given in Section Two-comonent lasma model 2.1. Gas-discharge model The gas-discharge is modelled by continuity equations for two charged secies, namely, electrons and ositive ions with article densities n e and n +, t n e + Γ e = source, t n + + Γ + = source, which are couled to Poisson s equation for the electric field in electrostatic aroximation, E = e ε 0 (n + n e ), E = Φ. Here, Φ is the electric otential, E is the electric field, e is the electron unite charge, ε 0 is the dielectric constant, Γ e and Γ + are the article current densities, which are described by drift and diffusion, where drift velocities are assumed to be linearly deendent on the local electric field with mobilities μ + μ e, Γ e = D e n e n e μ e E, Γ + = D + n + + n + μ + E. Two tyes of ionization rocesses are taken into account: the α rocess of electron imact ionization in the bulk of the gas, and the γ rocess of electron emission by ion imact onto the cathode. In a local field aroximation, the α rocess of ionization and rocess of recombination determine the source terms in the continuity Eqs. (1) and (2), source = Γ e ᾱ ( E ) βn + n e, ᾱ ( E ) = α 0 α ( E /E 0 ), which, due to charge conservation, is identical for both equations [1]. Constant β denotes the recombination coefficient. (1) (2) (3) (4) (5) (6) 2.2. Boundary conditions The system is axially symmetric. In dimensional units, cylindrical coordinates R and Z arametrize the directions arallel and orthogonal to the electrodes. The anode is located at Z = 0, and the cathode is at Z = H (Fig. 2). (Below, in dimensionless units, this corresonds to coordinates (r, z), z = 0 and z = h). Boundary conditions at the anode (Z = 0) describe the absence of ion emission. When diffusion at the anode is neglected, then the ion density vanishes, Γ + (R, 0,t)= 0 n + (R, 0,t)= 0. The boundary condition at the cathode, Z = H, describes γ - rocess of secondary electron emission, Γ e (R,H,t) = γ Γ + (R,H,t) μ e n e (R,H,t)= γμ + n + (R,H,t). Denoting alied DC voltage as U, due to gauge freedom, we set Φ(R,0,t)= 0, 2.3. Dimensional analysis Φ(R,H,t)= U. We introduce the following dimensionless time, coordinates and fields, τ = t, r = R, t 0 R 0 σ(r,τ)= n e(r,t), ρ(r,t)= n +(R,t), n 0 n 0 E(r,t)= E(R,t), φ(r,t)= Φ(R,t), (10) E 0 R 0 E 0 measuring quantities in terms of the intrinsic arameters of the system, R 0 = 1, t 0 = R 0, n 0 = ε 0α 0 E 0. (11) α 0 μ e E 0 e Here the dimensional R is exressed by coordinates (R, Z) and the dimensionless r by coordinates (r, z). The total alied voltage, discharge tube length, and diffusion coefficients (7) (8) (9)

3 116 I.R. Rafatov et al. / Physics Letters A 367 (2007) Fig. 2. Comutational domain. are rescaled as U = U, h= H, R 0 E 0 R 0 D e = D e, D + = D +. μ e E 0 R 0 μ e E 0 R 0 The mobility ratio μ of electrons and ions is (12) μ = μ +. (13) μ e Finally, the recombination coefficient is rescaled as β = βε 0 /(eμ e ) Physical arameters Our choice of arameters was guided by the exeriments [10]. We used shae and dimensions for the discharge tube as in Fig. 1, with tube hight H = 30 cm and tube diameter changing between 3 and 4.2 cm. Pressure is = 1 torr. Alied voltage is U = 400 V. We tested a number of estimations for the ionization coefficient (6) with classical Townsend aroximation of the form ᾱ(e) = A ex[ (B/E) s ] with different choices for arameters A, B, and s from [1,11], and more accurate aroximations for ᾱ from [12,13] and [14]. The better agreement with the exeriment [10] is rovided by the estimation roosed in [12, 13], ᾱ(e/n)/n = ex [ 72/(E/n) ] ex [ 187/(E/n) ] ex [ 700/(E/n) ] (14) ex [ /(E/n) ], where E/n isgivenintdwith1td= V, and ᾱ(e/n)/n is in. The range of E/n, where this formula is valid, is Td [13]. (Actual range of E/n from the numerical examle below is Td.) The gas density is taken as n = cm 3, which corresonds to a ressure = 1 torr at 273 K. For comarison, we resented results of calculation with coefficient ᾱ [14] ᾱ(e/n)/n E/n E/n = (E/n) 2 for E/n > 67.8Td, (E/n) 3 for E/n 67.8Td. (15) The ion mobility is calculated also as a function of the reduced electric field. We tested estimations for μ + from [15], 10 3 ( E/) V, E/ 60 cm torr μ + =, (16) ( ) (E/), E/>60 V 3/2 cm torr, E/ where μ + is in /(sv) and is in torr, and from [14] (which is closed to estimations from [12,16]), μ + = (1 + ( E/n) 1.5 ) 0.33 )n sv (17) where reduced electric field is given in Td. Estimations for the ion diffusion, according to Einstein s relations, can be aroximated by D + = 0.025μ + (18) (assuming ion temerature T + = ev) [14], and as in [15], D + = s. (19) The recombination coefficient is taken as β = cm 3 s 1 [1,15]. The electron yields of cathode materials are the least known data of gas discharge hysics. At the same time the modelling results may sensitively deend on the electron yield data [4,17 19]. Therefore, it is more aroriate in discharge models to calculate γ deendent on the discharge conditions rather than to use any constant value. There is an alternative aroach [18,19], where γ is treated as a fitting arameter to match the calculated electrical characteristics with their exerimental counterarts. For the secondary emission coefficient, we tested a constant aroximation γ = 0.07 [1,15], as well as estimations in the form of functions of the reduced electric field [12,16,20], where γ = [(E/n) c/30 000] (E/n) c / in [16], and (20) γ = 0.01(E/n) c 0.6 (21) in [20], with E/n given in Td and evaluated at the cathode. Calculations showed (and it is also stated in [8,21]) that discharge structure changes slightly at electronic temerature u to 5 ev, and T e = 1 ev can be used as a reasonable aroximation. The electron mobility and diffusion coefficients, assuming T e = 1 ev, are taken in the same way as in [15], μ e = sv, D e = s.

4 I.R. Rafatov et al. / Physics Letters A 367 (2007) Resulting dimensionless system and solution aroach The dimensionless system of equations has the form of τ σ (D e σ + Eσ) = D e σ + σ E α βρσ, τ ρ (D + ρ μeρ) = D e σ + σ E α βρσ, E = ρ σ, E = φ. (22) (23) (24) This system is solved using the finite element comutational ackage COMSOL Multihysics, by stationary nonlinear solver Direct (UMFPACK) [22]. Comutational domain is shown in Fig. 2. Boundary conditions described in Section 2.2 take the following form. At the anode, z = 0 (segment AB in Fig. 2), vanishing of the ion density (7) and homogeneous Neumann condition for the electrons, ρ(r,0,τ)= 0, z σ(r,0,τ)= 0. (25) At the cathode, z = h (segment CD in Fig. 2), condition of secondary electron emission (8) and homogeneous Neumann condition for the ion density, σ(r,h,τ)= γμρ(r,h,τ), z ρ(r,h,τ) = 0. (26) At the outer boundaries z = 0 and z = h, the electrodes are on the electric otential φ(r,0,τ) = 0 and φ(r,h,τ) = U, resectively. Finally, we used homogeneous Neumann conditions for the article densities and otential on the symmetry axis (zaxis, segment AD in Fig. 2) and on the wall of the discharge tube (segment BC). Calculations showed that solving of this system with stationary solver is very sensitive to initial estimate for the unknown variables. To rovide a convergence, we started from some aroriate initial conditions, using first a time-deendent solver, and then started the stationary Direct (UMFPACK), when the solution was about its steady state. To resolve stee layers near the electrodes, we used mesh adatation. The numerical convergence was checked by erforming the successive refinement of the sace grid. Iteration rocess was stoed when the relative error was smaller than 10 6 in one-dimensional and 10 4 in two-dimensional case. 3. Results Figs. 3(c, d) and 4 resent axial rofiles of the electric field magnitude and otential, E and Φ, and article number densities, n e and n +. Fig. 3 contains also rofiles of the onedimensional solutions, where the analysis is restricted to the longitudinal (normal to the electrodes) direction, assuming homogeneity in the transversal direction. Comarison of one- and two-dimensional solutions clearly shows the effect of the nonuniformity of the geometry. One-dimensional calculations, erformed under the resent arameter regime but for different discharge tube lengths (from 1 cm and higher u to 30 cm), showed that the cathode layer Fig. 3. Profiles of (a) electron and ion number densities, (b) electron and ion number densities in the cathode vicinity (zoom of the data lotted in anel (a)), (c) electric otential and electric field magnitude, (d) electric field magnitude (zoom). Pressure = 1 torr, alied voltage U = 400 V. 1D and 2D refer to one- and two-dimensional models. Numbering with 1 4 is exlained in the text (Section 3).

5 118 I.R. Rafatov et al. / Physics Letters A 367 (2007) Fig. 4. Axial values of electron and ion number densities. Asterisks indicate exerimental data [10]. Conditions are the same as in Fig. 3. is about several millimeters in width and does not ractically deend on the tube length. Hence, at tube length H = 30 cm, cathode layer occuies extremely small region in relation to the entire discharge dimensions, and solutions have extremely abrut rofiles near the cathode. This comlicates strongly the numerical solution. Fig. 3(b) illustrates extracted from the anel above Fig. 3(a) article densities rofiles in the cathode vicinity. In order to show the effect of different transort and ionization coefficients on the discharge roerties, we erformed one-dimensional calculations, using exressions (14), (16), and (19) for the ionization coefficient, ᾱ, ion mobility, μ +, and ion diffusion, D +, and secondary electron emission coefficient, γ = 0.07 (corresonding solution is numbered with 1 in Fig. 3); the same set of arameters but with ionization coefficient (15) (solution numbered by 2 ); the same set (14), (16), and (19), but not taking recombination (6) into account (solution 3, notice that the article number densities in Fig. 3(a) are noticeably higher in this case); secondary emission coefficient deendent on the reduced electric field (20) (solution 4 ). Solutions with ion mobility coefficient (17), diffusion coefficient (18), secondary yield estimation from [12] differ from the solution 1 very slightly, and they are not resented in Fig. 3. Eq. (21) for γ, in comarison with other estimations, led to the essentially smaller emission coefficient (about 0.023, e.g., value of γ followed from (20) is about 0.05), and seemed to be not aroriate for the resent conditions: calculated article number densities differed from the exerimental [10] ones. Concerning the two-dimensional calculations, they have been erformed using the same estimations for ᾱ, μ +, D +, and γ as for solution 1. Fig. 5 resents shaes of the electric field magnitude (a) and otential (b), the electron (c) and ion number densities (d). Notice that the radial rofiles of the discharge characteristics are ractically constants (as it is in the case of glow discharge in abnormal mode [15]). This tendency, esecially in the resent case when length of the discharge tube in forward direction Fig. 5. Shaes of the electric field magnitude (a), electric otential (b), electron number density (c), and ion number density (d). Conditions are the same as in Fig. 3.

6 I.R. Rafatov et al. / Physics Letters A 367 (2007) (30 cm) is much larger than its lateral dimensions (between 3 and 4.2 cm), is intensified by the aroximation for the boundary conditions (homogeneous Neumann conditions), adoted on the wall of the discharge tube. Numerical redictions for the electric otential and electric field (Figs. 3(c, d), 5(a, b)) demonstrate in general similarity with the revious references [1,15,21,23,24]. The otential sharly dros from about 125 V to 400 V which leads to a drastic difference in ion and electron number densities near the cathode where the negative sheath is formed. At the downstream of this sheath, the otential is steadily rising towards the anode which is ket at 0 V. As for the article number density rofiles (Figs. 4 and 5(c, d)), although the quasi-neutrality of the ositive column does not change, the net sace-charges are not constants along the axis. Electron and ion number densities (as well as electric field magnitude) in ositive column vary along the discharge tube resonding to its non-uniformity. It should be noted that we aly here a simle fluid model and no attemts of arameter fitting have been aimed. However, analysis of the solutions showed that the model can reroduce the essential features of the exeriment: numerical redictions of article number densities are in good qualitative and quantitative agreement with robe measurements (Fig. 4). This can be exlained by the fact that discharge tube is long, consequently the effect of boundaries (which cannot be adequately redicted by the resent model) is weak at the inner regions, where the tube is non-uniform (see Figs. 1 and 2) and where the robe measurements [10] have been erformed. Comarison of the numeric and measured electric fields [10] shows some discreancy. Moreover, actual value of the current in exeriment [10], which rovided voltage dro between the electrodes of U = 400 V, was higher, about 5 ma, while a total current, obtained here by integrating the current flux density j = e(γ + Γ e ) (where Γ + and Γ e are determined by Eqs. (4) and (5)) through the discharge cross section, is 2.3 ma. 4. Concluding remarks We have resented results of calculation of 30 cm long nonuniform DC glow discharge characteristics in an argon at ressure = 1 torr. We alied a two-dimensional two-comonent drift-diffusion lasma model. Results are comared with the ones obtained from the satially one-dimensional model. The effect of different estimations for the ionization source term, the article transort (drift and diffusion) coefficients, and secondary yield is tested. Comuted results exhibit good qualitative agreement with the classic glow discharge theory [1,23] and exerimental measurements [10]. References [1] Y.P. Raizer, Gas Discharge Physics, second corrected rinting, Sringer, Berlin, [2] A.L. Ward, Phys. Rev. 6 (1958) [3] D.B. Graves, K.E.A. Jensen, IEEE Trans. Plasma Sci. PS-14 (2) (1986) 78. [4] Z. Donkó, P. Hartmann, K. Kutasi, Plasma Sources Sci. Technol. 15 (2006) 178. [5] M. Mitchner, C.H. Kruger Jr., Partially Ionized Gases, Wiley Inerscience, New York, [6] A.M. Howatson, An Introduction to Gas Discharge, second ed., Pergamon Press, Oxford, [7] A.L. Ward, J. Al. Phys. 33 (1962) [8] Yu.P. Raizer, S.T. Surzhikov, High Tem. 26 (1988) 304. [9] Yu.P. Raizer, S.T. Surzhikov, High Tem. 28 (1990) 324. [10] D. Akbar, S. Bilikmen, Chin. Phys. Lett. 23 (2006) [11] S. Roy, B.P. Pandey, J. Poggie, D.V. Gaitonde, Phys. Plasmas 10 (2003) [12] A.V. Phels, Z.Lj. Petrović, Plasma Sources Sci. Technol. 8 (1999) R21. [13] D. Marić, M. Radmilović-Radenović, Z.Lj. Petrović, Eur. Phys. J. D35 (2005) 313. [14] G.K. Grubert, D. Loffhagen, D. Uhrlandt, Two-fluid modelling of an abnormal low-ressure glow discharge, Proc. Femlab Conference 2005, 6. [15] A. Fiala, L.C. Pitchford, J.P. Boeuf, Phys. Rev. E 49 (1994) [16] Z.Lj. Petrović, A.V. Phels, Phys. Rev. E 56 (1997) [17] M.M. Nikolić, A.R. Dordević, I. Stefanović, S. Vrhovac, Z.Lj. Petrović, IEEE Trans. Plasma Sci. PS 31 (2003) 717. [18] D. Marić, K. Kutasi, G. Malović, Z. Donkó, Z.Lj. Petrović, Eur. Phys. J. D 21 (2002) 73. [19] D. Marić, P. Hartmann, G. Malović, Z. Donkó, Z.Lj. Petrović, J. Phys. D: Al. Phys. 36 (2003) [20] Z. Donkó, Phys. Rev. E 64 (2001) [21] S.T. Surzhikov, J.S. Shang, J. Comut. Phys. 199 (2004) 437. [22] COMSOL Multihysics 3.2 (2006), COMSOL Inc., htt://www. comsol.com/. [23] A. Von Engel, Ionized Gases, Oxford Univ. Press, Oxford, [24] J.P. Boeuf, L.C. Pitchford, Phys. Rev. E 51 (1995) 1376.