Collisional radiative model of an argon atmospheric capillary surface-wave discharge

 View Online  Export Citation RESEARCH ARTICLE | DECEMBER 01 2004 Collisional radiative model of an argon atmospheric capillary surface-wave discharge  A. Yanguas-Gil; J. Cotrino; A. R. González-Elipe Phys. Plasmas 11, 5497–5506 (2004) https://doi.org/10.1063/1.1804972 Articles You May Be Interested In A simplified hydrokinetic model for a steady‐state microwave discharge sustained by traveling waves at atmospheric pressure conditions J. Appl. Phys. (October 1995) Characteristics of an atmospheric microwave-induced plasma generated in ambient air by an argon discharge excited in an open-ended dielectric discharge tube Phys. Plasmas (September 2002) Kinetic theory of surface waves in plasma jets Phys. Plasmas (February 2002) 01 July 2025 08:53:12 Collisional radiative model of an argon atmospheric capillary surface-wave discharge A. Yanguas-Gil, J. Cotrino, and A. R. González-Elipe Instituto de Ciencias de Materiales and Departamentos de Fisica Atómica, Molecular y Nuclear and Química Inorgánica (CSIC—Universidad de Sevilla), Sevilla, Spain (Received 20 April 2004; accepted 22 June 2004; published online 3 November 2004) The characteristics of a microwave surface-wave sustained plasma operated at atmospheric pressure in an open-ended dielectric tube are investigated theoretically as a first step in the development of a self-consistent model for these discharges. The plasma column is sustained in flowing argon. A surface-wave discharge that fills the whole radial cross section of the discharge tube is considered. With experimental electron temperature profiles [García et al., Spectrochim. Acta, Part B 55, 1733 (2000)] the numerical model is used to test the validity of the different approximations and to study the influence of the different kinetic processes and power loss mechanisms on the discharge. © 2004 American Institute of Physics.[DOI: 10.1063/1.1804972] I. INTRODUCTION Stationary surface wave (SW)discharges are well studied both theoretically and experimentally at low pressure, not only for argon but also for molecular gases such as oxygen or nitrogen1–8 due to the applications of these discharges in the fields of surface modification, thin film deposition, plasma remediation, or cold sterilization. SW discharges, especially those based on high frequency field applicators, are produced by launching an electromagnetic wave along the interface between the plasma and a dielectric vessel. Therefore, the dielectric wall and the plasma are the main propagating media for the wave. The surface wave dissipates energy as they propagate in plasma columns: the energy transfer to the discharge generally occurs through collisional processes. The physical modeling of these discharges at reduced pressures (with diffusion towards the wall being the principal loss mechanism of charged particles)has already reached an advanced stage. However, for pressures ranging from tens of torrs to atmospheric pressure SW discharges are not so well known; not only assumptions usually accepted at low pressure (for instance, neglecting collisional recombination processes in electron kinetics compared to wall losses)may be no longer valid in this pressure range, but also the accuracy of the approximations for atmospheric arc discharges (i.e., local thermodynamical equilibrium)must be verified. Compared with the low pressure discharges, the problem of developing theoretical models for SW plasma at intermediate and atmospheric pressure has been recently tackled: the models available are not only for argon but also for air, with a wide range of different approaches to the question.9–11 In two recent works analytical approaches to the SW dispersion relation have been developed.12,13 Until recently, the lack of experimental data was an important drawback when trying to set a comparison between models and experience. In the past years some experimental works in microwave atmospheric pressure discharges in general,14–17 and particularly on SW discharges,18–20 have been published, so that comparison is nowadays possible for these plasmas in the case of argon. In this work we consider a SW discharge that fills the whole radial cross section of the discharge tube. The production of electrons and excited atoms in a flowing nonequilibrium plasma generated from argon at atmospheric pressure will be determined by using a collisional radiative (CR) model which includes all important collisional and radiative processes. The aim of this work is to provide a further insight into fundamental questions related to the modeling of argon SW plasmas at atmospheric pressure, as a first step towards the development of a self-consistent model for these discharges. For that reason, the CR model described below has not been coupled to an energy balance equation and Maxwell equations as it is usually done,21–23 but instead we have used experimental electron temperature profiles that are available in literature19 in order to test the validity of the different approximations and to study the influence of the different kinetics processes and power loss mechanisms on the discharge. The model results are compared with experimental electron density data.19 The paper is organized as follows. Section II is devoted to study the axial model of the SW discharge, where the argon CR model, continuity equations of different species, electron balance energy equation, and attenuation coefficient of the SW are presented. Section III is dedicated to show the results of the model: populations and power dissipation, and the comparison with experimental data. Finally, in Sec. IV we present our conclusions. II. AXIAL MODEL OF THE SW DISCHARGE The model presented in this work has three different parts: a simple argon CR model has been developed, taking into account not only the electron impact processes but also the inelastic collisions induced by argon atoms. This model is coupled to a fluid model where the balance equations, the motion of the particles, and the energy conservation are considered. Finally, the electromagnetic equations are solved for the SW case, and the attenuation constant is derived through PHYSICS OF PLASMAS VOLUME 11, NUMBER 12 DECEMBER 2004 1070-664X/2004/11(12)/5497/10/$22.00 © 2004 American Institute of Physics5497 01 July 2025 08:53:12 the dispersion relation. As it has already been mentioned, the model proposed is not self-consistently solved, but it relies on the experimental data available for the electron temperature profiles. A. Argon collisional radiative model Although there are in the literature some extensive argon CR models,23–26 we have focused only on the main features of the argon kinetics and hence a four-level CR model for the atomic argon has been used. The atomic and molecular species considered are the neutral (level g)and singly ionized (level c)ground states of atomic argon, the Ar2 +ground state and the 3s2p54s(level s) and 3s2p54p(level p)excited configurations of neutral argon. The energies of the last two levels are calculated as the average of the real states pondered by their degeneracies. The number of species is therefore 6, five corresponding to the atomic and molecular argon, and the free electrons (see Table I). However, the densities of these six species are not independent as the following two constraints have been considered. The condition of plasma quasineutrality implies that ne=n++n关Ar2 +兴, where ne,n+, and n关Ar2 +兴are the electron, Ar+, and Ar2 +densities (i.e., the electron density must be the sum of the ion densities present in the discharge). Besides, the state equation relates the total argon density nAwith the total pressure (neglecting the free electron contribution) through the gas temperature T0: n++2n关Ar2 +兴+np+ns+ng=nA=p kBT0, where ngis the density of the neutral argon ground state and nsand npare those of the 4s(level s)and 4p(level p)excited configurations. Taking into account both relations, only four densities of the six species considered in the model are independent. One usual approach that can be found in literature22,25 is solving the CR model and the electron and heavy particle energy distribution function (EDF)self-consistently, generally by coupling the model to the Boltzmann equation for each species. However, when dealing with alternating electric fields, this approach becomes more complicated as it implies the solving of the time dependent Boltzmann equation. Although it is possible to overcome these difficulties at sufficiently low and high field frequencies,27 in our conditions the electron collision and field oscillation frequencies are of the same order of magnitude. The second approach is that followed by Nowakowska et al.,10 who considered the argon plasma in local thermodynamical equilibrium (LTE)and used the modified Saha equation to relate the electron density with the electron temperature. Although this approximation is valid at high electron temperatures (equal or greater than 1 eV),28 at lower temperatures the situation can be more complicated as the excited states and flow contributions to the electron density may be important. In this work the argon CR model has been used as a link between the experimental local electron temperature and the species populations, instead of assuming a LTE situation. However, we have not dealt with the problem of evaluating the different EDFs; instead, we have followed the approach of Benova et al.21 and have approximated the electron EDF by a Maxwellian distribution. This choice is supported by the fact that the electron–electron collision frequency is of the same order of magnitude than the neutral argon–electron collision and field oscillation frequencies. It has been shown27 that under these conditions the zeroth-order approximation of the electron EDF tends to a Maxwellian form. Also for heavy particles a Maxwellian EDF has been assumed, but with a temperature different to that of electrons due to the inefficiency of the energy transmission mechanisms (mainly elastic collisions)between electrons and heavy particles. In this way we deal with a two-temperature plasma. Hence, the EDFs for all the species present in our model are characterized by only two parameters: the electron temperature Teand the argon gas temperature T0. Three main different reaction paths have been considered in order to study the transitions between levels in this model: these are the electron impact, argon impact, and radiative processes. The electron impact processes included are (i)electron impact ionization and recombination from the ground state, e+Are+e+Ar +, (ii)electron impact ionization and recombination from an excited state, e+Ar* e+e+Ar +, and (iii)electron impact excitation and deexcitation, e+Ar* e+Ar* *. Each electron impact process is characterized by a rate coefficient Kij. For an upward or direct transition 共␧j⬎␧i兲, this rate coefficient can be obtained from its associated cross section ␴ ij and the electron EDF through the expression Kij =冑2 m 冕 ⌬␧ij ⬁ ␴ ij共␧兲␧1/2f共␧兲d␧, with the electron EDF normalization condition 兰f共␧兲d␧=1, and ⌬␧ij being the transition threshold energy and mthe electron mass. The effective level cross sections have been obtained as an average of the “real” levels’ cross sections weighted by the degeneracy of the transition initial level:24 TABLE I. Description of the Ar atomic levels. Level Configuration Degeneracy Energy (eV) g3s2p610 s3s2p54s12 11.65 p3s2p54p36 13.17 c3s2p56 15.76 5498 Phys. Plasmas, Vol. 11, No. 12, December 2004 Yanguas-Gil, Cotrino, and González-Elipe 01 July 2025 08:53:12 ␴ ij =兺 ␣␤ g ␣ ␴ ␣␤ 兺 ␣ g ␣ , covering the ␣ index the real states which belong to the i level and ␤ those of j. Under the Maxwellian assumption, the rate coefficient for the downward or inverse electron processes can be approximated applying the detailed balance principle30 Kji =gi gjexp 冉 ␧ij kBTe 冊 Kij, and Kci =gi gegc 冉 h2 2 ␲ mkBTe 冊 3/2 exp 冉 ␧ic kBTe 冊 Kic, gi,gc,gebeing the degeneracies of the corresponding species and kBthe Bolzmann’s constant. In atom impact processes, the transitions are induced by a neutral argon atom, and although they are usually neglected at low pressures,21 under circumstances where the neutral temperature is high enough their contribution to the plasma kinetics might be important.30 The atom impact processes considered are (i)atom impact ionization and three-body recombination from the ground state: Ar + Ar Ar + e+Ar +, (ii)atom impact ionization and three-body recombination from an excited state: Ar + Ar * Ar + e+Ar +, and (iii)atom impact excitation and deexcitation: Ar + Ar * Ar + Ar * * . The neutral argon impact direct transitions are characterized by the rate coefficient Kij A=冑2 M 冕 ⌬␧ij ⬁ ␴ ij A共␧兲␧1/2fA共␧兲d␧, Mbeing the argon mass, ␴ ij Aits associated cross section, and fA共␧兲the neutral argon energy distribution function. The inverse processes’ rate coefficients are given by30 Kji A=gi gjexp 冉 ␧ij kBT0 冊 Kij A and Kci A=gi gegc 冉 h2 2 ␲ mekTe 冊 3/2 exp 冉 ␧ic kBT0 冊 Kic A. Finally the radiative processes considered are (i)radiative recombination to the ground state, Ar++e→Ar + h ␯ , (ii)radiative recombination to an excited state, Ar++e→Ar * + h ␯ , and (iii)radiative deexcitation, Ar * * →Ar * + h ␯ . The radiative recombination rate coefficients can be written in the form24 Aci =冑2 m 冕 0 ⬁ ␧1/2qci共␧兲f共␧兲d␧, qci being the radiative recombination cross section. As photoionization has been neglected in this model, Aci are the net radiative recombination rate coefficients. The radiative deexcitation is characterized by the Einstein coefficient Aij. However, for the s→gradiative decay self-absorption must be taken into account due to the high population of the neutral argon ground state. To deal with self-absortion we have used Holstein’s theory,31 and so we have defined an effective Einstein coefficient given by Asg ef = ␬ sgAsg. ␬ sg is the so-called Holstein factor or escape factor, which is equal to unity when self-absorption can be neglected, and depends on pressure, gas temperature, and the discharge radius. Three gain and loss processes have been considered for the Ar2 +kinetics. (i)Molecular formation KMF, Ar++2Ar→Ar2 ++Ar, (ii)dissociative recombination KDR, Ar2 ++e→Ar + Ar, and (iii)molecular ion dissociation KMD, Ar2 ++e→Ar++Ar+e. The associative ionization from Ar metastables was found to be a second-order process compared to molecular formation, and therefore it has been neglected in this work. The cross sections used in this work have been taken from the following works. (i)Electron impact ionization: Drawin’s29 analytical formula with the coefficients given by Vlcek.24 (ii)Electron impact excitation and deexcitation: The cross sections for processes from ground state have been calculated using both the cross section and the coefficient given by Vlcek,24 whereas for the transition between the two excited states the theoretical results of Kimura et al.32 and experimental transition probabilities from Wiese et al.33 were used. (iii)Radiative recombination: expressions and coefficients given by Vlcek24 except for the recombination to the p level, where the expression has been fitted to the results of Duzy and Hyman.34 (iv)Atom impact processes: analytical expression proposed by Vlcek.24 (v)Ar2 +kinetics: Bultel et al.26 and references therein. Once defined all the rate coefficients, a collisional radiative balance equation is set for each argon level included in the model: 冏 ⳵ ne ⳵ t冏C=nengKgc +nensKsc +nenpKpc +共nA−ne兲ngKgc A +共nA−ne兲nsKsc A+共nA−ne兲npKpc A−ne 2n+共Kcp +Kcs +Kcg兲−共nA−ne兲nen+共Kcp A+Kcs A+Kcg A兲 −nen+共Acp +Acs +Acg兲, Phys. Plasmas, Vol. 11, No. 12, December 2004 Collisional radiative model of an argon…5499 01 July 2025 08:53:12 冏 ⳵ np ⳵ t冏C=nengKgp +nensKsp +ne 2n+Kcp +共nA−ne兲ngKgp A +共nA−ne兲nsKsp A+共nA−ne兲nen+Kcp A+nen+Acp −npne共Kpc +Kps +Kpg兲−np共nA−ne兲共Kpc A +Kps A+Kpg A兲−npAps, 冏 ⳵ ns ⳵ t冏C=nengKgs +nenpKps +ne 2n+Kcs +共nA−ne兲ngKgs A +共nA−ne兲npKps A+共nA−ne兲nen+Kcs A+nen+Acp +npAps −nsne共Ksc +Ksp +Ksg兲−ns共nA−ne兲 ⫻共Ksc A+Ksp A+Ksg A兲−ns ␬ sgAsg, 冏 ⳵ n关Ar2 +兴 ⳵ t冏C=n+ngKMF −n关Ar2 +兴ne共KMD +KDR兲. Here, n+and ngpopulation densities are obtained using the quasineutrality condition and the state equation. The main approximation of the CR model is admitting that the influence of the argon excited states with excitation energies greater than that of the 4pstates is negligible. Hence, a necessary condition (but not sufficient)for the validity of the CR model is that np⬍ns; that is, the 4plevel population must be smaller than that of the 4sstates. If the collisional radiative balance equations are added to each species continuity equation, we obtain a set of four partial differential equations which can be written in the form ⳵ ni ⳵ t=⵱共v ជ ini兲=冏 ⳵ ni ⳵ t冏c,共1兲 v ជ ibeing the macroscopic velocity associated to the species i. B. Continuity equation The CR model described above has been used to reproduce the axial behavior of an atmospheric argon SW discharge. The axial evolution of the different species is calculated from continuity equation (1). This equation is simplified if it is supposed that the neutral gas flow follows the Poiseuille law,9having thus its velocity a parabolic radial profile: u共r兲=2 ␾ ␲ a2 冉 1−r2 a2 冊 ,共2兲 where ␾ is the gas flow and ais the tube inner radius. The charged particles velocities are then obtained just by adding an additional term due to diffusion, which can be expressed using the Fick law. The quasineutrality condition, which implies the equality of flows, is taken into account by using not the free diffusion coefficients, but the ambipolar diffusion coefficient for both electrons and ions. Considering that the electron mobility is much higher than that of ions, this coefficient can be written as29 Da=kBT0 ␯ ciM 冉 1+Te T0 冊 , ␯ ci being the ion collision frequency. Then the electron and ion velocity can be written as v ជ =u共r兲e ˆz−Da ⵱ne ne.共3兲 When Eq. (3)is substituted in Eq. (1)the electron continuity equation yields ⳵ ni ⳵ t+u共r兲 ⳵ ne ⳵ z−⵱·共Da⵱ne兲=冏 ⳵ ni ⳵ t冏c. In stationary conditions and integrating on both sides any temporal or radial dependence can be eliminated. The partial differential equation thus turns into the following ordinary differential equation: udne dz =冏 ⳵ ne ⳵ t冏c+2 a2 冋 Dar ⳵ ne ⳵ r 册 0 a. The term 兩 ⳵ ne/ ⳵ r兩arepresents the electron losses in the walls, and no estimation of its value has been found for this kind of plasmas. In our model we are going to neglect its contribution compared to volume electron losses. The validity of this assumption will be discussed later, when comparing the model results with the experimental data available. The axial electron velocity component due to ambipolar diffusion also can be neglected when compared with the flow velocity term. All things considered, the equation for the species ican be expressed as udni dz =冏 ⳵ ni ⳵ t冏c.共4兲 These equations form a first-order ordinary differential equation system, but they are not a closed system as the axial evolution of the electron and gas temperature is still needed. C. Power balance When dealing with the energy balance the usual approximation is to admit that the electrons are mainly responsible for the absorption of the electromagnetic power, and that the energy taken by the electrons from the electromagnetic fields is mainly dissipated through collisions with heavy particles and heat conduction. This is the so-called channel model,35 and under this assumption the energy balance yields10 1 r d dr 冉 r␭edTe dr 冊 −QCR e+QEM e=0, where ␭eis the electron thermal conductivity, and QCR eand QEM eare the terms corresponding to collisional radiative losses and electromagnetic gain. The CR loss term can be expressed as the sum of the following terms: (i)electron–neutral elastic collision, Qela =共3m/M兲 ␯ eakB共Te−T0兲,(ii)electron–ion elastic collision, Qeli=共3m/M兲 ␯ eikB共Te−T0兲,(iii)electron excitation losses, Qexc=␧gs共Kgsng−Ksgns兲ne+␧sp共Kspns−Kpsnp兲ne 5500 Phys. Plasmas, Vol. 11, No. 12, December 2004 Yanguas-Gil, Cotrino, and González-Elipe 01 July 2025 08:53:12 +␧gp共Kgpng−Kpgnp兲ne,(iv)electron ionization losses, Qion=␧gc共Kgcng−Kcgne 2兲ne+␧sc共Kscns−Kcsne 2兲ne +␧pc共Kpcnp−Kcpne 2兲ne, and (v)electron radiative recombination losses Qrad=3 2kBTe共Acp+Acs+Acg兲ne 2. The terms Qexc,Qion, and Qrad can be easily obtained from the CR model, whereas ␯ ei and ␯ ea are the electron–ion and electron–neutral argon elastic collision frequencies, which have been calculated following the expressions given in Ref. 28. By radially averaging, and admitting the validity of the adiabatic wall approximation,10,28 the heat conduction term vanishes so that the power taken from the electromagnetic fields by the electron is balanced by the collisions with both neutrals and ions. Defining the power per unit length as L=2 ␲ 冕 0 a rQ共r兲dr, where ais the discharge radius, the energy balance can be finally expressed as LEM e=LCR e.共5兲 However, in this work an additional term will be considered in order to study its importance at high pressure: at atmospheric pressure heavy particle collisions are very effective, and one of the natural consequences of this fact is that argon neutrals and ions mean kinetic energies can be considered almost equal. Channel’s model states that ions inertia is so high that they are not able to gain power from high frequency alternating fields. But ions are still accelerated by the space-charge field that appears as a consequence of the charge separation inside the plasma. The existence of this field implies an acceleration of the ions towards the wall, gaining a kinetic energy that will be eventually released by means of collisions with the heavy particles inside the plasma. This process is supposed to be more important near the plasma wall, in the sheath, where the values of the spacecharge field are higher. Under these assumptions, one can obtain an estimation of the power released due to ion acceleration in the sheath; supposing that all the power gained by the ions is lost by means of elastic collisions, the power per unit length lost by ions, Lions, can be expressed as Lions =2 ␲ aJionseVw, where ais the discharge radius, jions is the ions flow density, and Vwis the plasma potential. As a first approximation jions can be approximated as Jions=n+uB, with n+the radial averaged ions density and uBthe Bohm’s velocity. Also, Vwcan be expressed as Vw=kBTe 2ln 冉 2 ␲ M m 冊 , using these two expressions an estimation of the importance of the contribution of the acceleration term can be obtained. From the previous assumptions, the power balance equation yields LEM =LCR e+Lions.共6兲 D. Electromagnetic power attenuation The attenuation of the electromagnetic power can also be obtained from Maxwell equations; by solving these equations it is possible to deduce an attenuation constant ␣ for the EM fields, which can be related to the EM power through Poynting’s theorem, yielding ␣ =−1 2 1 P dP dz .共7兲 Solving the electromagnetic field equations exactly would imply to know the axial and radial profiles of both electron density and mean kinetic energy, as plasma conductivity and dielectric permittivity depend on these parameters. The problem is simplified if the usual piecewise homogenous approximation for the axial variations of these parameters is accepted. Under these conditions, the SW electromagnetic field equations become separable for the radial and axial variables, and the equation for the radial electric field is given by ⵱t 2Ez+ 冉 ␻ 2 c2␧p+ ␥ 2 冊 Ez+ ␥ 2 ␻ 2 c2␧p+ ␥ 2 ⵱tEz·⵱tRe ␧p Re ␧p=0, where ␥ = ␣ +i ␤ is an eigenvalue that should be obtained from the dispersion equation that appears when fitting the solutions of the EM fields to the boundary conditions. In this equation the radial profiles of neand Teare still needed in order to obtain the plasma complex permittivity ␧p(Re ␧p denotes the real part of ␧p). However, when dealing with capillary SW it has been shown (see Ref. 1, for instance, and references therein)that as long as the condition ␤ aⰆ1is fulfilled, with ␤ the wave number and athe inner radius of the discharge, the dispersion relation is almost independent of the electron density radial profile. This means that it is possible to obtain the dispersion relation without calculating the exact electromagnetic field radial profiles, just treating the argon plasma as if it were radially homogeneous. Therefore, the dispersion relation which has been used is that of a piecewise homogeneous problem with three different media: the inner plasma of relative permittivity ␧1, the dielectric tube of relative permittivity ␧2, and the air that surrounds both, with ␧3=1. The CR model and the dispersion relation are coupled through the plasma permittivity ␧1, which can be expressed as36 ␧1=1− n 1+is, where n= ␻ eff 2/ ␻ 2and s= ␯ eff/ ␻ , with ␻ eff and ␯ eff the effective plasma frequency and collision frequency, and wthe EM fields’ oscillation frequency. By solving the Maxwell equations for the piecewise homogeneous problem one obtains the SW dispersion relation ⌬共ne,Te,T0,p,a,b兲=0 共8兲 formed in this case by a 4⫻4 determinant of firstand second-order complex modified Bessel functions, from Phys. Plasmas, Vol. 11, No. 12, December 2004 Collisional radiative model of an argon…5501 01 July 2025 08:53:12 which the attenuation constant ␣ is its solution. Therefore, in order to obtain the axial profile Eq. (4)is solved self-consistently for the four independent species considered (electrons, argon 4sand 4pexcited states, and Ar2 +). The power absorbed and the attenuation constant are obtained through the power balance equation [Eq. (6)] and the dispersion relation [Eq. (8)]. Consistency of the solution is tested comparing the power profiles obtained by the collisional balance and the resulting from Poynting’s theorem [Eq. (7)]. III. RESULTS The model has been solved for an atmospheric SW discharge sustained in a silica fuse capillary tube with a 0.5 mm inner radius, using a constant gas temperature and a linear profile of the electron temperature according to the experimental results obtained by García et al.19 Axial profiles have been obtained for different gas flows and gas temperatures in order to study the influence of these parameters. The origin of the axial zscale is located at the beginning of the plasma column, just after the surface-wave launcher. The axial model has been solved using a fourth-order Runge–Kutta method, and the EM power profiles have been obtained using both the collisional and the EM approaches described above. A. Populations and electron density The electron density profiles numerically obtained fits well to the experimental data available,19 from which we infer that wall losses do not seem hence to be an important recombination process. As it is shown in Fig. 1, in which numerical results are obtained with different initial conditions, the influence of initial conditions is not important. Electron densities obtained are higher than those that would be expected if LTE is supposed, as we will discuss below. The pand slevels profiles are also presented for a 1 l/min gas flow [Figs. 2(a)and 2(b)]. It should be noted that the plevel population is greatly affected by the decrease in both electron density and temperature, varying its population in some orders of magnitude as we approach to the end of the plasma column. Contrary to what happens to the atomic species, Ar2 +presents an opposite behavior [Fig. 2(c)], increasing its population as we move towards the end of the column but always with values one order of magnitude lower than those of the argon atomic ions. Due to plasma quasineutrality, the argon atomic ions follow the decreasing behavior of the electron density as we approach to the column end. Gas flow turns out to be an important parameter, as it affects the axial profile by increasing the plasma length. At a fixed distance from the surfatron, a flow increase leads to a FIG. 1. Electron density profile (T0=1300 K, ␾ =1 l/min). The origin of the zaxis is located just after the gap of the surface-wave launcher. FIG. 2. Species populations for T0=1300 K, ␾ =1 l/min. (a)4sconfiguration, (b)4pconfiguration, and (c)Ar2 +ions. zaxis as in Fig. 1. 5502 Phys. Plasmas, Vol. 11, No. 12, December 2004 Yanguas-Gil, Cotrino, and González-Elipe 01 July 2025 08:53:12 higher electron density, as it was experimentally obtained for microwave atmospheric plasmas in Ref. 14. The same trend is observed in the axial profiles of the sand pexcited levels. In Fig. 3 we present numerical electron density results obtained by using the same initial condition to solve the equations model, and the comparison with experimental data. Figure 4 shows the axial variation of excited states. Gas temperature has a smaller influence on electron density and the populations of the atomic argon excited states. For all of them slight decreases are obtained when reducing the gas temperature. Ar2 +, however, is strongly influenced by this parameter (Fig. 5), becoming more important when gas temperature is decreased, as it is very well known from the works on argon excimer lasers.37 An evaluation of the main processes that take place in the discharge yields that, contrary to what happens at low pressure, the gradient term due to gas flow present in electron balance equation (4)has an important effect as a populating mechanism for electrons. As it can be seen in Fig. 6(a), its influence becomes higher when moving towards the end of the discharge, where electron temperatures are lower. Therefore, electron densities in this region are higher than those expected if only collisional processes were taken into account. The influence of inelastic collisions due to neutral heavy particles is also important, atomic argon three-body recombination becoming the main electron loss mechanism in the region further from the SW launcher [Fig. 6(b)]. Although no direct experimental evidence has been found about this fact in SW discharges at atmospheric pressure, it must be mentioned that in the experiments carried out by Moon et al.14 on an argon atmospheric microwave discharge, the electron density was fitted with good agreement supposing a recombination term proportional to ne 2, instead of the ne 3term that would correspond to the electron induced three-body recombination.The study of the main kinetic processes that affect the populations of argon excited states reveals that Ar 4pand 4s levels are strongly coupled with each other, being the electron collisional excitation and deexcitation and the p→sradiative decay the most important gain and loss processes for both levels. B. Power dissipation In Fig. 7 the axial profile of the EM power absorbed by the different processes considered shows that elastic collisions are the main loss process for electrons. Contrary to FIG. 3. Gas flow influence on electron density 共T0=1300 K兲.zaxis as in Fig. 1. FIG. 4. Gas flow influence on argon excited states 共T0=1300 K兲.(a)4s configuration and (b)4pconfiguration. zaxis as in Fig. 1. FIG. 5. Gas temperature influence on Ar2 +density 共 ␾ =1 l/min兲.zaxis as in Fig. 1. Phys. Plasmas, Vol. 11, No. 12, December 2004 Collisional radiative model of an argon…5503 01 July 2025 08:53:12 what happens at low pressure, the influence of electron–ion elastic collisions is higher than electron–neutral collisions: the high electron density values and the low temperatures increases the electron–ion collision frequency, whereas the electron–neutral elastic collision frequency is affected by the Ramsauer minimum of the argon elastic cross section. Compared to the elastic losses, the influence of inelastic processes in power absorption by electrons turns out to be negligible, a result that is in agreement with the channel model.35 If the ion contribution to the electromagnetic power absorption is neglected, it is possible to obtain the total power needed to sustain the discharge just by integrating the total power per unit length absorbed by electrons. The integration yields values of total incident power lower than 10 W, which are almost an order of magnitude below the nominal experimental values used by García et al.19 However, if the power loss contribution due to the ions acceleration by the space-charge field is considered, the total power needed increases up to 65 W: as it is shown in Fig. 7, our estimation to this contribution is higher than the total electron contribution all along the discharge, and therefore it has an strong influence on the calculation of the incident electromagnetic power, approaching its value to the experimental one. The electromagnetic power profile has been obtained also solving the SW dispersion relation. In Fig. 8 the axial variation of the attenuation constant ␣ is presented for two different gas temperatures. Results show that, although electron density is barely affected by gas temperature, the attenuation constant presents an important variation with that parameter, mainly due to the influence of gas temperature on electron collision frequency through the gas density. Taking into account the relation between incident EM power and the attenuation constant, ␣ =−1 2 1 P dP dz , it is possible to obtain a normalized axial profile of the EM power from ␣ . As it can be seen in Fig. 9, the attenuation of the incident EM power takes place in an scale that agrees well with the column length experimentally obtained, hence providing an indirect support to the electron density profiles calculated using the CR model. The power per unit length derived from the power profile of Fig. 9 is presented in Fig. 10. Although only a norFIG. 6. Probabilities of the different electron kinetic processes (T0 =1300 K, ␾ =1 l/min).(a)Gain processes and (b)loss processes. zaxis as in Fig. 1. FIG. 7. Power absorbed per unit length (T0=1300 K, ␾ =1 l/min).The contribution of inelastic processes has not been represented as it is negligible compared with the elastic contribution. zaxis as in Fig. 1. FIG. 8. Axial variation of the attenuation constant ␣ for two different gas temperatures 共 ␾ =1 l/min兲.zaxis as in Fig. 1. 5504 Phys. Plasmas, Vol. 11, No. 12, December 2004 Yanguas-Gil, Cotrino, and González-Elipe 01 July 2025 08:53:12