Power Transfer to Gas Heating in Pure N2 and in N2âO2 Plasmas

Article pubs.acs.org/JPCC

Power Transfer to Gas Heating in Pure N2 and in N2−O2 Plasmas Carlos D. Pintassilgo*,†,‡ and Vasco Guerra† †

Instituto de Plasmas e Fusão Nuclear, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal Departamento de Engenharia Física, Faculdade de Engenharia, Universidade do Porto, R. Dr. Roberto Frias, 4200-465 Porto, Portugal

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‡

ABSTRACT: A time-dependent kinetic model is developed to study the energy transfer to gas heating in N2 and N2−O2 plasmas. The model is based on the coupled solutions to the gas thermal balance equation and a system of rate balance equations for the most important neutral (including vibrational kinetics) and ionic heavy species produced in these plasmas. A complete set of gas heating mechanisms is taken into account, together with gas cooling by heat conduction to the wall. Calculations are performed for a cylindrical geometry and fractional concentrations of oxygen molecules [O2]/Ng (Ng being the gas number density) up to 80%. Modeling simulations are carried out for fixed values of the reduced electric field E/Ng and electron density ne with the purpose of focusing on the role of adding molecular oxygen O2 into the N2−O2 mixture. It is shown that the fractional discharge power and energy transferred to the translational mode (gas heating) increase with [O2]/Ng. The present work also shows that when the variation of E/Ng and ne with the mixture composition is considered, the gas temperature has a maximum for [O2]/Ng ∼ 20%, which is qualitatively in line with experimental observations. Gas heating mechanisms are identified and discussed in detail. reported in our previous publications2 provided a comprehensive description of the most important gas heating mechanisms for the pulse and for the afterglow situation. Modeling simulations2 have shown that the fast gas heating before the first millisecond comes mainly from O2 dissociation by electron impact collisions and quenching of nitrogen electronically excited states by molecular oxygen, as observed by Popov.5,6 For longer times, an important additional contribution to gas heating results from (i) vibrational excitation, through vibrational−translational (V−T) energy exchanges in N2−O collisions; (ii) chemical exothermic reactions, such as N + NO → N2 + O; and (iii) recombination of oxygen atoms at the wall. These three heating mechanisms continue to be dominant in the afterglow, explaining the relatively slow decrease of the gas temperature measured along the post-discharge.4 The study concerning the competition between all important gas heating mechanisms in air plasmas, coming from electron, vibration, chemical, and surface kinetics, was continued by further simulations3 where we systematically analyzed the role of the reduced electric field, electron density, pressure, and tube radius on the values of the power transfer to gas heating and on the values of the gas temperature. This investigation included at the same time a detailed description of the fractional discharge power transferred to gas heating. These results are however limited to the situation of air plasmas, corresponding to a fixed

1. INTRODUCTION Modeling of the gas temperature in plasmas is of major importance. For instance, the value of this parameter is essential to determining temperature-dependent rate constants for reactions in the gas phase and, ultimately, the plasma composition and the concentration of the different active species. In the particular case of exothermic reactions, the corresponding rate constant influences the value of the gas temperature, which in turn, may affect the value of the rate constant. This interplay requires therefore a coupled analysis between the plasma chemistry and gas heating mechanisms. Moreover, in the case of plasma discharges in molecular gases such as N2 and O2, considered in this work, vibrational excitation plays a crucial role in the overall kinetics. The cross section for excitation of the manifold of vibrational levels is very high in N2, causing an efficient energy transfer from the electrons to vibrations.1 Part of this energy may be returned to the electrons via superelastic collisions, or transferred to chemical reactions and to the translational mode. Hence, a detailed study of these different energy channels, being all coupled, needs to take into account simultaneously electron, vibrational, and chemical kinetics. Here, we are interested in the role of these different kinetics on the energy transfer to gas heating in N2−O2 mixtures and in pure N2. We have recently addressed this issue for air (N2−20% O2) plasma discharges at low pressures.2,3 We have developed a model2 to explain time-resolved measurements of the gas temperature obtained for a microsecond pulsed discharge in air and the corresponding post-discharge.4 The accurate results © 2016 American Chemical Society

Received: May 31, 2016 Revised: August 8, 2016 Published: September 2, 2016 21184

DOI: 10.1021/acs.jpcc.6b05463 J. Phys. Chem. C 2016, 120, 21184−21201

Article

The Journal of Physical Chemistry C

< 50% in another paper.20 These two papers consider constant values for the reduced electric field and electron density. While the first one19 focuses principally on the total gas heating mechanisms, assuming constant gas temperature, the second one20 does not specify the role of the different heating channels for different N2−O2 mixture compositions. Hence, the fundamental aspects concerning gas heating in nitrogen−oxygen mixtures are still in general poorly known with the exception of air plasmas. Therefore, it is noteworthy to check if a continuous increase in the fractional concentration of oxygen in the mixture leads or not to more efficient gas heating mechanisms and, in consequence, to a faster heating and/or larger values of the gas temperature. On one hand, one may expect that kind of behavior with the enhancement of electron impact collisions with O2 and subsequent production of oxygen atoms. On the other hand, important amounts of oxygen atoms can eventually contribute to a very efficient depletion of the N2 vibrational distribution function by (V−T) N2−O processes, which will no longer contribute to gas heating.2 In parallel, if the relative concentration of N2 in the mixture becomes smaller, the energy transfer from vibrationally or electronically excited states of nitrogen to the translational mode (gas heating) diminishes.6 A full picture of the temporal variation of these possible effects consists of one of the other purposes of this work, including the answer to the question of whether there is a specific range of fractional concentrations of O2 that leads to more efficient energy transfers to gas heating. The organization of the present paper is as follows. Section 2 describes the modeling procedure. Sections 3.1 and 3.2 present and discuss in detail the results for N2 and N2−O2 mixtures, respectively. Finally, section 4 summarizes the main conclusions of this work.

initial fractional concentration of 20% of molecular oxygen in a N2−O2 mixture. In this work, we focus on the time-dependent description of the gas temperature and fractional discharge power channeled into gas heating considering a wide range of fractional concentrations of O2 and the situation of pure N2. In spite of the intensive research work carried out over the past decades in these gases, the theoretical study of the gas heating mechanisms in these plasmas has not received too much attention. For instance, with respect to pure N2, the time evolution of the energy transfer to the translational mode has been carried out by Boeuf and Kunhardt7 although without the calculation of the gas temperature and only for times below 1 ms, where the kinetics of nitrogen atoms may be discarded. That work showed an important contribution to gas heating from the pooling reactions N2(A) + N2(A) → N2(B) + N2 and N2(A) + N2(A) → N2(C) + N2, followed, for longer times, by V−V collisions between vibrationally excited N2 molecules. Furthermore, recent simulations of the gas temperature in N2 plasmas considered atmospheric pressure conditions involving the analysis of its radial profile8 and its evolution in order to study the fast anode arc reattachment.9 In addition, selfconsistent calculations of microwave discharges in N2 at low pressures10 have correctly simulated the value of the gas temperature, together with a discussion of the main gas heating mechanisms, but considering only a steady-state situation. Time-dependent studies of gas heating in pure N2 discharges at intermediate values of pressure, 100 and 150 Torr, have also been carried out.11,12 For times ranging between 1 and 10 ms the main gas heating sources are found to be vibrational− vibrational (V−V) energy exchanges in N2−N2 collisions and the energy release in V−T processes in N2−N collisions.11 Modeling predictions12 and measurements at earlier times for the gas temperature show that gas heating is caused essentially by an energy transfer from electronic excited states of nitrogen, such as N2(A), N2(B), N(2D), and N(2P), to gas heating. One of the goals of this paper is to consider both the submillisecond and over millisecond time scales in order to analyze the temporal evolution of the different gas heating mechanisms in pure N2 within these time ranges. The present work also provides the fractional discharge power transferred to gas heating, but without the restriction of considering a negligible role for the nitrogen atoms, as considered by Boeuf and Kunhardt.7 In what concerns N2−O2 mixtures, the study of the energy transfer to gas heating has been mainly dedicated to the specific situation of air.6,12−17 Within this group of papers, as in many other related works, the effect of fast gas heating has been widely studied. However, the same does not happen with gas heating in other N2−O2 mixture compositions. This subject has been investigated by other authors18−20 although without too much detail. As a matter of fact, on the steady-state situation considered concerning a low pressure (2 Torr) glow discharge,18 the gas temperature is shown to have a maximum for a fractional concentration of O2 of the order of 20%, decreasing for larger values of molecular oxygen. In spite of the very detailed description of the kinetic model presented in that work,18 the discussion on the main gas heating mechanisms in that paper is relatively absent. Additionally, in the timedependent modeling study19 the total gas heating rates are found to increase progressively with [O2]/Ng (where Ng denotes the gas number density) until the situation of pure O2 for times up to 1 s. A similar effect is predicted for [O2]/Ng

2. MODELING DETAILS We adopt in this work the experimental conditions concerning a glow discharge in N2−O2 produced at a constant pressure of 266 Pa (2 Torr) in a cylindrical discharge tube with a radius of 0.8 cm.18 As extensively studied in the past,18,21−24 this kind of discharge involves a very complex plasma chemistry, with a strong interplay between electron, vibrational, and chemical kinetics. As in our previous works25,26 the electron kinetics is described from the solutions to the homogeneous electron Boltzmann equation under the classical two-term approximation.24,27 This equation is solved for different N2−O2 mixture compositions for a given value of the reduced electric field E/ Ng and vibrational temperature Tv of the electronic ground state of nitrogen molecules N2(X1Σg+,v). This vibrational temperature characterizes the degree of vibrational excitation of N2, obtained from the Treanor-like distribution28 that best fits the vibrational distribution function (VDF) in the lowest vibrational levels. On the other hand, the VDFs of O2 molecules are noticeably less excited than those of N2.21 Moreover, the vibrational temperature of O2 molecules is found to be close to the gas temperature. Therefore, we have considered in the present model that the VDFs of O2 are close to the Treanor one. The Boltzmann equation is solved by using the same electron cross sections considered in those previous works. The electron cross sections for N2 molecules are close to the set proposed by Pitchford and Phelps29 and available at the ISTLisbon database at the LXCat,30 whereas the ones concerning O2 molecules are taken from Phelps.31 In addition, we have 21185


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processes with constant probabilities, regardless the mixture composition. These vibrational and chemical processes are essentially the same as those we have considered in previous works concerning N2−O2 mixtures,25,26,34 and the reader should refer to these papers for more details. Note that we are considering here 46 vibrational levels for N2(X,v), obtained from the two-term polynomial expansion from spectroscopic data,37 while other works38,39 report 61 and 59 vibrational levels for the electronic ground state of N2. While the description with 61 vibrational levels for molecular nitrogen results from the potential curve according to the Rydberg− Klein−Rees (RKR) method presented by da Silva et al.,38 the 59 vibrational levels considered in Laporta et al.39 are obtained from a phenomenological potential energy recently reported.40 This number of vibrational levels is fewer than the 68 levels found for the Morse curve adopted in Laporta et al.40 As shown by da Silva et al.38 for the case of a microwave discharge in pure N2, the VDF has essentially the same shape when 46 or 61 vibrational levels are considered. The main difference occurs at the tail of the VDF, which is slightly more populated for energy values larger than 8 eV, mostly due to V− V energy-exchange reactions. Hence, the present analysis with 46 vibrational levels may underestimate the role of V−V processes. However, this effect is expected to be negligible in the present work since the populations of the VDF are in general small for energies larger than 8 eV. Furthermore, since the calculations carried out in that work for a post-discharge situation indicate that the time-dependent kinetics of the electronically excited states of molecular nitrogen N2(A, B, C), as well as that the atomic states N(4S, 2D, 2P) are similar, we expect a reliable temporal description of these species and of their contribution to gas heating. Nonetheless, it is worth mentioning that work is underway to couple the dynamics of gas heating with the more detailed description of vibrational kinetics for N2 and O2 reported recently by Coche et al.41 The coupling between electron, vibrational, and chemical kinetics is ensured through the following iterative process: (i) the electron energy distribution function (EEDF) is obtained from the solutions to the electron Boltzmann equation for specific values of E/Ng, Tv, and different fractional compositions of the mixture, as in our previous work;21 (ii) the corresponding electron parameters and rate coefficients determined from the EEDF are used as input parameters in the system of rate balance equations for the heavy species for a given electron density value ne; (iii) convergence is achieved when the value of Tv estimated after solving this system of rate balance equations and the fractional mixture composition are the same as the one considered as an input parameter in the Boltzmann equation. This criterion is a consequence of the one-to-one relationship between the values of Tv and the electron density for a specific value of E/Ng.24 Note that the present calculations do not take into account superelastic electronic collisions. Their influence in shaping the EEDF is more important under afterglow collisions.42 However, electron impact processes during the afterglow are not considered in our calculations, due to the relatively long time scales under investigation, and any modifications in the relaxation of the EEDF should not have any noticeable influence on the present results. The gas temperature T is described for a cylindrical geometry under the assumption of a parabolic gas temperature profile across the discharge tube with radius R

considered in the present simulations the decrease of the cross sections in N2 for the electron−vibration (e−V) processes σ(v, w) with w (where w > v) between excited vibrational levels, using18,32 σ(v, v + n) = σ(0, n)(1 + av)−1, with n = 1 up to 45 and a = 0.05.18 This effect has been recently analyzed33 for pulsed discharges in nitrogen using a very complete set of e−V electron cross sections. With respect to our previous works25,26,34 the incorporation of this effect constitutes a more detailed description of e−V processes since we used before an identical shape and magnitude for the cross section for the transition 0 → (w − v), but with a different energy threshold accounting for the anharmonicity of the oscillator. In the case of the present calculations, we have verified that both descriptions lead to very similar values for the VDF and gas temperature. In spite of the above-mentioned scaling law we have now introduced, it is worth mentioning that the cross sections we are considering here do not have the detail of the set reported in Colonna et al.33 for N2, or recently presented in Celiberto et al.35 for N2, O2, and other molecules. In those works, the electron cross sections include single quantum transitions for a wide range of vibrational levels and multiquantum transitions from v = 0 as well as from higher vibrational states. Electron cross sections leading to the dissociation of N2 and O2 from vibrationally excited states are also reported in Celiberto et al.35 The results obtained using this complete set of electron-impact collisions for N2 were compared by Colonna et al.33 and Capitelli et al.36 to the ones obtained in a similar way as in this work. It was shown in Colonna et al.33 that the results are similar to the ones obtained with the scaling law used in this work. On the other hand, the results reported in Capitelli et al.36 reveal that for a pressure of ∼750 Pa (5.6 Torr) the intermediate levels of the VDF at a time t = 1 ms are more populated, as a consequence of more efficient e−V processes concerning v → w transitions. This result indicates that the results presented in this work for these vibrational levels may be somewhat underestimated. The kinetics of heavy species is described by a zerodimensional model involving a coupled system of timedependent rate balance equations for the most important heavy species produced in low-pressure N2−O2 plasmas. For this purpose, we consider 45 vibrationally excited states of the electronic ground state of molecular nitrogen N2(X1Σg+,v) and molecular oxygen O2(X3Σg−), the electronically excited states of N2(A3Σu+, B3Πg, B′3Σu−, C3Πu, a′1Σu−, a1Πg, w1Δu) and O2(a1Δg, b1Σg+), both ground and excited states of atomic nitrogen N(4S, 2D, 2P), and atomic oxygen in the ground state O(3P) and excited state O(1D). Other chemical neutral species NO(X2Π, A2Σ+, B2Π), NO2(X, A), and O3 produced by chemical reactions are also considered in the calculations, as well as the main positive ions N2+, N4+, O2+, O+, and NO+ and the negative ion O−. Vibrational kinetics takes into account electron−vibration (e−V) collisions, vibration−translation (V−T) energy exchanges in N2−N2, N2−O2, N2−N, and N2−O collisions, and vibration−vibration (V−V) N2−N2 and N2−O2 processes.21 Furthermore, chemical reactions between vibrationally excited N2 molecules and active species produced in the discharge are also included in the system of rate balance equations. The present model further includes the deactivation of vibrationally excited molecules N2(X,v) at the wall and describes the recombination of nitrogen and oxygen atoms as first-order 21186


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plasmas.2,34 Nonetheless, besides a thorough analysis and insight on the effect of the relative concentration of O2 for gas heating in these mixtures, we discuss in section 3 the main physical aspects related to gas heating when the reduced electric field and the electron density change with [O2]/Ng, according to the requirement of a quasi-neutral discharge for a given current. This corresponds to a situation where the total ionization rate exactly compensates the total loss rate of the electrons. The molar heat capacity and the thermal conduction are input parameters in eq 1. These two parameters depend on the gas temperature, being determined at the same time as eq 1 is numerically solved. Moreover, as in our previous work in air plasmas,2 we have verified that the heat capacity and the thermal conduction for different relative mixture compositions may be obtained by a linear combination of the values of these parameters in N2 and in O2, accounting for their relative fractional concentration in N2−O2. These two parameters are calculated as a function of the gas temperature from expressions taken from other papers44−46 and are also reported in our previous work.2 Although the basis of these expressions do not include the detail reported in Capitelli et al.47 where the degrees of freedom that are not in equilibrium with Tg are treated in the balance directly through kinetic equations, they provide a satisfactory agreement with experiment for the conditions under study.2 In more detail, the values for the molar heat capacity are 29.13 and 29.39 J mol −1 K −1 respectively for pure N2 and pure O2 when the gas temperature is 300 K. For a gas temperature equal to 600 K, these two values increase respectively up to 30.11 and 32.09 J mol−1 K−1. In what concerns the values of the thermal conductivity, they are 25.35 × 10−3 and 26.63 × 10−3 W m−1 K−1 respectively for pure N2 and pure O2 when Tg = 300 K and increase up to 45.47 × 10−3 and 50.59 × 10−3 W m−1 K−1 for a gas temperature of 600 K. For the situations considered in this work, involving pure N2 and fractional N2−O2 mixture compositions increasing up to [O2]/Ng = 80%, the ratio cp/λg, relevant for the timedependent study of Tg provided by eq 1, varies from ∼1.8 up to 2. Therefore, the impact of the variation of these parameters in the present calculations is almost negligible. The list of the most important gas heating mechanisms in pure N2 and in N2−O2 mixtures under the conditions considered in this work is presented in Table 1 and in Table 2, respectively, with the corresponding expressions incorporated in the term Qin of eq 1 and values concerning the energy transferred to gas heating. Note that this list of processes does not correspond to all mechanisms included in the term Qin. Such a comprehensive description has already been conducted in our previous work.2 Nonetheless, we point out below some important aspects regarding the reactions reported in Tables 1 and 2. Processes (R1)−(R8) (Table 1) refer to gas heating mechanisms in pure N2, while processes (R9)−(R22) (Table 2) occur in N2−O2 mixtures. With respect to the first group, it is worth noting that the pooling reactions (R3) and (R4) are usually important for fast gas heating in pure N2, occurring earlier than vibrational excitation (R1) and (R2).7 As in other papers6,9,15 we have considered an available energy of 4 eV in process (R3) corresponding to N2(A) + N2(A) → N2(B) + N2(X, v = 2) instead of ... →N2(B) + N2(X, v = 8) with an energy release of 2 eV.7,10 The energy transferred to gas heating from the deactivation of metastables N2(A) and N2(a′) to the wall (R5) depends on

⎛ r ⎞2 T (r , t ) = T0(t ) − (T0(t ) − Tw )⎜ ⎟ ⎝R⎠

with T0(t) and Tw denoting, respectively, the gas temperature at the tube axis and at the wall. If we further assume a radially averaged gas temperature Tg given by Tg(t ) =

1 πR2

2π

∫0 ∫0

R

T (r , t )r d θ d r

the dynamics of gas heating may be described by the solutions to the time-dependent gas thermal balance equation under isobaric conditions:2 nmcp

∂Tg ∂t

=

8λg (Tw − Tg) R2

+ Q in

(1)

In eq 1, nm represents the molar density, while cp is the molar heat capacity at constant pressure. Heat conduction is the dominant cooling mechanism, with λg denoting the thermal conductivity. Finally, the term Qin represents all the gas heating mechanisms. With respect to N2−O2 plasmas, Qin must include the following processes:2 (i) elastic collisions of electrons with N2 and O2 molecules; (ii) electron-impact rotational excitation of N2 and O2 molecules, followed by the rapid rotational− translational relaxation; (iii) nonresonant V−V energy exchanges in N2−N2 and N2−O2 collisions; (iv) V−T energy exchanges in N2−N2, N2−O2, N2−O, and N2−N collisions; (v) vibrational deactivation of N2(X,v) at the wall; (vi) exothermic chemical reactions; (vii) diffusion and subsequent deactivation of molecular and atomic metastables states on the wall; (viii) recombination of N and O atoms at the wall; (ix) nitrogen and oxygen dissociation by electron impact collisions through predissociative states N2* and O2*; and (x) electron−ion recombination. Since many of the rate coefficients associated with these mechanisms depend on the gas temperature, eq 1 must be solved simultaneously with the coupled system of time-varying kinetic master equations for the heavy species, where the electronic rate coefficients are obtained from the homogeneous electron Boltzmann equation. This system is numerically solved for the range time between 0.1 and 100 ms with NAG routine D02EJF which integrates a stiff system of first-order ordinary differential equations over an interval with suitable initial conditions, using a variable-order and variable-step method.43 For the purpose of this work, we have considered the situation of pure N2, as well as the three following fractional concentrations of O2 in nitrogen−oxygen mixtures, [O2]/Ng (Ng being the gas number density): 20, 50, and 80%. At the beginning, for a time t = 0 s, there is no vibrational or electronic excitation and the populations of these two molecules are in the electronic ground level. Therefore, the initial conditions for these four situations are [N2(X, v = 0)]/Ng = 1, 0.8, 0.5, and 0.2, respectively, and [O2(X, w = 0)]/Ng = 0, 0.2, 0.5, and 0.8. The calculations are carried out for a constant reduced electric field E/Ng of 100 Td and for an electron density ne equal to 1010 cm−3, corresponding to typical values of a glow discharge. By considering constant values for these two discharge parameters, we want to focus essentially on the role of the fractional concentration of O2 in a N2−O2 mixture in what concerns its gas heating. For this reason, we do not include in the present model a self-consistent calculation of E/ Ng and ne, as in our previous works in N2−O225,26 and in air 21187


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Rate coefficients are given in cm s and Tg is in K. See text for further details. is the rate coefficient for these V−V processes; ωN2 = 4.443 × 10 s and χeN2 = 6.073 × 10 are the spectroscopic constants for the anharmonic Morse oscillator describing the N2 vibrational levels, with EvN2 = ℏωN2[(v + 1/2) − χeN2(v + 1/2)2].37 cPNv,w2−N is the rate coefficient for the V−T N2−N processes, where multiquantum transitions are considered. dDN2 represents the diffusion coefficient for these two metastable states; Λ is the diffusion length and β denotes the thermal accommodation coefficient.

2, 8

their diffusion coefficients and on the thermal accommodation factor β. We assume in the present calculations a value β = 0.5, corresponding to a situation where half of the available energy is returned to the gas phase. As shown previously2 the impact of this input parameter on the modeling results for the gas temperature is relatively small when gas heating from volume processes is dominant. In what concerns reaction (R6) N2(B) + N2 → N2(A) + N2, it has been shown50 that the quenching of N2(B) by N2 involves a significant amount of energy that is transferred to vibrational excitation of N2(A) and N2(X). For this reason, we have assumed for the reference model a negligible contribution from this reaction to gas heating. In a study on glow discharges in N2 at atmospheric pressure,8 it was considered that 85% of the energy release in this reaction goes into vibration. Since the energy difference between N2(B) and N2(A) electronically excited states is 1.18 eV, this corresponds to an available energy for heating of 0.18 eV. In a recent study on nanosecond pulsed discharges in nitrogen,12 it was considered that a value of 0.86 eV is transferred to gas heating from this reaction. Owing to these different values, we have analyzed the sensitivity of the present model when the available energy to gas heating from this reaction is (i) 0.18 eV, (ii) twice this value 0.36 eV, and (iii) the upper limit value of 1.18 eV. For the same reasons pointed out for reaction (R6), we have also assumed for the reference model a negligible contribution for gas heating from processes (R7) and (R8) concerning the quenching of N(2D) and N(2P) metastable states of atomic nitrogen. Nonetheless, in order to evaluate the effect of this assumption on the calculations, we have also considered the upper limit values of 2.28 and 3.58 eV, corresponding respectively to the difference in energy between N(2D) and N(2P) to the ground state N(4S). Note that these upper limits have been considered recently.12 As far as the situation of pure N2 is concerned, let us note that electron impact processes as e + N2 → e + N(4S) + N(2D) with an available energy2 lower than 1 eV or the recombination of electrons with N2+ play a negligible role in gas heating in the conditions of this work. This is essentially due to the relatively low value considered here for the reduced electric field and electron density. With respect to the main gas heating mechanisms in N2−O2 mixtures listed in Table 2, they are essentially the same in our previous research works on air plasmas. In brief, processes (R9)−(R14) are responsible for the so-called fast gas heating in these mixtures,6,15,51,52 which start to take place before vibrational excitation processes (R15) and (R16). Electron impact collisions (R9) and (R10) are responsible for the dissociation of O2 through the predissociative states of O2(A3Σu+, C3Δu, c1Σu−) with energy thresholds of ∼5.9 eV and O2(B3Σu−) with 8.34 eV, respectively. The corresponding available energies for gas heating results from the difference between these energy thresholds and the energies of 5.12 and 7.08 eV, associated respectively with reaction products O(3P) + O(3P) and O(3P) + O(1D). Furthermore, the energy transferred to gas heating from the quenching of electronically excited molecules N2* by O2 (R11)−(R14) corresponds to the difference between the threshold energy of a given N2* electronically excited state and the energy of 5.12 eV, related to products O(3P) + O(3P). This same amount of energy is available on the reassociation of oxygen atoms at the wall (R18). Note that in our calculations we are assuming that the heterogeneous process (R18) leads to oxygen molecules on the ground state. However, very detailed

14 −1

b w−1,w Pv,v−1

3 −1 a

[N(2 P)][N2](6 × 10−14)ΔE

N(2 P) + N2 → N( 4S) + N2 (R8)

ΔE = 0 eV (ref value) or 3.58 eV (see text)

−3

2, 8

ΔE = 0 eV (ref value) or 2.38 eV (see text)

[N( D)][N2] × 10

N(2 D) + N2 → N( 4S) + N2 (R7)

2

−13

[N2(B)][N2](2.85 × 10−11)ΔE N2(B) + N2 → N2(A) + N2 (R6)

exp(− 510/Tg)ΔE

2, 8

1, 2

Λ2

DN2

[N2(A)][N2(A)](1.5 × 10

ΔE N2(1 − β) [N2(A , a′)] diffusion of N2(A) or N2(a′) to the wall (R5)d

ΔE = 0 eV (ref value) or 0.18, 0.36, and 1.18 eV (see text)

ΔE = 0.4 eV N2(A) + N2(A) → N2(C) + N2(X)

N2−N

N2−N2

−10

(R4)

)ΔE

ΔE = 4 eV [N2(A)][N2(A)](7.7 × 10−11)ΔE N2(A) + N2(A) → N2(B) + N2(X) (R3)

ΔE N2 = 6.17 eV for N2(A) and 8.4 eV for N2(a′)

1, 2

6, 15

1, 2 v

w

V−T (R2)c

[N] ∑ {[N2(X , v)] ∑ PvN, w2 − N}ΔEv , w

2

ΔEv , w = ℏω N2(v − w)[1 − χeN (v + w + 1)]

1, 2 2

ΔEvw, v, w−+1 1 = 2ℏω N2χeN (w − v)

w v

∑ [N2(X, v)] ∑ [N2(X, w − V−V

(R1)b

available energy for gas heating, ΔE (eV) corresponding expression in term Qin of eq 1 (eV s−1 cm3) process

Table 1. Most Important Gas Heating Mechanisms in N2 Plasmas for the Conditions of the Present Worka

1)]Pvw, v−−1,1wΔEvw, v, w−+1 1

ref


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The Journal of Physical Chemistry C Table 2. Most Important Gas Heating Mechanisms in N2−O2 Plasmas for the Conditions of the Present Worka process (R9)

b

(R10)

6

ΔE = 2.28 eV

2, 6

)ΔE

ΔE = 3.4 eV

2, 6

)ΔE

ΔE = 3.6 eV

2, 6

ΔE = 3.9 eV

2, 6

ΔEv , v − 1 = ℏω N2(1 − v)

2, 21

ΔEv ,13 = Ev − Ev = 13

2, 21

1

ne[O2 ]k(E /Ng)ΔE

e + O2 → e + O2 * → e + O( P) + O( D)

N2(w) + O2 → N2(X) + O + O

(R14)

ΔE = 1.26 eV

3

N2(a) + O2 → N2(X) + O + O

(R13)

6

ne[O2 ]k(E /Ng)ΔE

N2(a′) + O2 → N2(X) + O + O

(R12)

ΔE = 0.8 eV

3

N2(B) + O2 → N2(X) + O + O

(R11)

available energy for gas heating, ΔE (eV)

3

e + O2 → e + O2 * → e + O( P) + O( P) c

corresponding expression in term Qin of eq 1 (eV s−1 cm3)

−10

[N2(B)][O2 ](3 × 10

[N2(a′)][O2 ](2.8 × 10

−10

[N2(a)][O2 ](4.3 × 10 −10

[N2(w)][O2 ](10

∑ {[N2(X,

(R15)d

V−T

(R16)

N2(X, v ≥ 13) + O → NO(X) + N( 4S)

N2−O

)ΔE

−11

)ΔE

v)]PvN, v2−−1O

− [N2(X, v −

1)]PvN−21,−vO}[O]ΔEv , v − 1

v

ref

45

[N2(X, v)][O](10−13)ΔEv ,13

∑ v = 13

(R17) (R18) (R19)

[N( 4S)][NO](1.05 × 10−12)Tg 0.5ΔE

ΔE = 2.45 eV

48

O + wall → (1/2)O2

0.5[O]υOΔEO(1 − β)

ΔEO = 5.12 eV

2

diffusion of O2 (b) to the wall

[O2 (b)]υO2ΔEO2(1 − β)

ΔEO2 = 1.63 eV

2

ΔE = 1.38 eV

16

ΔE = 1.38 eV

49

ΔE = 0.33 eV

49

1

3

−11

1

) exp(107/Tg)ΔE

(R20)

O( D) + N2 → O( P) + N2

[O( D)][N2](1.8 × 10

(R21)

O(1D) + O2 → O(3P) + O2

[O(1D)][O2 ](10−12)ΔE

(R22) a

e

N( 4S) + NO(X) → N2(X, v ∼ 3) + O

O(1D) + O2 → O(3P) + O2 (b) 3

−1

−11

1

[O( D)][O2 ](2.6 × 10

) exp(67/Tg)ΔE

O2* represents the predissociative states of O2(A Σu , C3Δu, c1Σu−); k(E/Ng) is the rate N2−O represents the predissociative excited state O2(B3Σu−). dPv,v−1 is the rate coefficient for the

b

3

+

Rate coefficients are given in cm s and Tg is in K. coefficient obtained from the Boltzmann equation. cO2* V−T N2−O processes. eυO is the recombination frequency of oxygen atoms at the wall, determined by the recombination frequency γO and their thermal velocity.

studies53 on oxygen recombination on quartz surfaces show that this process forms initially highly vibrationally excited states of O2. Hence, the available energy from this process to gas and surface heating may be lower than 5.12 eV, which may cause an overestimated contribution of (R18) to gas heating in the present calculations. Note also that these authors have also found that the energy transferred as heat flux to the surface is less than 40% of the total energy. This fraction of energy is similar to the value of 50% considered here for the thermal accommodation coefficient. Concerning process (R19) also related with wall effects, the deactivation of O2(b) giving O2(X), with an energy difference of 1.63 eV, is assumed to be a first-order process with a deactivation probability of 2 × 10−2.54 Finally, a fraction of about ∼25−30% of the available energy in chemical reactions (R17), (R20), and (R21) is spent respectively in N2 and O2 vibrational excitation.6,16,48 Owing to the difference of 1.96 eV between O(1D) and O(3P) indicated in the previous paragraph, reactions (R20) and R(21) have then an available energy for gas heating equal to 1.38 eV.

Figure 1. Temporal variation of the radially averaged gas temperature Tg (full curves) and the gas temperature at the axis T0 (dashed curves) in a plasma discharge in N2 with a pressure p = 2 Torr (266 Pa) in a tube radius equal to 0.8 cm for a reduced electric field E/Ng of 100 Td and an electron density ne equal to 1010 cm−3.

correspond to the ones experimentally considered for a glow discharge in N2−O2 (including the situation of pure nitrogen) by Gordiets et al.18 However, the aim of the present work is not to make an accurate simulation of that specific experimental setup, but to provide a very detailed insight and broad overview on fundamental aspects of gas heating mechanisms and their influence on the values of the gas temperature in pure N2 and in different N2−O2 mixture compositions, as discussed further in section 3.2. Figure 1 reveals that the gas temperature increases smoothly during the first milliseconds and remains roughly constant afterward up to a time of ∼10 ms. For longer times, there is an acceleration of gas heating, as a result of the onset of gas

3. RESULTS AND DISCUSSION 3.1. Gas Heating in Pure N2. Figure 1 shows the predicted results for the temporal evolution of the radially averaged and on-axis values2 of the gas temperature Tg(t) and T0(t) = (2Tg − Tw) (as shown, for instance, in refs 1 and 2) for a discharge in pure N2 at a pressure of 2 Torr produced in a tube discharge with a radius R equal to 0.8 cm when the reduced field E/Ng is 100 Td (1 townsend (Td) = 10−17 V cm2) and the electron density ne is 1010 cm−3 for the time range 0.1−100 ms. We have considered in these calculations a value of 300 K for the wall temperature Tw. These values of pressure and tube radius 21189


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This effect occurs at earlier times, where the fractional population of N2(v = 0) is still very large. In fact, modeling results show that [N2(v = 0)]/[N2] is equal to 0.984 and 0.841, respectively, at 0.1 and 1 ms. Hence, endothermic processes N2(v > 1) + N2(0) → N2(v − 1) + N2(1) are favored. Moreover, these processes cause an increase in the population of vibrational level N2(v = 1), leading then to endothermic processes N2(v > 2) + N2(1) → N2(v − 1) + N2(2) and so on. A similar time-dependent contribution to gas cooling has been pointed out in for a pulsed discharge in pure N2 produced at a pressure of 150 Torr11 as well as in the case of a nanosecond discharge in atmospheric pressure air.14 For longer times, the well-known V−V pumping-up mechanism starts to take place.1 In consequence, intermediate vibrational levels become more populated and the overall contribution of nonresonant V−V energy exchanges in N2−N2 collisions results in exothermic processes. This effect is illustrated in Figure 2 for a time t > 2 ms, where we may also observe a continuous increase of the temporal contribution of these V−V processes to gas heating. This effect is followed by an efficient contribution from V−T relaxation processes in N2−N collisions (R2). Figure 2 shows indeed that this gas heating mechanism becomes the most important one for t > 30 ms. This behavior is a result of a temporal increase of nitrogen atoms (not shown in a figure) combined with the fact that the rates of these V−T processes increase both with the vibrational quantum number and with the gas temperature.10 Therefore, the observed increase of the gas temperature reported in Figure 1 leads to larger rate coefficients for these processes, which in turn become even more efficient to gas heating. Figure 2 shows that, under the conditions considered here, the most important heating channels are processes (R1)−(R4). This result is in line with modeling predictions10 for gas heating mechanisms, under stationary conditions, concerning a microwave discharge in pure N2 at a pressure of 0.5 Torr with and an electron density of ∼1010 cm−3. It was also shown10 that elastic collisions or V−T energy exchanges in N2−N2 collisions have a relative contribution to gas heating of the order of 5%. Under the conditions of the present work, the percentage of each of these two processes is of the order of ∼1−2%. A similar relative contribution is found in our calculations for the recombination of nitrogen atoms at the wall. Due to these relatively low values, the gas heating rates for these processes are not plotted in Figure 2. Moreover, as mentioned in section 2, other gas heating mechanisms in N2 such as electron−ion recombination and nitrogen dissociation by electron impact collisions to predissociative states e + N2 → e + N2* → e + N(4S) + N(2D) have a negligible role on the conditions of the present work. This is a consequence of the relatively low values of the reduced electric field and electron density considered in the calculations. Due to the importance of vibrational excitation of N2 molecules under the conditions of this work, we plot in Figure 3 the time-dependent kinetics of the VDF, [N2(X,v)]/[N2], for 1, 10, and 100 ms. Figure 3 shows the typical temporal evolution of the VDF, which starts by the combined effect of e−V and V−V exchanges at low vibrational levels, followed by near-resonant V−V exchanges at intermediate levels, which tend to form a plateau in this region, and V−T N2−N exchanges at the high levels. In a recent study6 of fast gas heating, where vibrational excitation plays a minor role, it was shown that the relative contribution of pooling mechanisms to gas heating is almost 90% in the situation of pure N2 for a pressure of 0.4 Torr and a

heating through vibrationally excited levels (see below). At a time of 100 ms, the plasma discharge has not reached yet a steady-state situation (which occurs at ∼1 s, not shown in a figure) and the values of the gas temperature are relatively low. The radially averaged gas temperature is 343 K, whereas the gas temperature at the axis is 386 K. Figure 2 plots the heating rates of the most important gas heating mechanisms for a pure N2 discharge under the same

Figure 2. Time-dependent evolution of heating rates of the most important gas heating mechanisms in a N2 discharge when p = 2 Torr, R = 0.8 cm, E/Ng = 100 Td, and ne = 1010 cm−3, as follows: V−V N2− N2 nonresonant collisions (R1); V−T N2−N collisions (R2); pooling reactions N2(A) + N2(A) → N2(B,C) + N2(X) ((R3) and (R4)); deactivation of metastables N2(A) and N2(a′) at the wall (R5).

conditions reported in Figure 1. Before the first millisecond, gas heating comes essentially from the role of the electronically excited state N2(A3Σu+) (hereafter referred to as N2(A); a similar notation will be used for the other electronically excited states) through the global contribution of the pooling reactions N2(A) + N2(A) → N2(B) + N2(X) (R3) and N2(A) + N2(A) → N2(C) + N2(X) (R4). This effect is a result of an almost monotonic increase of N2(A) with time up to a time of 1 ms (not shown in a figure), where its populating mechanisms are mainly the quenching of electronically excited N2(B) molecules by N2, via reaction N2(B) + N2 → N2(A) + N2 and, to a lesser extent, electron impact collisions e + N2→ e + N2(A). Due to the difference in the rate coefficients of reactions (R3) and (R4) and to the corresponding available energies transferred to gas heating, the heat rate of process (R3) is about 5 times larger than that of (R4). Figure 2 shows also a smaller contribution to gas heating from the diffusion and subsequent deactivation of this metastable state (R5), as well as that of N2(a′1Σu−), at the wall. For longer times, the vibrationally excited levels N2(X,v) and the atoms N(4S) become sufficiently populated, playing an important role in the overall kinetics. Therefore, N2(A) starts to be depleted not only by the pooling reactions (R3) and (R4) but also by collisions of this species through processes N2(A) + N2(5 ≤ v ≤ 14) → N2(B) + N2(X) and N(4S)+ N2(A) → N(2P) + N2(6 ≤ v ≤ 9). For this reason, we can observe in Figure 2 a decrease in the contribution of processes (R3) and (R4) to gas heating, accompanied at the same time by an enhancement of the contribution of reactions (R1) and (R2) corresponding, respectively, to nonresonant V− V energy exchanges in N2−N2 collisions and V−T relaxation processes in N2−N collisions. With respect to process (R1) it is very interesting to observe that these mechanisms may contribute to a slight gas cooling. 21190


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simulations reveal that the value of Tg increases about 10% for a time of 100 ms when the corresponding rate coefficient is multiplied by 5. Under this condition, modeling also shows the interesting effect of a slight decrease of Tg of ∼4 K at 1 ms, as a result of the gas cooling effect that occurs at this time, as mentioned before. On the other hand, if the rate coefficients of V−V collisions are decreased by a factor of 5, the values of Tg are found to decrease by only ∼5% for 100 ms. For t = 1 ms, the value of Tg is practically the same. In what concerns these sensitivity tests for V−T collisions, the influence of the rate coefficients has a larger impact on the radially averaged values of the gas temperature. In fact, if the rate coefficient of this process is modified by a factor of 5 there is an increase in Tg by about 60 K at 100 ms, corresponding to ∼17%. When the V−T rate coefficients are decreased by a factor of 5, the value of Tg decreases about 20 K. Hence, we may conclude that an increase of the gas temperature is favored by larger V−T rate coefficients. Although concerning different discharge conditions, the investigation regarding the phenomena of constricted glow discharges in N2 at atmospheric pressure8 showed that gas heating is also governed mainly by V−T N2−N relaxation processes for the case of a high-current constricted discharge, where the electron density is ∼3 × 1012 cm−3 and the reduced electric field E/Ng is ∼40 Td. In contrast, under low-current constricted discharge conditions, corresponding to an electron density of 1012 cm−3 and E/Ng < 20 Td, modeling simulations8 revealed that the dominant gas heating channel is the quenching of the electronically excited state N2(B) by N2 molecules, through reaction N2(B) + N2 → N2(A) + N2 (R6). It is assumed in that work that this reaction releases 0.18 eV into gas heating. Note that this value is 15% of the upper limit of 1.18 eV, corresponding to the difference in the energy thresholds of N2(B) and N2(A) states. In our previous study in air plasmas2 we have checked that the amount of energy spent on gas heating on this process has a minor effect on the total gas heating rates and consequently on the values for the gas temperature. This result comes from the fact that the most important gas heating channels in air plasmas,2 which involve O2 molecules (see section 3.2), have larger heating rates than process (R6). However, in the case of pure N2 a detailed analysis of the role of this process into gas heating requires further investigation. Furthermore, in a very recent study12 on nanosecond pulse discharges in nitrogen and air at a pressure of 100 Torr, the quenching of N2(B) by N2 molecules was found to be a possible heating mechanism to describe experimental measurements for the gas temperature in pure N2 at the microsecond time scale. In that work, it was considered that the energy release into gas heating from process (R6) is 0.86 eV. This value arises from the assumption of a vibrational excitation of N2(X,v) as follows: N2(B) + N2 → N2(A) + N2(X, v = 4). Among other values reported for the energy release to gas heating involving electronically excited states of nitrogen, it was also shown12 that the quenching of metastable atoms N(2D) and N(2P) by N2 (R7) and (R8) molecules may constitute one of the most important gas heating channels for time scales within the range 10−100 μs. The energy values transferred to gas heating from quenching processes N(2D) + N2 → N(4S) + N2 (R7) and N(2P) + N2 → N(4S) + N2 (R8) were assumed to be 2.38 and 3.58 eV.12 This assumption corresponds to a situation where all the available energy in these processes is

Figure 3. Temporal evolution of the VDF [N2(X,v)]/[N2] in a plasma discharge in N2 with a pressure p = 2 Torr (266 Pa) in a tube radius equal to 0.8 cm for a reduced electric field E/Ng of 100 Td and an electron density ne equal to 1010 cm−3.

current density of 0.8 A cm−2. Other modeling works dealing with N2 plasma discharges at higher pressures8,9,11 have also observed a similar role from vibrational excitation into gas heating. In fact, besides the gas cooling due to V−V N2−N2 collisions (R1) for earlier times discussed above, the timedependent analysis carried out for vibrational energy exchanges in nitrogen at a pressure of 150 Torr showed11 that these processes started to play an important role to gas heating after a time of ∼2 ms. Modeling results11 also reported an increasing contribution from V−T N2−N energy exchanges (R2) to gas heating with time, which become dominant beyond 6 ms. This behavior is quite comparable to the one plotted in Figure 2 for longer times. The importance of these V−T processes has also been pointed out in a time-dependent modeling study of the energy transfer in a nitrogen discharge.7 However, at the time of that study, the corresponding rate coefficients were not well-known. Therefore, this mechanism was not included in that work. Even if the conditions considered in that work are different (pressure of 100 Torr, electron density equal to 1012 cm−3, and E/Ng ranging from 30 to 90 Td) from the ones adopted here, let us note that the simulations presented in that study also report a relatively important contribution to gas heating from quenching processes (R3) and (R4) before a time of 0.1 ms. For times up to 1 ms, these two processes remain significant for gas heating, and an increase of the contribution of V−V and V−T N2−N2 collisions is observed. Moreover, another study of gas heating mechanisms dealing with an arc plasma torch at atmospheric pressure with nitrogen,9 shows that the gas heating processes (R1) and (R2) are also the most important ones, followed by the pooling processes (R3) and (R4) at times longer than 0.1 ms. That study shows a more important contribution to gas heating for earlier times than in our calculations, which is due to the fact that V−T relaxation time scales inversely proportional with gas pressure.55 Hence, for low degrees of ionization,10 of the order of 10−6, gas heating in nitrogen discharges is determined by V− V and V−T energy exchanges between vibrationally excited molecules and molecules and atoms. In order to get further insight on the role of these two processes on gas heating, we have carried out sensitivity tests on their rate coefficients to determine their influence on the values of the gas temperature. Within this purpose, we have increased and decreased the rate coefficients by a factor of 5. With respect to nonresonant V−V collisions, modeling 21191


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To clarify the role of considering these upper limit values for the available energy concerning gas heating processes (R6)− (R8), let us indicate (not shown in a figure) that the gas heating rate from process (R6) increases from ∼45 000 to 75 000 K/s between 1 and 10 ms. Within the same time range, the combined effect of both processes (R7) and (R8) varies from ∼3000 to 11 000 K/s. Note that these two gas heating processes only start to be relevant after 1 ms, following the kinetics of N(4S). At a time of ∼1 ms, gas heating rates from the quenching of N2(B) by N2 (R6) are about 3 times larger than those resulting from the pooling mechanisms, as shown in Figure 2. At a time of ∼10 ms and beyond, the gas heating rate of (R6) is larger than the contributions from (R7) and (R8), which in turn are larger than the ones for pooling reactions, reported in Figure 2. This behavior is qualitatively in agreement with the one observed for these gas heating mechanisms even if the experimental conditions are different.12 An overall picture is depicted in Figure 5, where we plot the fractional power transferred to gas heating ηR with respect to

only spent on gas heating without any energy transferred to vibrational excitation. From the previous discussion, it is of major importance to analyze the impact of considering these values for the energy release to gas heating involving the quenching processes (R6)− (R8) on the present calculations. Note that in our reference modeling results we have considered that these processes do not contribute to gas heating, and that the corresponding available energy is entirely transferred to vibrational excitation. Figure 4 reports this analysis by comparing the time-dependent

Figure 4. Temporal variation of the radially averaged gas temperature Tg under the same conditions as in Figure 1, calculated using different values from the energy release to gas temperature from reaction (R6) N2(B) + N2 → N2(A) + N2: (A) 0 eV (reference values), (B) 0.18 eV, (C) 0.36 eV, and (D) 1.18 eV plus a full contribution of the available energy from the quenching processes (R7) N(2D) + N2 → N(4S) + N2 and (R8) N(2P) + N2 → N(4S) + N2 energy to gas heating (see text).

evolution of the radially averaged value of the gas temperature Tg of our reference model (curve A of Figure 4 and already plotted in Figure 1) to modeling results when the energy transferred to gas heating from process (R6) N2(B) + N2 → N2(A) + N2(X) is 0.18 eV8 (Figure 4B) and twice this value, 0.36 eV (Figure 4C). Figure 4D also shows modeling results when all the available energy from quenching processes (R6)− (R8) goes to gas heating. Note that this last assumption corresponds to an upper limit for the values of the gas temperature. Figure 4 clearly illustrates that gas heating is accelerated with an additional contribution from processes (R6)−(R8) involving the electronically excited state N2(B), as well as metastables N(2D) and N(2P). With respect to the quenching of N2(B) by N2 molecules (R6), we may observe during the first millisecond a slight increase of the gas temperature, with respect to the reference values, when the energy spent on gas heating from this processes is 0.18 eV. For this situation, the values Tg are about 10 K larger than those obtained as reference values in the time interval 1−10 ms. This difference diminishes to ∼5 K for longer times. A similar behavior may be observed when a value of 0.36 eV is assumed for the energy release into gas heating from process (R6), leading to slightly larger values for Tg. Furthermore, Figure 4 reveals a significant increase in the values of Tg when we consider not only N2(B) + N2 → N2(A) + N2(X), but also additional heating mechanisms coming from the quenching of the atomic metastables N(2D) and N(2P) by N2, where the available energy is completely spent on gas heating. Under these conditions, gas heating is faster and the values of Tg increase about 30−40 K after the first millisecond.

Figure 5. Fractional discharge power ηR transferred to gas heating for the same case as in Figure 4.

the discharge power for the same conditions reported in Figure 4. As in our previous work,3 this parameter is determined by taking into account the total gas heating rates, which is divided by the electron power absorbed from the electric field, obtained from the electron Boltzmann equation. Figure 5 shows then the time-dependent variation of ηR for the conditions reported in Figures 1 and 2 (reference case), as well as for different hypotheses for the available energies resulting from gas heating processes (R6)−(R8). Figure 5 reveals that ηR is low for the reference modeling values, having values always lower than 2%. When the available energy to gas heating concerning process (R6) is 0.18 and 0.36 eV (curves B and C of Figure 4) there is, as expected, an increase of the fractional power to gas heating. Under these assumptions, ηR increases up to ∼4%. Therefore, we may observe that, in the case of pure N2, the energy transferred to gas heating from this process plays a certain role, causing an increase in the gas temperature of ∼10 K, as pointed out above. In what concerns the upper limit case where it is assumed that all the energy in excess regarding processes (R6)−(R8) is spent on gas heating (see curve D), Figure 5 reports a value of the fractional power transferred to gas heating that increases with time up to almost 10%. This case corresponds to a significant enhancement of the power spent on gas heating 21192


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These modeling results were obtained from the solutions to the electron Boltzmann equation for a reduced electric field of 100 Td and an input value of the N2 vibrational temperature Tv equal to 5000 K. We have carried out calculations for [O2]/Ng < 80%. The study of N2−O2 mixtures with larger relative concentrations of oxygen is out of the scope of this work. Nevertheless, we discuss below some qualitative aspects that may occur on gas heating as [O2]/Ng → 100%. Figure 6 describes some important points in a N2−O2 plasma discharge regarding the effect of adding oxygen into the mixture. Figure 6 shows that, for [O2]/Ng < 40%, the dominant energy loss channel for electrons comes from the vibrational excitation of N2 through inelastic collisions, PIV(N2). At the same time, the well-known effect of superelastic collisions involving N2 molecules, PSV(N2), plays an important role, being an energy source for electrons. With the diminution of N2 introduced in the discharge, these two processes become less important. Nonetheless, they correspond respectively to about 20 and 10% of the discharge power for the larger values of [O2]/Ng considered here. Under this situation, the electron power spent in O2 vibrational excitation, PIV(O2), is still relatively low, being smaller than the electron energy losses associated with N2 vibrational excitation. This behavior is a direct consequence of the larger magnitude and shape of the electron cross sections for vibrational excitation of N2 with respect to those of O2.30 Figure 6 further shows an important increase of the electron power lost in O2 electronic excitation, dissociation, and ionization, PIY(O2), as O2 is added into the mixture. This behavior is accompanied by an expected decrease in the electron power losses on excitation, dissociation, and ionization of N2 molecules. Figure 6 also reveals that the discharge power increases smoothly with [O2]/Ng. This effect results from the difference in the magnitude of the electron cross sections for nitrogen and oxygen, particularly due to the strong maximum for vibrational excitation in N2, at electron energies around 2 eV, which causes a sharp decrease in the EEDF at that point.21 Therefore, an increase in [O2]/Ng at constant E/Ng produces an enhancement of the high energy region of the EEDF. It is important to mention that we have also included in our calculations the power losses in elastic collisions with N2 and O2 molecules, as well as in their rotational excitation. However, since these values are very small (∼10−11 and 10−13 eV s−1 cm3 in elastic collisions with N2 and O2, respectively, and about 1 order of magnitude lower for rotational excitation), they are not plotted in Figure 6. For the same reason, the power absorbed by the electrons through superelastic e−V collisions with O2, being of the order of 10−13 eV s−1 cm3, is not shown in Figure 6. Note also that the power gained by the electrons from the electric field and superelastic collisions with N2 and O2 molecules is exactly compensated by the total electron power lost by the electrons in all the inelastic processes (vibrational, electronic excitation, dissociation, and ionization) reported in Figure 6, in addition to the electron power losses associated with elastic collisions and rotational excitation. The key points described in Figure 6 concerning electron collisions in N2−O2 plasmas are important to understanding the behavior of gas heating mechanisms for different fractional compositions of this mixture. In fact, we may observe in Figure 6 that the increase of the electron power losses in the electronic excitation of oxygen molecules with [O2]/Ng is more important than the corresponding enhancement of the discharge power. Therefore, processes coming from or involving electronically

from dissociation and electronic excitation of molecules with respect to the reference case considered for this work. Finally, it is worth mentioning that the predicted values for the gas temperature reported in Figure 1 for the reference case, as well as the ones shown in Figure 3, when additional gas heating is considered from other possible heating mechanisms, may seem relatively low. As a matter of fact, in Figures 1 and 3 modeling simulations show that the Tg is lower than 400 K until a time of 100 ms. Note that we are considering a situation with heat conduction to the wall. Consequently, the value of the tube radius R plays an important role in the calculations due to the term 8λg(Tg − Tw)/R2 in eq 1. As discussed in a previous paper,3 larger values for R lead to smaller cooling rates. For instance, in the reference situation of this work, reported in Figure 1, with R = 0.8 cm, we obtain Tg = 315 and 343 K, respectively, at times t of 10 and 100 ms. However, if we consider a tube radius of 1.2 cm in our calculations (not shown in a figure), modeling shows that the radially averaged values predicted for the gas temperature Tg increase up to 351 and 390 K for t = 10 and 100 ms. On the other hand, with the exception of ref 10, other research works in pure N2 have considered plasma discharges where the pressure is higher than ∼1 Torr. As shown in our previous modeling work3 for the gas temperature in air plasmas, an increase of the pressure causes an increase of the total power transfer to gas heating per unit volume. A similar behavior can be observed in pure N2 plasmas. Our calculations predict indeed that the gas temperature Tg is 385 and 505 K at t = 10 and 100 ms, respectively, if the value of the pressure changes from 1 to 10 Torr, keeping all the other input values as for the reference case. This effect is even more pronounced for a pressure of 20 Torr, where we obtain Tg = 514 and 589 K, respectively, for times of 10 and 100 ms. 3.2. Gas Heating in N2−O2 Plasmas. We start the analysis of gas heating in N2−O2 discharges by plotting in Figure 6 the electron power transferred per electron at unit gas density as a function of the relative concentration of oxygen molecules [O2]/Ng in the mixture.

Figure 6. Electron power transferred per electron at unit gas density as a function of [O2]/Ng when E/Ng = 100 Td for a N2 vibrational temperature Tv of 5000 K, with the following notation: upper full curve, discharge power, corresponding to the power gained by the electrons from the electric field; PIV(N2), electron power lost in N2 vibrational excitation; PSV(N2), power received by the electrons from e−V superelastic collisions with N2; PY(N2) and PY(O2), electron power lost in the electronic excitation, dissociation, and ionization of N2 and O2; PIV(O2), electron power lost in O2 vibrational excitation. 21193


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Figure 7 shows an important acceleration of gas heating as [O2]/Ng increases. Moreover, Figure 7 reveals that the values of Tg become larger with the fractional concentration of oxygen. In the first milliseconds, the gas temperature for [O2]/Ng = 20% is larger than in the situation of pure N2 by about 50 K. For longer times, this difference increases up to almost 80 K. Figure 7 further shows that when [O2]/Ng = 50% the gas temperature increases faster than in the case of [O2]/Ng = 20%. For the larger value of 80% considered in this work for the fractional concentration of O2, a significant enhancement of the gas temperature Tg is observed. The predicted values for this parameter are found to be ∼100 K larger than in the situation for [O2]/Ng = 50% after a time of 0.5 ms. These results can be explained from the analysis of the heating rates of the most important gas heating mechanisms when E/Ng = 100 Td, ne = 1010 cm−3, and p = 2 Torr for [O2]/ Ng = 20, 50, and 80%, which are plotted in parts a, b, and c, respectively, of Figure 8. The results reported in Figure 8a are quite similar to those we have obtained in our previous studies on air plasmas.2 Before the first millisecond, gas heating comes mainly from processes (R9) e + O2 → e + O(3P) + O(3P) and (R10) e + O2 → e + O( 3P) + O( 1 D) and by the quenching of N2 electronically excited states by oxygen molecules, N2(B, a′, a, w) + O → N2 + O + O. Note that the sum of the heating rates regarding these processes is larger than the ones reported in Figure 2 for pure N2. For this reason, gas heating is accelerated with the addition of O2 into the mixture even for earlier times. At times longer than 1 ms, vibrational excitation of N2 and plasma chemistry start to contribute significantly to gas heating through the energy exchanges in V−T N2−O collisions and from the exothermic reaction reaction (R17) N + NO → N2(X, v ∼ 3) + O. Figure 8a also shows a relatively important contribution to gas heating from the recombination of oxygen atoms at the wall and through process (R16) N2(X, v ≥ 13) + O → NO(X) + N(4S). Other gas heating processes, such as N2(A) + O → NO(X) + N(2D) and (R21) O(1D) + O2 → O(3P) + O2(a), have heating rates slightly smaller than 104 K/s. Thereby, they are not plotted in Figure 8. It is also important to mention that the contribution of V−T N2−N collisions to gas heating in the different N2−O2 mixtures considered in this work is always vanishingly small. In what concerns nonresonant collisions in V−V N2−N2 collisions, their relative contribution to gas heating in the mixture is always less than 2%; nonetheless, let us say that our calculations predict, as in the case of pure N2, a slight contribution to gas cooling around a time of ∼1 ms. With respect to nonresonant energy exchanges in collisions in V−V N2−O2, the contribution of these processes to gas heating is always negligible, even for the larger values of [O2]/Ng considered here. Note that other relevant gas heating mechanisms in pure N2, such as the pooling reactions or the diffusion of metastables N2(A) and N2(a′), are no longer significant in the mixture. This is due to the important role played by the quenching of these two excited species by atomic or molecular oxygen.25,26 The population of oxygen atoms is larger than that of nitrogen atoms by about 1 order of magnitude in air plasmas and other N2−O2 mixtures.25,56 Therefore, besides the role of the corresponding rate constants, V−T N2−O collisions involving oxygen atoms are much more efficient for gas heating in N2−O2 mixtures than the contribution of V−T N2−N processes. For the same reason, the reassociation of oxygen atoms at the wall and subsequent energy release to gas heating

excited states of O2 are expected to play an important role in gas heating as the amount of molecular oxygen introduced in the discharge increases. This effect has been observed in the study of gas heating in air-like plasmas, corresponding to N2−20% O2 mixtures, where electron impact with O2 molecules through predissociative states, via processes (R9) e + O2 → e + O(3P) + O(3P) and (R10) e + O2 → e + O(3P) + O(1D), reported in Table 2, are found to be very important to explain the so-called fast gas heating phenomena.5,6 A similar effect is observed for the quenching of N2 electronically excited states N2(B), N2(a′), N2(a), and N2(w) by oxygen molecules, through reactions (R11)−(R14) (see Table 2), represented here by N2(B, a′, a, w) + O2 → N2 + O + O on the nanosecond5,6,52 and microsecond time scales.2,3 Furthermore, when vibrational excitation of N2 molecules plays an important role, as in the present case involving times longer than 1 ms, an increase in the density of oxygen atoms leads to energy exchanges in V−T N2−O collisions (R15) which are very efficient for gas heating in air plasmas.2,3,12 At the same time, recombination of oxygen atoms at the wall (R18) may be also an important gas heating source.2 It follows from this discussion that gas heating from additional processes resulting from the introduction of oxygen is quite efficient even for [O2]/Ng = 20%. For this fractional oxygen concentration, the electron power involved in vibrational and electronic excitation of N2 is still larger than the contribution from electron collisions with O2 depicted in Figure 6. Owing to the fact that the electron power transferred to the electronic excitation of O2, dissociation, and ionization (curve PIY(O2) in Figure 6) increases about 1 order of magnitude when [O2]/Ng varies from 20 to 80%, important effects on gas heating are expected to occur within this range for the fractional composition on the mixture. Figure 7 plots the predicted values for the radially average values for the temporal evolution of the gas temperature Tg

Figure 7. Temporal evolution of the radially averaged gas temperature Tg when E/Ng = 100 Td and ne = 1010 cm−3 in a plasma discharge in N2 and in the mixtures 80% N2−20% O2; 50% N2−50% O2, and 20% N2−80% O2, with p = 2 Torr and R = 0.8 cm.

when the relative concentration of oxygen molecules on the mixture [O2]/Ng increases up to 80% for the fixed values E/Ng = 100 Td and ne = 1010 cm−3, at a constant pressure of 2 Torr and for a tube of radius 0.8 cm. As in section 3.1, we have considered for these calculations a constant value for the wall temperature equal to 300 K. Figure 7 also shows the calculated values for the case of pure N2, already reported in Figure 1. 21194


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Figure 8. Temporal evolution of the heating rates of the most important gas heating mechanisms in a discharge in (a) 80% N2−20% O2, (b) 50% N2−50% O2, and (c) 20% N2−80% O2 when p = 2 Torr, E/Ng = 100 Td, and ne = 1010 cm−3, as follows: e + O2 → e + O2* → e + O(3P) + O(3P) (R9) and ... → e + O(3P) + O(1D) (R10); N2(B, a′, a, w) + O2 →N2* + O2 →N2(X) + O + O (R11)−(R14); V−T N2−O processes (R15); N2(X, v ≥ 13) + O → NO(X) + N(4S) (R16); N(4S) + NO(X) → N2(X, v ∼ 3) + O (R17); O + wall → (1/2)O2 (R18); deactivation of metastable O2(b) at the wall (R19).

is much more important than the recombination of atomic nitrogen. Figure 8a indicates that the total gas heating rate at 100 ms is about ∼2 × 105 K/s, being almost 1 order of magnitude larger than the values obtained for pure N2 (see Figure 2). Therefore, energy transfer to gas heating is more efficient than in pure N2 for the same values of E/Ng = 100 Td and ne = 1010 cm−3. Note that under these conditions, as depicted in Figure 6, the discharge power remains essentially the same, while an important amount of electron power is spent in inelastic collisions (electronic excitation, dissociation, and ionization) with O2. The concentration of oxygen atoms has been found to increase with the fractional concentration of molecular oxygen in the mixture almost to a situation of a discharge in pure O2.18,21 Therefore, the role of O(3P) becomes more important in the overall kinetics. In what concerns gas heating, this effect is clearly seen in Figure 8b,c, where the heating channels (R15) V−T N2−O collisions and (R18) O + wall → (1/2)O2 involving this species are dominant. As can be seen in Figure 8b, V−T N2−O collisions are still the most important gas heating process for longer times when [O2]/Ng = 50%. Note that for this situation, as shown in Figure 6, the electron energy spent in N2 vibrational excitation continues to be significant. At the same time, the electron power transferred to the electronically excited states of O2 starts to be the most important electron energy loss channel. Figure 8b further shows that the total gas heating rate is ∼3 × 105 K/s for longer times. Since this value is only slightly larger than the one obtained for [O2]/Ng = 20%, the predicted increase for Tg

reported in Figure 7 when the fractional concentration of O2 increases to 50% is not very significant. In contrast, Figure 8c indicates that, for the larger value of [O2]/Ng = 80% considered here, the total gas heating rate is ∼5 × 105 K/s, causing an important increase in the magnitude of the gas temperature (see Figure 7). For this value of the fractional concentration of O2, gas heating comes mainly from the recombination of oxygen atoms at the wall. Besides a smaller contribution from V−T N2−O processes, Figure 8c also reveals that the diffusion of the electronically excited O2(b) to the wall and subsequent deactivation plays a relatively important role in gas heating. Note that the discharge power is mostly channeled into inelastic collisions with O2, as shown in Figure 6. This behavior leads to an increase of the gas heating rate concerning processes (R9) e + O2 → e + O(3P) + O(3P) and (R10) e + O2 → e + O(3P) + O(1D) (with a relative percentage contribution to gas heating of ∼5%) and to a negligible contribution to gas heating from reactions N2(B, a′, a, w) + O → N2 + O + O. The present discussion shows the crucial role played by V−T N2−O collisions on gas heating. Therefore, it is pertinent to check the influence of their rate coefficients on the values of Tg, as we did in section 3.1 for V−T N2−N processes. Increasing the rate coefficient of V−T N2−O collisions by a factor of 5 causes a relative increase of Tg of the order of 20% for [O2]/Ng = 20 and 50% for the longer times. For [O2]/Ng = 80%, the relative increase of Tg is lower, being ∼11%. This result is a consequence of the lower contributions to gas heating from V− T processes for this higher value of the fractional concentration of O2 in the mixture, as shown in Figure 8c. If the rate coefficients are decreased by a factor of 5, the values of Tg are 21195


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Figure 9. Temporal evolution of the VDF [N2(X,v)]/[N2] in a plasma discharge with a pressure p = 2 Torr in a tube radius equal to 0.8 cm for a reduced electric field E/Ng of 100 Td and an electron density ne equal to 1010 cm−3 for the following mixtures: (a) 80% N2−20% O2, (b) 50% N2− 50% O2, and (c) 20% N2−80% O2.

considered a fixed value for the reduced electric field of 100 Td, but for a very low value for the electron density, equal to 1 cm−3. Owing to the relevance of vibrational collision processes involving N2, the temporal behavior of the VDF [N2(X,v)]/ [N2] is shown in Figure 9 for the different N2−O2 mixture compositions considered here. We may observe an increase in the excitation of the lower vibrational levels for the earlier time, followed by an important depletion of the tail of the VDF as a result of very efficient V−T N2−O collisions. This effect limits the appearance of a plateau, as shown in Figure 3. In order to complete the present analysis, it is interesting to connect the information provided by Figure 6 with the results reported in Figure 7, through the analysis of the fractional power transferred to gas heating, as we did previously for pure N2 in Figure 5. The temporal evolution of this parameter is depicted in Figure 10 for [O2]/Ng= 20, 50, and 80%. We have also plotted in Figure 10 the obtained results for the case of pure N2, already reported in Figure 2. Figure 10 clearly shows an increase of ηR with the fractional concentration of oxygen in the mixture. At earlier times, the fractional discharge power transferred to gas heating is slightly lower than 5% for the fractional concentrations of oxygen equal to 20 and 50%. After the first milliseconds, ηR increases to ∼15 and ∼20% when [O2]/Ng = 20 and 50%, respectively. The value of ηR becomes larger for [O2]/Ng = 80%, varying from ∼6 to ∼30% within the time scale considered in our simulations. This behavior is a consequence of the competing role of the most important gas heating mechanisms with [O2 ]/N g previously discussed. As a matter of fact, the visible enhance-

found to decrease by about 7% when [O2]/Ng = 20 and 50% and by ∼3% for [O2]/Ng = 80%. Hence, as in the case of pure N2, we see the strong dependence of the gas temperature on these processes, particularly when this parameter increases. Other gas heating mechanisms with lower relative percentage contributions (and therefore not represented in Figure 8c) involve collisions of O(1D) atoms with N2 and O2 molecules through reactions (R20) O(1D) + N2 → O(3P) + N2 and (R21) O(1D) + O2 → O(3P) + O2, with both having the same value of energy release to gas heating (see Table 2). However, due to the difference in the corresponding rate coefficients, the first one has a relative contribution to gas heating of about 3% for longer times, while reaction (R21) has a slightly lower value of ∼1%. A similar relative contribution is found for reaction (R22) O(1D) + O2 → O(3P) + O2(b). Moreover, the contribution to gas heating from the exothermic reaction (R17) N + NO → N2(X, v ∼ 3) + O becomes negligible as a result of the significant decrease in the populations of N and NO for [O2]/Ng = 80%.18,21 An increase of the gas heating rates with the oxygen content (up to 20%) in the mixture has also been observed for a pressure of 0.4 Torr and current densities of 0.8 and 1 A cm−2.5,6 It was shown in those works that the total gas heating rate increases by about a factor of 4. Although concerning different experimental conditions, this result is in line with the one we obtained here from the comparison between Figures 2 and 8a. In another modeling study regarding gas heating at atmospheric pressure, the gas heating rate was found to increase in general with the concentration of O2 until a time of ∼10 s.19 The calculations presented in that paper also 21196


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from the energy exchanges involving V−T O2−O processes and nonresonant O2−O2 collisions. However, such a study is far beyond the scope of the present paper. Nonetheless, let us point out a qualitative analysis on the possible effects that might occur in pure O2. A relatively recent work on V−T O2−O collisions57 shows that the corresponding rate coefficients remain almost constant (∼10−11 cm3 s−1) for a gas temperature ranging between 300 and 1000 K. The contribution of these processes to gas heating is expected to be small since an increase of the gas temperature will not lead to an increase of these coefficients, in opposition to what we have observed during this work for N2−N and N2− O collisions. Furthermore, recent experimental measurements in low-pressure inductively coupled plasmas in pure oxygen reveal a very long plateau after an initial strong depopulation of the first O2(w) vibrational levels.58 Ongoing modeling work by Annusova (private communications) involving this experiment condition have shown that this effect does not seem to have any noticeable effect in the plasma chemistry and that the gas heating rates concerning V−T O2−O collisions are at least 1 order of magnitude lower than the recombination of oxygen atoms at the wall. In the situation of pure O2 or with [O2]/Ng ≲ 1, the gas heating rates of process (R18) O + wall → (1/2)O2 will probably be larger than the ones we obtained for [O2]/Ng = 80%, while the contribution to gas heating from the energy stored in the vibrational excitation is most likely small. At the same time, diffusion of electronically excited states, namely O2(b), and subsequent energy release to the gas may also become more important. This discussion has been dedicated essentially to the analysis of the competing role between the different gas heating processes when the mixture composition varies from pure N2 up to [O2]/Ng = 80%. Within this goal, we have focused the present study on the time-dependent variation of the gas temperature, the corresponding gas heating rates, and discharge power for different fractional concentrations of O2, while the values of the reduced electric field and electron density are kept constant. However, in a more realistic situation, these two parameters are not imposed. In fact, the discharge sustaining electric field may be obtained from the balance between the total rate of ionization, including stepwise ionization by electron impact and associative ionization, and the total rate of electron losses. Moreover, the electron density may be determined for a given current I through the equation I/πR2 = enevd (where e denotes the absolute value of the electron charge and vd is the electron drift velocity calculated from the Boltzmann equation). Such a complete self-consistent analysis has been already carried out for air plasmas2 and is currently underway for N2 plasmas in order to explain time-resolved experimental measurements for different pulsed discharge currents ranging from 25 to 150 mA.59 These ongoing calculations also include the detailed description of the energy transfer during the pulse by calculating the energy carried by each of the active species as a function of time. Nonetheless, we can advance here a discussion on the influence of the mixture composition by considering in the present calculations the experimental values of the reduced electric field and electron density measured for a N2−O2 glow discharge18 with the same conditions of pressure and tube radius adopted here.

Figure 10. Fractional discharge power ηR transferred to gas heating in N2, 80% N2−20% O2; 50% N2−50% O2, and 20% N2−80% O2 plasmas with p = 2 Torr and R = 0.8 cm when the reduced electric field E/Ng is 100 Td and for an electron density of 1010 cm−3.

ment of the total gas heating rates with the concentration of oxygen in the mixture is not followed by the smooth increase observed in Figure 5 for the discharge power. In the conditions of our work, gas heating in pure N2 comes mainly from the energy stored in the vibrational excitation of nitrogen. As the amount of O2 is increased in the mixture, the electron energy transferred and stored in the electronically excited states starts to play an important and increasing role in gas heating, starting from the dissociation of oxygen molecules by electron impact (R9) e + O2 → e + O(3P) + O(3P) and (R10) e + O2 → e + O(3P) + O(1D) and reactions (R11)−(R14) N2(B, a′, a, w) + O2 → N2 + O + O. Subsequently, the combined effect of vibrational excitation of N2 and oxygen atoms, whose populations increase with time, may lead to very important contributions to gas heating from V−T N2−O energy exchanges. In parallel, for larger values of [O2]/Ng the role of oxygen atoms becomes more important and their reassociation at the wall constitutes an additional relevant gas heating channel. An enhancement of the fractional discharge power transferred to gas heating with the relative concentration of O2 in the mixture has been also predicted in other works. It was shown that the fractional power ηR increases from 30 to 50% when [O2]/Ng varies from 2 to 50%.20 However, the conditions of the present work are quite different from the ones in that work, where a pressure of 300 Torr and atmospheric pressure, electron densities of 1014 and 1015 cm−3, and a reduced electric field of 1000 Td were considered.20 For those conditions, the most important gas heating mechanisms were found to be the quenching of electronically excited states by O2 molecules and electron−ion recombination. Note that, under the conditions of our work, the contribution to gas heating from this latter process is vanishingly small, in particular due to the relatively low value of 100 Td and electron density of 1010 cm−3 considered in our calculations. We have considered in this work an upper limit for the fractional concentration of oxygen equal to 80% for the modeling simulations. Under these conditions, as indicated by Figure 6, the electron energy losses involving the vibrational excitation of oxygen molecules is still relatively small in comparison to other inelastic processes. A detailed study concerning larger values of [O2]/Ng > 80% or even the situation of pure O2 would require a more detailed description of O2(X,v) kinetics, including the contribution to gas heating 21197


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this effect on the simulated results for the gas temperature, we have considered a value for γ0 5 times larger than the reference value of 2 × 10−3 when [O2]/Ng = 80%. The corresponding results show that the gas temperature at the axis decreases from 604 to 572 K. Second, we have considered in the present calculations a value of 0.5 for the thermal accommodation coefficient.2,10 As indicated in our previous calculations,2 the impact of this parameter is rather small in situations where the role of the wall in gas heating is not dominant. However, that is not the case for a N2−O2 mixture composition with [O2]/Ng = 80%, where besides the major contribution to gas heating from the recombination of oxygen atoms at the wall (see Figure 8c) the diffusion of the electronically excited state O2(b) to the wall and subsequent energy transfer to the gas have also an important role. In order to conclude the present discussion, we have analyzed the sensitivity of our model predictions for this case, when [O2]/Ng = 80%, by considering the thermal accommodation coefficient β equal to 0.7 and 0.9. This means respectively that 70 and 90% of the energy corresponding associated with the recombination of oxygen atoms and with the diffusion of O2(b) are dissipated on the wall, while the remaining energy percentage is spent on gas heating. Modeling reveals that, under these assumptions, the gas temperature decreases to 576 and 546 K, respectively. Therefore, we can verify that a modification in the values of the two input parameters γ0 and β are responsible for an additional decrease of the gas temperature when [O2]/Ng = 80%. This effect may constitute a possible explanation for the values of the gas temperature measured for the larger values of the fractional concentration of O2.18

Considering, for example, the situation concerning a current of 80 mA, the experimental values of E/Ng were found to be 70, 90, 78, and 66 Td, respectively, for [O2]/Ng = 0, 20, 50 and 80%.18 Under these conditions, the electron density remains almost unchanged, being 2 × 1010 cm−3 for pure N2, and decreasing to a constant value of 1.8 × 1010 cm−3 for [O2]/Ng > 10%. Moreover, these experimental measurements include also the experimental values of the wall temperature, Tw, which were found to be 367, 385, 367, and 360 K for [O2]/Ng = 0, 20, 50 and 80%, respectively.60 Using these different values of E/Ng, ne, and Tw as input parameters in the solutions to the thermal balance equation, eq 1, for the corresponding N2−O2 mixture compositions, we obtain for times longer than 1 s (when steady-state conditions are achieved) values for the gas temperature at the axis of the tube, T0, equal to 555, 657, 643, and 604 K, respectively, when [O2]/Ng = 0, 20, 50, and 80%. These results are qualitatively in line with the ones measured for the gas temperature in the central region of the tube, which are approximately 800, 900, 800, and 700 K for these fractional oxygen concentrations.18 As O2 is added into the mixture, an enhancement of the gas temperature up to [O2]/Ng ∼ 20% is observed. For progressively larger values of the fractional concentration of molecular oxygen, the gas temperature decreases. In addition to the gas heating mechanism discussed in this section, this behavior is essentially due to a combined effect of the values of E/Ng and Tw for different N2−O2 mixture compositions. In fact, for the above-mentioned values of the maintenance reduced electric field when [O2]/Ng = 0, 20, 50, and 80%, the corresponding values of the discharge power per electron at unit gas density are respectively 55, 92, 85, and 71 (in 10−10 eV cm3 s−1 units). In parallel, as we have shown in our previous publication,2 a variation in the value of Tw leads to a similar variation of the gas temperature, as a direct consequence of the cooling term which depends on the difference (Tg − Tw). Thus, for the situation corresponding to [O2]/Ng = 20%, the discharge power is larger and the cooling term is smaller, explaining the higher value obtained for the gas temperature for this fractional concentration of O2 when we use the measured parameters E/Ng and ne. The predicted values are however lower than the measured ones. Note that these experimental measurements, obtained by means of optical emission spectroscopy of the second positive system of N2, N2(C3Πu → N2(B3Πg), are determined from the rotational distribution of this emission band, having in general uncertainties of the order of 10−20%.25,60 Therefore, these calculations predict values for the gas temperature that are near the lower limit of the error bar. For the case of [O2]/Ng = 80%, the experimental value of the gas temperature is slightly lower than in the case of pure N2.18 This relative difference, not predicted by the results described above for the situation of 80 mA, where we obtained 555 and 604 K, respectively, for pure N2 and [O2]/Ng = 80%, may have two reasons. First, the probability of losses of nitrogen and oxygen atoms at the wall may change with the mixture composition.61 In particular, with respect to the recombination of oxygen atoms at the wall, shown here to be the dominant gas heating mechanism for [O2]/Ng = 80% (see Figure 8c), the corresponding coefficient γ0 can increase by a factor of 2−5 as [O2]/Ng → 100%.49 This kind of effect is not taken into account in our modeling simulations since we have considered constant recombination coefficients independently of the plasma composition. In order to know the possible impact of

4. CONCLUSIONS We have presented throughout this paper a modeling study of the time-dependent variation of the gas temperature in N2 and N2−O2 plasma discharges with a systematic analysis of the corresponding heating mechanisms. This analysis includes the fractional discharge power transferred to gas heating. Modeling calculations have shown that, in both N2 and N2− O2 plasmas, gas heating results initially from energy transfer to the translational mode from electronically excited states, usually designated by fast gas heating, followed by an additional contribution from vibrationally excited molecules for times longer than 1 ms. In what concerns discharges in pure N2, the present study shows that the effect of fast gas heating has its origin in the pooling reactions N2(A) + N2(A) → N2(B) + N2(X) and N2(A) + N2(A) → N2(C) + N2(X) in accordance with other works.7,10 Our modeling results further indicate that the quenching of electronically excited species N2(B), N(2D), and N(2P) by N2 molecules may constitute an important extra gas heating source if the corresponding available energy is mainly released into the translational mode, as it was recently assumed.12 The contribution of reaction N2(B) + N2 → N2(A) + N2(X) when intermediate fractions of its energy are transferred to gas heating is also studied here. The present study shows that an important energy transfer from the vibrational to the translational modes of N2 occurs after a few milliseconds. This effect begins with the contribution of nonresonant V−V collisions between nitrogen molecules and continues with very efficient energy exchanges in V−T N2−N collisions. Our results show that the fractional 21198



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discharge power transferred to gas heating in pure N2 is of the order of 2%. This value is found to increase up to 10% if all the available energy from the quenching processes N2(B) + N2→ N2(A) + N2(X), N(2D) + N2→ N(4S) + N2, and N(2P) + N2→ N(4S) + N2 is spent on gas heating. In any case, a very significant fraction of the available energy is stored in active species, such as N2(X,v), N2*, and N atoms. With respect to N2−O2 mixtures, the most important gas heating mechanisms are shown to be significantly different than in the situation of pure N2. In fact, modeling shows that when [O2]/Ng = 20 and 50%, fast gas heating results essentially from the quenching of electronically excited states of N2 by oxygen molecules N2(B, a′, a, w) + O2 → N2 + O + O, as well as through electron impact processes e + O2 → e + O(3P) + O(3P) and e + O2 → e + O(3P) + O(1D). Beyond 1 ms, our calculations indicate additional gas heating from the exothermic reaction N + NO → N2(X, v ∼ 3) + O, V−T N2−O collisions, and recombination of oxygen atoms at the wall. These two latter heating mechanisms continue to be dominant for the higher value of [O2]/Ng = 80% considered here. The present simulations predict an increase of the gas temperature with the fractional concentration of O2 in the mixture if the values of the reduced electric field and electron density are kept constant. The present calculations show that, under these conditions, the discharge power has almost the same value, increasing smoothly with [O2]/Ng, while the total gas heating rate increases significantly with this parameter. This effect is illustrated by an increase of the fractional discharge power transferred to gas heating from 15 to 30% when [O2]/Ng varies from 20 to 80%. This behavior has also been observed in other modeling studies on N2−O2 mixtures,6,19,20 although concerning different experimental conditions. Our modeling predictions further reveal that when we take into account the measured variation of the reduced electric field, electron density, and wall temperature with [O2]/Ng, the gas temperature has its maximum for a fractional concentration of O2 of about 20%, as observed experimentally.18 We show here that this is mainly a consequence of a twofold effect resulting from a higher discharge power and lower gas cooling due to heat conduction to the wall. In summary, the simulations performed by the present model provide the physical grounds for the understanding of gas heating mechanisms in N2 and N2−O2 plasmas and describe correctly, from a qualitative point of view, experimental measurements. However, the accuracy of these theoretical predictions can be improved through the comparison with more experimental data for the gas temperature in different working discharge conditions.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was partially supported by the Portuguese FCT, Fundaçaõ para a Ciência e a Tecnologia, under Project UID/ FIS/50010/2013. 21199


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