Modelling of an Atmospheric–Pressure Air Glow Discharge Operating in High–Gas Temperature Regimes: The Role of the Associative Ionization Reactions Involving Excited Atoms

: A model of a stationary glow ‐ type discharge in atmospheric ‐ pressure air operated in high ‐ gas ‐ temperature regimes (1000 K < T g < 6000 K), with a focus on the role of associative ionization reactions involving N( 2 D, 2 P) ‐ excited atoms, is developed. Thermal dissociation of vibrationally excited nitrogen molecules, as well as electronic excitation from all the vibrational levels of the nitrogen molecules, is also accounted for. The calculations show that the near ‐ threshold associative ionization reaction, N( 2 D) + O( 3 P)  NO + + e, is the major ionization mechanism in air at 2500 K < T g < 4500 K while the ionization of NO molecules by electron impact is the dominant mechanism at lower gas temperatures and the high ‐ threshold associative ionization reaction involving ground ‐ state atoms dominates at higher temperatures. The exoergic associative ionization reaction, N( 2 P) + O( 3 P)  NO + + e, also speeds up the ionization at the highest temperature values. The vibrational excitation of the gas significantly accelerates the production of N 2 (A 3 ∑ u+ ) molecules, which in turn increases the densities of excited N( 2 D, 2 P) atoms. Because the electron energy required for the excitation of the N 2 (A 3 ∑ u+ ) state from N 2 (X 1 ∑ g+ , v) molecules (e.g., 6.2 eV for v = 0) is considerably lower than the ionization energy (9.27 eV) of the NO molecules, the reduced electric field begins to noticeably fall at T g > 2500 K. The calculated plasma parameters agree with the available experimental data.


Introduction
Several kinetic schemes have been proposed in the literature for modelling atmospheric pressure nonequilibrium air discharges, such as streamers [1,2], low-current arc and glow discharges at rest (or in low gas flows) [3][4][5][6], glow discharges in fast gas flows [7][8][9], and high-current pulsed discharges [10][11][12][13][14][15]. It is found that the ionization by electron impact of O2 and N2 molecules dominates at lowgas temperature (Tg is below the range of 1000-2000 K [10,14,15]), while the electron-impact ionization of NO molecules is written as: where R refers to the reactions used in the model. This electron-impact ionization is favored by its low ionization energy of 9.27 eV) and dominates at high gas temperatures [3,4,14,15]. At Tg > 4500 K, the associative ionization in ground-state atomic collisions can be shown as: The ionization rate is independent of the reduced electric field, because the densities of both O( 3 P) and N( 4 S) depend only on Tg [3,4,10,13,15], which becomes the dominant ionization mechanism in air. This transition leads to a sharp drop in the reduced electric field [10].
The high efficiency of associative ionization reactions involving nitrogen and oxygen atoms is explained by a decrease in the ionization threshold at the expense of the energy of the resultant chemical bond. For instance, reaction (R13) has an activation barrier of 2.76 eV for ground-state reactants and is therefore assumed to be the dominant ionization mechanism in high-gas-temperature air, where the reduced electric field is low, (i.e., E/N < 20 Td, where E is the electric field strength and N is the neutral gas density). As the reduced field increases, both vibrational and electronic excitations come into play. As a result, more efficient channels for associative ionization may appear [16]. However, despite the great interest and importance of ionization processes in hot air discharges, the influence of the electronically excited states of reactants on associative ionization processes is yet unclear, as they are not routinely considered in air kinetic models [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. To answer this question, it is necessary to carry out more comprehensive investigations. This paper is an effort to address this issue.
Recently, a kinetic model [17] that considered associative ionization reactions involving N( 2 D, 2 P)-excited atoms was presented. The model was applied to the modelling of nonequilibrium air plasmas created by glow discharges in a fast longitudinal air flow. High-speed gas flows were used to provide sufficient cooling of discharges at high pressures. If the gas residence time in the discharge was small as compared with the vibrational-to-translational (V-T) energy relaxation time, gas heating was almost suppressed and the molecular gas was in a strongly nonequilibrium state [8,9]. This state was characterized by a level of vibrational energy that considerably exceeded its equilibrium value. On the other hand, for discharge conditions that the gas residence time in the discharge was larger than the V-T relaxation time (e.g., ambient air discharges), the molecular gas changed to a state close to the thermodynamically equilibrium one with a higher gas temperature, as in an arc discharge. In this work, a kinetic model [17] was used for simulation of stationary glow discharges operated in ambient air at high-gas-temperature regimes (1000 K < Tg < 6000 K) over a wide current density range. The major difference between the kinetic schemes proposed in the literature for modelling atmospheric-pressure air glow discharges [3][4][5][6][7][8][9] and the one used in this work is that the present model takes into account ionization processes involving excited-state atomic collisions. The results indicated a strong impact of these reactions under the conditions considered. The calculated plasma parameters agreed with the available experimental data.

Modeling of an Atmospheric-Pressure Air Glow Discharge Operated in High-Gas-Temperature Regimes
A nitrogen-oxygen (N2-20% O2) mixture was considered. A kinetic model included processes involving positive irons (i.e., NO + , N2 + , O2 + , and O + ) and negative irons (O -, O2 -, and O3 -), neutral species (i.e., N2(X 1 ∑g + , v), N2(A 3 ∑u + ), N2(B 3 g), N2(a' 1 ∑u -), N2(C 3 u), N( 2 D), N( 4 S), N( 2 P), O( 3 P), O( 1 D), O( 1 S), O2, and NO), and electrons (e). As a distinctive feature, the model incorporated an exoergic associative ionization reaction with the participation of N( 2 P) atoms [18,19] and a near-threshold reaction with the participation of N( 2 D) atoms with a low activation barrier of 0.38 eV [20,21], which can be described as: Note that associative ionization reactions with the participation of N( 2 D, 2 P) metastable atoms were also included in the kinetic model presented in [22] for modelling nonequilibrium discharges in oxygen-nitrogen mixtures. However the present conditions (i.e., high gas temperature (1000 K < Tg < 6000 K) and vibrationally excited air discharges) are quite different to those conditions (i.e., low gas temperature (200 K < Tg < 500 K) and vibrationally unexcited gas) considered in [22]. Since a nonnegligible amount of electron energy lost in vibrational excitation can be recovered by electrons in superelastic collisions, the present model also included the effects of these collisions on the enhancing of the tail of the electron energy distribution function which can lead to an increase of some electronimpact rate coefficients by several orders of magnitude [23]. Furthermore, the thermal dissociation of vibrationally excited nitrogen molecules, as well as the electronic excitation from all the vibrational levels of N2(X 1 ∑g + , v) molecules, was also considered. The main reactions are presented in Appendix (Table A1).
It should be noted that the proposed reaction kinetic scheme did not include three-body reactions for the generation of N4 + and O4 + . These cluster ions play only a dominant role in atmospheric-pressure air discharges at low gas temperature (Tg < 900 K). Gas heating beyond 900 K results in the total decomposition of such cluster ions through the processes, which are reverse to three-body reactions for the generation of N4 + and O4 + , because of the low dissociation energy of these ions [13].
The In stationary high-pressure plasmas, this approximation is usually justified at a long-time scale for diffusive losses, rather than at a relatively short-time scale, to achieve local equilibrium [24]. The rate for the production of the N2(A 3 ∑u + ) state by cascading was assumed to be equal to the sum of the rates for the production by the electron impact of the N2(B 3 g) and N2(C 3 u) states [3,25]. The plasma quasineutrality equation was used to identify the density of NO + -the dominating ion under the conditions considered. The densities of N2(X 1 ∑g + , v) and O2 were given by the condition of conservation of N and O nuclei. A pressure (p = 1 atm) was assumed in the calculations. The model was complemented with the balance equation depicting the mean vibrational energy of nitrogen molecules as well as the mean kinetic energy of the gas. The time of V-T relaxation of N2(X 1 ∑g + , v) on O( 3 P) was taken from [26][27][28]. The "fast" gas heating in chemical reactions [29] was also taken into account for reactions, where energy release was considered accompanied by the exothermic energy value on the right side of the reactions in Table A1 (Appendix). More details in the equations of the model and the treatment of the source terms can be found in [17]. The radial heat loss by thermal conduction-the dominant discharge cooling mechanism under the present conditions [5]-was calculated, considering a parabolic radial profile for the gas temperature in a discharge column with a characteristic radius R of approximately 1 mm (e.g., [30]). The translational thermal conductivity of heavy particles was taken from [31].

Results and Discussion
Low-current discharges in atmospheric-pressure air have been studied in a number of experiments (see recent reviews [32][33][34] and references therein). In most of the experiments, for which current density values j are available [3,[35][36][37][38][39][40][41][42][43][44][45], the current discharge radius was inferred by the emission spectroscopy of the N2 C state. However, the emission discharge and the current discharge radii may be not equal, and both their radii and their ratios depend on experimental conditions [5,46]. As a result, variation of cathode materials (e.g., glow discharges with liquid [35,40,41], plasma [36,37], or metal [3,38,39,[42][43][44][45] cathodes), stabilization modes (with or without tubes and flows as well as swirl versus axial flows [38]), or the electrode gaps and of discharge currents causes a large scatter of discharge parameter data in terms of the current density. In any case, since the C state of N2 is produced by electron-impact excitation, the available current density values are representative of regions with high electron number density. As a consequence, the model output gives estimations of discharge variables evaluated at the center of the discharge column.  Figure 1 shows the electron temperature, the gas temperature, and the vibrational temperature of the N2(X 1 ∑g + , v) molecules versus the discharge current density for radii R of 1 and 2 mm. Experimental data are also shown in Figure 1. The difference between the vibrational and gas temperatures increased substantially with a decrease in j (or in R), due to the strong dependence of the V-T energy relaxation rate on Tg. Considering the uncertainty in the experimentally inferred j values, the results of the calculations for R = 1.0 mm were in fairly good agreement with experimental data [39,41,[43][44][45]. On the other hand, the calculated and measured distributions did not coincide for the larger R value. The electron temperature gently decreased with values around 7000-9000 K, as the discharge current density increased, which is typical of glow-type discharges in molecular gases (e.g., [3,4]). Figure 2 presents the thermal dissociation for vibrationally excited molecules versus the discharge current density calculated for R = 1.0 mm. For comparative purposes, the electron-impact dissociation from all the vibrational levels of N2(X 1 ∑g + , v) is also shown. As it can be seen, differences with several orders of magnitude between both dissociation rates existed across the whole range of current densities, thus suggesting that the electron-impact dissociation process is not important under the considered low-electric-field conditions. Furthermore, thermal dissociation of vibrationally excited molecules dominated the production of N( 4 S) atoms at j < 1 A/cm 2 . At higher j values (Tg > 2000 K), the production of N( 4 S) was mainly controlled by the exchange reaction (R72).  The number densities of various neutral particles versus the discharge current density calculated for R = 1.0 mm are shown in Figure 3. At low current densities (j = 0.1 A/cm 2 ), the concentration of NO was mainly controlled by reactions (R49), (R61), (R62), and (R66). In particular, the production of NO was strongly affected by collisions with N2(A 3 ∑u + ) molecules produced by the electron impact from all the vibrational levels of N2(X 1 ∑g + , v). Therefore, the densities of NO at low j values were greater than those corresponding to local thermodynamic equilibrium at Tg. At higher current densities (j > 1 A/cm 2 ), the densities of NO were governed by reactions with the participation of ground-electronic-state particles reactions (R62) and (R72), balanced by their inverse reactions (R73) and (R61), respectively. These results agreed with the inferences made in [10]. The density of N2(A 3 ∑u + ) molecules was moderately high (5 × 10 17 -8 × 10 18 m -3 ) under the conditions considered (although it was at least one order of magnitude lower than that found in atmospheric-pressure nitrogen discharges [47], because N2(A 3 ∑u + ) molecules were efficiently quenched by oxygen), favored by the vibrational nonequilibrium state of the discharge. Note that the rate of population of the N2(A 3 ∑u + ) (and other states) by electron impact became significantly higher, as the degree of vibrational excitation of N2(X 1 ∑g + , v) molecules increased, which in turn increased the densities of excited N( 2 D) and N( 2 P) atoms via quenching reactions (R49) and (R50), respectively. The presence of N2(A 3 ∑u + ) also caused a rise in the density of O( 3 P) atoms, because O2 molecules were also dissociated via the quenching reaction (R44). Note also the considerable dissociation degree of nitrogen and oxygen molecules in the discharge. As quoted before, the former was mainly decomposed by thermal dissociation enhanced by vibrational nonequilibrium at j < 1 A/cm 2 and by the exchange reaction (R72) for higher j values, while the latter was mainly decomposed by the exchange reaction (R62) at j > 1 A/cm 2 .  Figure 4 depicts various mechanisms for the production and loss of electrons versus the discharge current density calculated for R = 1.0 mm. It can be seen that at low current densities (j < 1 A/cm 2 ), the production of electrons was dominated by the electron-impact ionization of NO molecules via the reaction (R3) [3,4,[8][9][10]. However, an increase in the current density values (and thus in the gas temperature) progressively changed the ionization kinetics of the discharge at j > 1 A/cm 2 (Tg > 2500 K) according to Figure 1, mainly via the following reactions: , (R49) . (R16) The rate coefficient of reaction (R16) was strongly dependent on Tg and independent of E/N. For higher values of j, the increase in the population of N( 2 P) also speeded up the ionization due to the growing importance of the reactions: . (R18) According to the obtained results, the associative ionization reaction involving N( 2 D)-excited atoms is the major ionization mechanisms in air at 2500 K < Tg < 4500 K. These results are quite different from those found in the literature for similar discharge conditions (e.g., [3,4,10]), which indicate that the electron-impact ionization of NO molecules via reaction (R3) is the major ionization mechanism in air at 1000 K < Tg < 4500 K. At j > 10 A/cm 2 (Tg > 4500 K), the dominant ionization mechanism became a high-threshold associative ionization reaction in collisions between groundstate atoms indicated by reaction (R13) [10]. However, the reduced electric field still governed the electron density through electron loss, rather than through ionization [10]. The electron-ionrecombination reaction (R23) dominated the loss of electrons across the whole current density range. Since the gas temperature was rather high, there was negligible attachment under the conditions considered. Figure 5. The electron number density in the plasma versus the discharge current density calculated for R = 1.0 mm.
The calculation and experimental results of the electron number density in plasma versus the discharge current density for R = 1.0 mm [36,37] are shown in Figure 5. Good agreement of the calculation results with the experiment results was observed. There was also an increase of one order of magnitude in the calculated electron number density, when the discharge current density was increased by the same amount [7]. Figure 6 presents the reduced electric field (1 Td  10 -21 V m 2 ) versus the discharge current density (i.e., the E/N-j characteristic curve) calculated for R = 1.0 mm. Solid and dashed lines are the results of calculations with and without the associative ionization reactions (R16) and (R18), respectively. The characteristic curve given by the solid line follows the changes in the ionization mechanisms shown in Figure 4. It was seen that noticeable differences between both curves exist at current density values of 1-10 A/cm 2 or when 2500 < Tg < 4500 K (as shown in Figure 1), i.e., when the associative ionization reaction (R16) becomes the major ionization mechanism in air. It should be noted that the N2(X 1 ∑g + , v) vibrationally excited molecules were not directly involved in the reactions of associative ionization but they significantly accelerated the production of N2(A 3 ∑u + ) molecules in reaction (R9), which in turn increased the densities of excited nitrogen N( 2 D) and N( 2 P) atoms via reactions (R49) and (R50), respectively, participating in the associative ionization reactions (R16) and (R18). Since the electron energy required for the excitation of the N2(A 3 ∑u + ) state from N2(X 1 ∑g + , v) molecules (e.g., 6.2 eV for v = 0) was considerably lower than the ionization energy (9.27 eV) of the NO molecules, the reduced electric field began to noticeably fall at Tg > 2500 K when the change in the dominant ionization mechanism took place (as shown in Figure 4). That is, long before that the high-threshold associative ionization reaction (R13) began to significantly contribute to the charged particle balance at Tg > 4000 K, as was previously reported in the literature (e.g., [3,4,10]).   Figure 7 shows the calculation and experimental results of the reduced electric field versus the discharge current density for R = 1.0 mm [3,37,38,42]. It was seen that the calculated E/N-j characteristic showed fairly good agreement with the experimental data, considering the gas temperature variation in various experiments

Conclusions
A global model of a stationary glow-type discharge in atmospheric-pressure air was developed. The assumptions of temporal and spatial homogeneity inherent in the global model approach were accurately enough under the conditions considered. Relevant kinetic processes in nitrogen-oxygen (N2-20% O2) mixtures at a gas temperature range (1000 K < Tg < 6000 K) were considered. A distinctive feature of the model is that it takes into account associative ionization reactions involving excited atoms, which are not routinely considered in the literature. Thermal dissociation of vibrationally excited nitrogen molecules, as well as electronic excitation from all the vibrational levels of nitrogen molecules was also taken into account. The results of calculations suggested a strong impact of the electronically excited sates of reactants on associative ionization reactions in atomic collisions in hot air. It was shown that the near-threshold associative ionization reaction involving N( 2 D) atoms progressively replaced the ionization of NO molecules by electron impact at Tg > 2500 K, becoming the main ionization mechanism in air up to 4000-4500 K. As a consequence, the reduced electric field of the discharge began to noticeably fall at Tg > 2500 K, when the change in the dominant ionization mechanism took place. The exoergic associative ionization reaction involving N( 2 P) atoms also speeded up the ionization at the highest temperature values. The role of processes concerning vibrationally excited nitrogen molecules played a significant role in the air kinetics under the analyzed conditions. The calculated plasma parameters agreed with the available experimental data. Acknowledgments: E.C. thanks CONICET for his doctoral fellowship. L.P. is a member of CONICET.

Conflicts of Interest:
The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results. Table A1. List of reactions. Te and Tg are in Kelvin. Rate coefficients obtained with the help of the BOLSIG+ code were corrected in order to account for gains in electron energy in superelastic collisions [17]. Calculations were performed using rates for reactions (R22) and (R23) taken from [56].