# State-to-state modeling of NO production by nanosecond electrical discharge in a nitrogen-oxygen mixture at atmospheric pressure

I Armenise, V Laporta, F. Esposito

► To cite this version:

I Armenise, V Laporta, F. Esposito. State-to-state modeling of NO production by nanosecond electrical discharge in a nitrogen-oxygen mixture at atmospheric pressure. Plasma Sources Science and Technology, 2025, 34 (8), pp.085012. 10.1088/1361-6595/adf996. hal-05245085

HAL Id: hal-05245085

https://hal.science/hal-05245085v1

Submitted on 8 Sep 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

**PAPER • OPEN ACCESS**

# State-to-state modeling of NO production by nanosecond electrical discharge in a nitrogen-oxygen mixture at atmospheric pressure

To cite this article: I Armenise et al 2025 *Plasma Sources Sci. Technol.* 34 085012

View the [article online](#) for updates and enhancements.

## You may also like

- *Numerical investigation of stability of low-current needle-to-plane negative corona discharges in air*
  N G C Ferreira, P G C Almeida, A Eivazpour Taher et al.

- *Influence of large biased surfaces on the plasma parameters in the extraction region of a negative hydrogen ion source*
  D Yordanov, D Wünderlich, C Wimmer et al.

- *Temporally and spatially resolved continuum radiation between 600 and 1000 nm from nanosecond discharge in water: implications for understanding the initiation mystery*
  Milan Šimek, Petr Bílek, Garima Arora et al.
# State-to-state modeling of NO production by nanosecond electrical discharge in a nitrogen-oxygen mixture at atmospheric pressure

I Armenise<fnref n="1" />,*, V Laporta<fnref n="1" />,<fnref n="2" /> and F Esposito<fnref n="1" />

E-mail: iole.armenise@cnr.it

Received 2 May 2025, revised 17 July 2025

Accepted for publication 8 August 2025

Published 21 August 2025

<figure><img src="image_3.png" /></figure>

CrossMark

## Abstract

The vibrational non-equilibrium of a N₂/N/O₂/O/NO/e⁻ mixture, primarily composed of nitrogen, is numerically investigated under discharge and post-discharge conditions, with a focus on the formation of NO molecules. The study is conducted using a reduced electric field of 74 Td and a pulse duration of 200 ns. The numerical model accounts for both heavy particle collisions and electron-heavy particle interactions to accurately capture the complex plasma behavior. The main objectives are to examine the roles of electron-vibrational (e-V) and electron-dissociation (e-D) reactions, as well as to evaluate the effects of slight variations in operating temperature (T = 300, 380, and 500 K), gas recirculation, and pulse repetition frequency. The analysis is performed using a state-to-state model implemented in an in-house, zero-dimensional, time-dependent code, coupled with a Boltzmann equation solver. The results underscore the critical importance of including the full set of vibrational resolved e-V and e-D reactions to avoid underestimating the impact of the discharge. The effect of the temperature, in the investigated range, is clear during the post-discharge and especially at its end, where the atomic densities increase with the temperature and the opposite happens to NO. When the gas recirculation is taken into account, during the discharge—post-discharge cycle, a certain percentage of new gas is entered exactly at the initial set conditions causing a delay in the physical-chemical evolution of the gas itself. Finally, the role of the multiple pulses is studied, observing an enhancement of NO formation for sufficiently high repetition rate.

Keywords: air cold plasma, heavy particle collisions, electron collisions, nitrogen fixation, vibrational state-to-state simulations

<figure><img src="image_4.png" /></figure>

Original content from this work may be used under the terms of the [Creative Commons Attribution 4.0 licence](https://creativecommons.org/licenses/by/4.0/). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

<footnote n="1">¹ CNR ISTP (Consiglio Nazionale delle Ricerche, Istituto per la Scienza e Tecnologia dei Plasmi), Sede Secondaria di Bari, Via Amendola 122/D, 70126 Bari, Italy</footnote>

<footnote n="2">² Institut de Physique de Rennes, Université de Rennes, 263 avenue du Général Leclerc, Rennes, France</footnote>

<footnote>* Author to whom any correspondence should be addressed.</footnote>
# 1. Introduction

The study of a vibrational non equilibrium nitrogen-oxygen mixture in a large temperature and pressure range has always been the focus of the interest of many research groups due to the possible applications in a variety of different fields, among which air plasma [1–5] and shock waves [6–11] can be mentioned.

The interest of the present work is in the investigation of a nitrogen-oxygen mixture at atmospheric temperature and pressure because it is of paramount importance, indeed it enters in the description of many different problems in fields ranging from the plasma medicine [12–15] to the plasma agriculture [16–21] and, more in general, in fields in which NO formation and/or nitrogen fixation are key features [21–28], as discussed in [29, 30] and, on a wider temperature range, in [31], where applications related to the hypersonic flight were mentioned and studied too.

The authors of the present article have contributed to the detailed knowledge of nitrogen-oxygen mixture behavior, with computational molecular dynamics studies about heavy particle [29, 32–39] and electron-heavy particle [40–45] collisions and with state-to-state (StS) vibrational kinetics models [46–49].

The present work is mainly devoted to the improvement of the StS vibrational kinetics model and to its application to a particular electrical discharge—post-discharge problem, paying special attention to the O + N₂(v) and N + O₂(v) collisions, as a needful step toward the ultimate goal of controlling the N₂ fixation and NO formation, in order to produce reactive nitrogen by means of a process alternative to the conventional Haber-Bosch [21, 24, 25] one, with a significantly lower carbon footprint. It is in line with previous works [30, 31], in which the interpolations of rate coefficients of the reactions due to O + N₂(v) and N + O₂(v) collisions, calculated by means of a quasi-classical trajectory (QCT) method, were performed and implemented in StS vibrational kinetic codes, used for thoroughly investigating the effects of these reactions in a N₂/N/O₂/O/NO mixture under different conditions.

The current improvement of the StS vibrational kinetics model consists in the implementation of the electron-heavy particle electron-vibration (e-V) and electron-dissociation (e-D) reactions in a structured way, rather than in a preliminary way as it was done in the past [50], and it also includes the possibility of simulating a gas flow in a zero-dimensional code.

With this perspective, the adopted model, with its different tasks and including the input parameters description, is explained in the next section (section 2), whereas in section 3 the results are deeply discussed paying particular attention to the roles of the different electrons-heavy particle collisions, of the temperature, of the gas recirculation simulation and, finally, of the pulse repetition. Finally, in section 4 the conclusions are drawn.

# 2. The model

The aim of the present work is to apply the StS vibrational model to the numerical simulation of a discharge—post-discharge cycle at atmospheric temperature and pressure.

An in-house zero-dimensional time-dependent numerical code solves the StS vibrational kinetics [30]:

$$ \frac{dn_i}{dt} = \sum_k (\nu_{i,k}^p - \nu_{i,k}^r) \left[ R_{f,k} \prod_l n_l \nu_{l,k}^r - R_{b,k} \prod_l n_l \nu_{l,k}^p \right] \\ i = 1, \dots, N_{sp}, \qquad (1) $$

by returning the number densities $n_i$ of the different $N_{sp}$ species, i.e. the number densities of the vibrational levels of the molecules, of the atoms and of the electrons. In the system (1), $k$ is the index of the reaction, so that for each reaction $k$ the forward and backward rate coefficients are $R_{f,k}$ and $R_{b,k}$, whereas the reactant and the product stoichiometric coefficients of the species $i$ are $\nu_{i,k}^r$ and $\nu_{i,k}^p$.

Table 1 summarizes the elementary processes considered in the kinetics modeling of N₂/N/O₂/O/NO/e⁻ mixture. The reactions include heavy particle collisions—such as atom-molecule VT, direct dissociation reactions, molecule-molecule VT and VV, exchange Zeldovich reactions—and electron-heavy particle collisions.

The cross sections and the rate coefficients for the processes in table 1 have different origins. Specifically, some of the heavy particle collisions rates have been obtained by cross sections calculated, in turn, thanks to a QCT method [34, 35, 39]. In particular, the atom-molecule VT and the direct dissociation rates of the processes (2), (3), (6) and (7) have been calculated starting from QCT simulations [51, 52]. Also the exchange and dissociation rates of the reactions (10) and (12) have been calculated starting from QCT simulations [29, 37], as well as the exchange, dissociation and atom-molecule VT rates of the processes (11), (13) and (14) [29, 31, 33]. The interpolation of the rates of the O + N₂ and N + O₂ collisions at low temperature is in [30].

On the other side, the nitrogen molecule-molecule VT rates of the process (4) have been calculated by Billing and Fisher [53, 54] and the oxygen molecule-molecule VT and VV rates of the processes (8) and (9) by Billing and Kolesnick [55, 56], all of them through semiclassical methods, as discussed in [46, 47].

Finally, the nitrogen VV rates of the process (5) have an experimental origin and have been taken from the work of Doroshenko et al [57].

At this stage it is important to mention that the ladder climbing model [48, 49] allows to include the dissociation-recombination reactions also in the VV (processes (5) and (9)) and molecule-molecule VT (processes (4) and (8)) processes
<table><caption>**Table 1.** List of the elementary processes considered in the model.</caption><thead><tr><th>Process</th><th>Method</th><th>References</th><th>Label</th></tr></thead><tbody><tr><td colspan="4"><strong>Nitrogen VV, VT, dissociation-recombination reactions</strong></td></tr><tr><td>N + N<sub>2</sub>(v) ⇌ N + N<sub>2</sub>(w)</td><td>QCT</td><td>[51, 52]</td><td>(2)</td></tr><tr><td>N + N<sub>2</sub>(v) ⇌ 3 N</td><td>QCT</td><td>[51, 52]</td><td>(3)</td></tr><tr><td>N<sub>2</sub> + N<sub>2</sub>(v) ⇌ N<sub>2</sub> + N<sub>2</sub>(v-1)</td><td>Semiclassical</td><td>[53, 54]</td><td>(4)</td></tr><tr><td>N<sub>2</sub>(v<sub>1</sub>) + N<sub>2</sub>(v<sub>2</sub>) ⇌ N<sub>2</sub>(v<sub>1</sub> + 1) + N<sub>2</sub>(v<sub>2</sub>-1)</td><td>Experimental</td><td>[57]</td><td>(5)</td></tr><tr><td colspan="4"><strong>Oxygen VV, VT, dissociation-recombination reactions</strong></td></tr><tr><td>O + O<sub>2</sub>(v) ⇌ O + O<sub>2</sub>(w)</td><td>QCT</td><td>[51, 52]</td><td>(6)</td></tr><tr><td>O + O<sub>2</sub>(v) ⇌ 3 O</td><td>QCT</td><td>[51, 52]</td><td>(7)</td></tr><tr><td>O<sub>2</sub> + O<sub>2</sub>(v) ⇌ O<sub>2</sub> + O<sub>2</sub>(v-1)</td><td>Semiclassical</td><td>[55, 56]</td><td>(8)</td></tr><tr><td>O<sub>2</sub>(v<sub>1</sub>) + O<sub>2</sub>(v<sub>2</sub>) ⇌ O<sub>2</sub>(v<sub>1</sub> + 1) + O<sub>2</sub>(v<sub>2</sub>-1)</td><td>Semiclassical</td><td>[55, 56]</td><td>(9)</td></tr><tr><td colspan="4"><strong>Zeldovich reactions</strong></td></tr><tr><td>O + N<sub>2</sub>(v) ⇌ NO(w) + N</td><td>QCT</td><td>[29, 37]</td><td>(10)</td></tr><tr><td>N + O<sub>2</sub>(v) ⇌ NO(w) + O</td><td>QCT</td><td>[29, 31, 33]</td><td>(11)</td></tr><tr><td colspan="4"><strong>Nitrogen–Oxygen dissociation-recombination reactions, VT</strong></td></tr><tr><td>O + N<sub>2</sub>(v) ⇌ O + 2 N</td><td>QCT</td><td>[29, 37]</td><td>(12)</td></tr><tr><td>N + O<sub>2</sub>(v) ⇌ N + 2 O</td><td>QCT</td><td>[29, 31, 33]</td><td>(13)</td></tr><tr><td>N + O<sub>2</sub>(v) ⇌ N + O<sub>2</sub>(w)</td><td>QCT</td><td>[29, 31, 33]</td><td>(14)</td></tr><tr><td colspan="4"><strong>e-V collisions</strong></td></tr><tr><td>e<sup>-</sup> + N<sub>2</sub>(v) ⇌ e<sup>-</sup> + N<sub>2</sub>(w)</td><td>R-matrix/LCP</td><td>[40]</td><td>(15)</td></tr><tr><td>e<sup>-</sup> + O<sub>2</sub>(v) ⇌ e<sup>-</sup> + O<sub>2</sub>(w)</td><td>R-matrix/LCP</td><td>[41]</td><td>(16)</td></tr><tr><td>e<sup>-</sup> + NO(v) ⇌ e<sup>-</sup> + NO(w)</td><td>R-matrix/LCP</td><td>[43]</td><td>(17)</td></tr><tr><td colspan="4"><strong>e-D collisions</strong></td></tr><tr><td>e<sup>-</sup> + N<sub>2</sub>(v) → e<sup>-</sup> + N + N</td><td>R-matrix/LCP</td><td>[40]</td><td>(18)</td></tr><tr><td>e<sup>-</sup> + O<sub>2</sub>(v) → e<sup>-</sup> + O + O</td><td>R-matrix/LCP</td><td>[42]</td><td>(19)</td></tr><tr><td>e<sup>-</sup> + NO(v) → e<sup>-</sup> + N + O</td><td>R-matrix/LCP</td><td>[44]</td><td>(20)</td></tr></tbody></table>

by adding a pseudo-level above the last vibrational level of the corresponding bound molecule.

Among all the rate coefficients of the heavy particle kinetic processes, it is important to report in this paper the plots of the forward and backward StS rate coefficients of the two Zeldovich reactions (10) and (11) versus the reactant molecule vibrational level, v, at selected NO vibrational levels, w, and at T = 380 K because they are not shown in the cited references (in [30] the higher rates at T = 500 K are shown) and because they will be useful in the discussion of some results in section 3.

In figures 1 and 2 vibrational StS rate coefficients for the reactive processes (10) and (11) are shown, respectively, calculated in [29] at low temperature (in the range 300–1000 K). The importance of using StS rates instead of initially state-selected ones in chemical kinetic simulations has been already stressed in [30]. In the cited [29] the comparison of thermal rate coefficients is also shown with experimental data, and it is excellent on the whole temperature range explored, from 300 to 5000 K. Even StS rate coefficient comparison with quantum-mechanical results (concerning O + N<sub>2</sub> only) in [58] on the same potential energy surface set is very good. This is

not surprising, due to the heavy masses involved and the temperature not lower than 300 K. It is important to underline that this is generally true for reaction rates of air species, but not necessarily for vibrationally inelastic processes at low temperature, for which QCT method shows important issues [29].

All the rate coefficients of the electron-heavy particle collisions adopted in the system of equation (1) results from the cross sections calculated by Laporta et al [40–44] and the electron energy distribution function (EEDF) obtained, at each temporal step of the system (1), by solving the Boltzmann equation [50, 59–61]. In particular, the cross sections of the e-V and e-D collisions involving N<sub>2</sub> (processes (15), (18)) are in [40]; the ones related to the e-V collisions of O<sub>2</sub> (16) are in [41], whereas the ones corresponding to the e-D of O<sub>2</sub> (19) are in [42]; finally, about the electron-NO collisions, the cross sections of the e-V processes (17) are in [43] and the cross sections of the e-D reactions (20) are in [44].

Concerning the electron-molecule scattering, the calculations of vibrationally state-resolved cross sections were carried out within the framework of the Local Complex Potential (LCP) model [62, 63]. The potential energy curves for the target molecules and resonant states were obtained using MRCI
<figure><img src="image_5.png" /><figcaption>**Figure 1.** Forward (left panel) and backward (right panel) state-to-state rate coefficients of the O + N₂(v) ⇌ NO(w) + N Zeldovich reactions (10) versus the N₂ vibrational level, *v*, at selected NO vibrational levels, *w*, and at *T* = 380 K.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 2.** Forward (left panel) and backward (right panel) state-to-state rate coefficients of the N + O₂(v) ⇌ NO(w) + O Zeldovich reactions (11) versus the O₂ vibrational level, *v*, at selected NO vibrational levels, *w*, and at *T* = 380 K.</figcaption></figure>

model implemented in the ab initio quantum chemistry codes MOLPRO [64] and the R-matrix approach [65] which reveals a powerful method to study resonant electron scattering at low energies. Discussion on the validity of the calculated cross sections for NO molecule can be found in the original articles in [42–44].

For the three molecules considered in the present work, N₂, O₂ and NO, the e-V cross sections from *v* = 0 to *v* = 1 and from *v* = 0 to *v* = 10 are plotted in figure 3, whereas the e-D cross sections from *v* = 0 and from *v* = 10 are reported in figure 4.

As mentioned, these cross sections enter in an in-house Boltzmann solver [50]. The terms taken into account in the Boltzmann equation are the flux of the electrons in the energy space driven by the applied electric field (this term is considered only during the discharge, see the following section 2.3), the flux of the electrons in the energy space driven by the elastic collisions, the terms due to the vibrational inelastic and superelastic collisions, in line with [50, 59–61, 66].

The Boltzmann equation is discretized both in the time and in the electron energy. The electron energy discretization

consists in dividing the 10 eV energy range in 1000 cells to get an energy step lower than the lowest vibrational energy quantum jump in the vibrational ladders of the three considered molecules. The time discretization corresponds to the time discretization of the system of equation (1). At each time step the Boltzmann equation is solved in the energy space, the obtained EEDF enters in the calculation of the rate coefficients of the electron-heavy particle collisions that, in turn, enter in the equation (1); the cycle is repeated for each time step up to the final time. By the way, at the time *t* = 0 the initial EEDF is fixed to the electron equilibrium Boltzmann distribution at the initial temperature *T*.

Before to conclude this Section about the chemical and kinetic processes included in the system (1), it is useful to point out that at this stage it was considered more important to investigate the vibrational kinetics of the molecular ground electronic states, whose rates are known with great detail and accuracy, than to take into account for the excited electronic states, whose corresponding quantities are poorly known. More in detail, in [29–31] there is an evidence that NO vibrational kinetics is strongly influenced by the level of
<figure><img src="image_7.png" /><figcaption>Figure 3. Some state-selected cross sections for the e-V collisions (15–17) as a function of the electron energy.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 4. Some state-selected cross sections for the e-D reactions (18–20) as a function of the electron energy.</figcaption></figure>

detail of the input data. In the future it is planned to add all further relevant processes with reliable rates available in the literature, also because the electronically excited states and the kinetic processes associated with O$_{3}$ and N$_{x}$O$_{y}$ species could modify the plasma behavior.

The effects of the wall are neglected both because the simulation is referred to (see the following section 2.3. Input parameters) a plasma spot generated by a discharge in a point to plane geometry far enough from the reactor walls [67] and because of the gas recirculation (see section 2.2. Gas recirculation simulation). However, this approximation will be overcome in future works to check its correctness and/or its weakness.

## 2.1. Pseudo-first order decomposition and formation constants

To a better understanding of the results, it is also useful to analyze the N$_{2}$, O$_{2}$ and NO pseudo-first order decomposition constants, $k_{N2,dec}$ (s$^{-1}$), $k_{O2,dec}$ (s$^{-1}$), $k_{NO,dec}$ (s$^{-1}$), as well as the NO formation constants, $k_{NO,form}$ (s$^{-1}$). At each numerical step, that means at each $\Delta t$, they are directly derived by both the code output, i.e. the species number densities, and the rate

coefficients of the relevant reactions [31]:

$$k_{N2,dec} = \frac{\sum_v [N_2(v)][N] K_v^{NN2,d}}{n_{tot}} + \frac{[N_2(v_{max})][N_2] K_{vmax}^{N2N2VT,d}}{n_{tot}} + \frac{\sum_w [N_2(v_{max})][N_2(w+1)] K_w^{N2N2VV,d}}{n_{tot}} + \frac{\sum_v \sum_w [N_2(v)][O] K_{v,w}^{ON2,NOF}}{n_{tot}} + \frac{\sum_v [N_2(v)][O] K_v^{ON2,d}}{n_{tot}} + \frac{\sum_v [N_2(v)][n_e] K_v^{N2e,d}}{n_{tot}}, \quad (21)$$

$$k_{O2,dec} = \frac{\sum_v [O_2(v)][O] K_v^{O2,d}}{n_{tot}} + \frac{[O_2(v_{max})][O_2] K_{vmax}^{O2O2VT,d}}{n_{tot}} + \frac{\sum_w [O_2(v_{max})][O_2(w+1)] K_w^{O2O2VV,d}}{n_{tot}} + \frac{\sum_v \sum_w [O_2(v)][N] K_{v,w}^{NO2,NOF}}{n_{tot}} + \frac{\sum_v [O_2(v)][N] K_v^{NO2,d}}{n_{tot}} + \frac{\sum_v [O_2(v)][n_e] K_v^{O2e,d}}{n_{tot}}, \quad (22)$$

$$k_{NO,dec} = \frac{\sum_w \sum_v [NO(w)][N] K_{w,v}^{ON2,NOd}}{n_{tot}} + \frac{\sum_w \sum_v [NO(w)][O] K_{w,v}^{NO2,NOd}}{n_{tot}} + \frac{\sum_v [NO(v)][n_e] K_v^{NOe,d}}{n_{tot}}, \quad (23)$$

$$k_{NO,form} = \frac{\sum_v \sum_w [N_2(v)][O] K_{v,w}^{ON2,NOF}}{n_{tot}} + \frac{\sum_v \sum_w [O_2(v)][N] K_{v,w}^{NO2,NOF}}{n_{tot}}, \quad (24)$$

In the previous equations, $n_{tot}$ is the total number density. In the equations (21) and (22) the different terms on the right
hand side are due to, in order, the direct dissociation by atom-molecule collisions, the ladder climbing model dissociation by VT molecule-molecule collisions, the ladder climbing model dissociation by VV collisions, the exchange reaction by either the O + N₂(v) (for equation (21)) or the N + O₂(v) (for equation (22)) collisions, the direct dissociation by either the O + N₂(v) (for equation (21)) or the N + O₂(v) (for equation (22)) collisions and, finally, the direct dissociation e-D by electron-molecule collisions. Likewise, the first two terms on the right hand side of equations (23) and (24) are due to the backward (equation (23)) and forward (equation (24)) exchange reactions (10) and (11), whereas the last term of equation (23) is due to the e-D reactions (20). As expected kNO,form is the sum of the fourth terms of equations (21) and (22).

Finally, the last terms of equations (21)–(23), the terms due to the e-D reactions, are a novelty with respect to the corresponding decomposition constants in the [31].

## 2.2. Gas recirculation simulation

In the framework of the model discussion, it is important to stress that usually in experimental setups the gas recirculates, whereas the adopted numerical model is a zero-dimensional one.

To simulate gas recirculation in the zero-dimensional model used in this work, the approach consist in updating the initial condition for each time step $t + \Delta t$ based on a weighted combination of the current solution at time t and the initial condition at $t = 0$ s. Specifically, a fraction $\epsilon$, where $0 \le \epsilon \le 1$, of the current state is replaced by the corresponding fraction of the initial state, as follows:

$$n_i(t) = n_i(t) * (1 - \epsilon) + n_i(t = 0) * \epsilon. \\ i = 1, \dots, N_{sp}. \qquad (25)$$

Here $\epsilon$ represents the gas replacement fraction due to flow.

The case $\epsilon = 0$ obviously corresponds to the case in which the gas does not recirculates and the whole updated solution for the number densities at the time t, $n_i(t)$, is taken as initial condition of the time step $t + \Delta t$.

It is clear that, on one side, $\epsilon$ must be a dimensionless number and, on the other side, it must be proportional to the flow rate (in $cm^3 s^{-1}$):

$$\epsilon = \frac{\text{flow rate}}{V} \Delta t \qquad (26)$$

where V is the volume (in $cm^3$) in which the experimental measurements are taken (see the following section 2.3 Input parameters).

In the adopted model a constant recirculation is imposed because the flow in the suggested experiment ([67, 68]) is fixed and maintained at $3.3 cm^3 s^{-1}$ (200 sccm) and also because constant temperature is considered (see the next section 2.3 Input parameters).

However if a variable temperature was considered, assuming an approximately constant pressure, the conservation of

the mass flow rate ($\rho(t_1)v(t_1)A = \rho(t_2)v(t_2)A$, with A sectional area and $\rho$ mass density of the flow) would imply that the average gas velocity $v$ (therefore the recirculation too), due to the ideal gas law, should vary proportionally to $T$: $v(t_1)/T(t_1) = v(t_2)/T(t_2)$ [69]. This is also in line with equation (3) of [70].

## 2.3. Input parameters

The conditions under which the numerical code is run are those corresponding to an experiment [68] carried out in the reactor shown in the [67], in which the discharge is developed in ‘an asymmetric point to plane geometry based on a tungsten needle (HV) and a brass disk (GND) electrode’. Moreover, the main body of the reactor is composed of stainless steel [68].

In the electric discharge [68] the pulse duration and the repetition rate are, respectively, 200 ns and 200 Hz, the used mixture is a 95% N₂–5% O₂ one at 760 Torr (101 325 $Nm^{-2}$), the peak reduced electric field during the breakdown is $E/n_{tot}$ about 74 Td (74 × $10^{-21}$ $Vm^2$), the peak electron density is about $10^{15}$ part $cm^{-3}$, the flow rate in the reactor is a $3.3 cm^3 s^{-1}$ one, the volume of the measurements is $2.355 \times 10^{-3}$ $cm^3$. In particular, the value of $E/n_{tot}$ was obtained by means of time resolved emission spectra of FNS and SPS band of nitrogen [68].

The input parameters of the numerical model are step by step modified to understand their role on the final results. They are summarized in table 2.

The initial percentage of the oxygen, the sum of O₂ and O, is always set to 5%, the other part of the mixture is mostly nitrogen, the sum of N₂ and N, except for minor NO and electron percentages. The initial NO percentage is set to $10^{-5}$. The initial electron percentage, $e^{-}$%, is calculated starting from the peak electron density and it is shown in the table 2, together with the electron number density normalized to the total number density, $n_e/n_{tot}$. Currently the code runs with a constant electron density; further development is planned to include electron attachment/recombination reactions and this could affect the simulation results.

Moreover, at the starting time, $t = 0$ s, the N₂, O₂ and NO vibrational distributions are always set equal to their corresponding equilibrium Boltzmann distributions; at $T = 500$ K, they are the blue curves 1 of the left panel plots (a), (c), (e) of the following figure 5 (a detailed analysis is performed in section 3.1).

Beside the initial mixture composition, it is important to point out the other adopted parameters.

About the pressure, the atmospheric one (101 325 $Nm^{-2}$ or 760 Torr) is chosen in all the numerical test cases.

The initial mixture and the working temperatures were firstly fixed to 500 K, then to 380 K and finally to 300 K. Among these temperatures, the most realistic one should be 380 K, value obtained thanks to rotational temperature measurements from the SPS(0,0) band emission in the experiment of [68]. This is the reason why 380 K was chosen in place of 400 K in the numerical test cases.

To be precise, during the discharge the experimental researchers [68] measured an increasing temperature reaching
<table><caption>**Table 2.** Test cases for different electrical discharge and post-discharge simulations. P = 101 325 Nm⁻² (760 Torr); E/n<sub>tot</sub> = 74 Td (74 × 10⁻²¹ Vm²); O₂ + O = 5%; NO = 10⁻⁵%.</caption><thead><tr><th>T(K)</th><th>e<sup>-</sup> (%)</th><th>n<sub>e</sub>/n<sub>tot</sub></th><th>E(V cm<sup>-1</sup>)</th><th>ε</th><th>Considered processes</th><th>Test no.</th></tr></thead><tbody><tr><td rowspan="4">500</td><td rowspan="4">7 × 10<sup>-3</sup></td><td rowspan="4">7 × 10<sup>-5</sup></td><td rowspan="4">1.09 × 10<sup>4</sup></td><td rowspan="4">0 * Δt</td><td>(2)–(14)<br>e-V: v = 0, w = 1–10<br>e-V: v, w = v + 1</td><td>TC1</td></tr><tr><td>(2)–(14)<br>e-V: v = 0, w = 1-last<br>e-V: v, w = v + 1</td><td>TC2</td></tr><tr><td>(2)–(14)<br>e-V: (15)–(17)</td><td>TC3</td></tr><tr><td>(2)–(14)<br>e-V: (15)–(17)<br>e-D: (18)–(20)</td><td>TC4</td></tr><tr><td>380</td><td>5.2 × 10<sup>-3</sup></td><td>5.2 × 10<sup>-5</sup></td><td>1.43 × 10<sup>4</sup></td><td>0 * Δt<br>1400 * Δt</td><td>(2)–(14)<br>(15)–(20)</td><td>TC5<br>TC6</td></tr><tr><td>300</td><td>4.1 × 10<sup>-3</sup></td><td>4.1 × 10<sup>-5</sup></td><td>1.81 × 10<sup>4</sup></td><td>0 * Δt</td><td>(2)–(14)<br>(15)–(20)</td><td>TC7</td></tr></tbody></table>

380 K at the end of the discharge itself; in the post-discharge the temperature should decrease. The numerical code used in this work was developed to run at constant *T*, however the way to improve it taking into account for a variable temperature is under investigation.

It is interesting to observe that, in the case of DBD in a deionized water-air interface, an electronically excited state rotational temperature (used to estimate the gas temperature) of approximately 380 K is reached after about 20 ns and one of about 500 K is reached after about 40 ns [71].

Starting from the chosen E/n<sub>tot</sub> value and from the parameters *T* and *P* considered in the numerical test cases, different values of the electric field *E* are obtained and then used in the code to solve the Boltzmann equation and to find the rates of the electron-heavy particle collisions. They are reported in table 2.

In any case the electric field is applied only in the first 2 × 10⁻⁷ s, whereas for *t* = 2 × 10⁻⁷–5 × 10⁻³ s it is set *E* = 0 V cm⁻¹, corresponding to the pulse duration (200 ns) and repetition rate (200 Hz).

# 3. Results and discussion

## 3.1. The role of electrons

Before starting the results discussion, it is important to mention that in every numerical simulation performed in this study (see table 2), both in the discharge and in the post-discharge, all the processes (2)–(14) (see table 1) involving heavy particles are considered.

About the kinetic processes and reactions due to the electron-heavy particle collisions (15)–(20), they are implemented in our in-house numerical code that solves system (1) for the first time in the framework of this work. To better investigate the role of these different collisions (15)–(20), in

the first four test cases of table 2, the ones related to *T* = 500 K, they are included one step at a time: this means that, among the e-V kinetic processes (15)–(17), not all the possible coupling *v*–*w* between the vibrational level *v* of the reactant and the vibrational level *w* of the product are necessarily considered and that the e-D reactions (18)–(20) are taken into account only in the last test case TC4.

In particular, in the cases TC1 and TC2 the e-V kinetic processes (15)–(17) are partially included: in TC1 the e-V collisions from a reactant molecule in the vibrational ground state *v* = 0 to a product molecule in a vibrational level in the range *w* = 1–10 as well as monoquanta e-V processes (*v*, *w* = *v* + 1) are considered, whereas in TC2 the e-V collisions from a reactant molecule in the vibrational ground state *v* = 0 to a product molecule in whatever vibrational level *w* and monoquanta e-V processes (*v*, *w* = *v* + 1) are considered. On the other side, in TC3 all the e-V kinetic processes (15)–(17) are included. Finally, in TC4 all the e-V kinetic processes (15)–(17) as well as all the e-D reactions (18)–(20) are taken into account.

Results obtained in these first four test cases (TC1–TC4) are reported in the figures 5–7.

In figure 5 the N₂, O₂ and NO vibrational distributions both at the end of the discharge (left panel plots), i.e. at *t* = 2 × 10⁻⁷ s, and at the end of the post-discharge (right panel plots), i.e. at *t* = 5 × 10⁻³ s, are plotted. The correspondence between the vibrational energies of the molecules, E<sub>vib</sub>, and their vibrational levels, *v*, is in the supplementary material of [31]. The different curves, 2–5 in the left panel plots and 1–4 in the right panel plots, correspond to the test cases TC1–TC4; as already mentioned in the previous section 2.3, the blue curves 1 of the left panel plots are the initial (*t* = 0 s) N₂, O₂ and NO equilibrium Boltzmann distributions, reported to show the vibrational distributions time evolution.

By comparing the different curves of each molecule at the end of the discharge, the distributions of N₂, O₂ and NO on
<figure><img src="image_9.png" /><figcaption>**Figure 5.** N₂ (a), (b), O₂ (c), (d) and NO (e), (f) vibrational distributions at the end of the discharge (a), (c), (e) and at the end of the post-discharge (b), (d), (f). The e-V processes and e-D reactions are included as in the test cases TC1–TC4 of table 2 (see the different curves). *T* = 500 K.</figcaption></figure>

their medium-high vibrational levels are higher when a lar-
ger number of e-V collisions are considered (see curves 2–4).
Moreover, the introduction of the e-D reactions (pink curves
5, to be compared with violet curves 4 representing the case in
which all the e-V processes are included in the solution) affects
to a minimal extent the N₂ and O₂ distributions at the end of

the discharge whereas it strongly impacts on the NO distribu-
tion. The whole NO vibrational distribution for TC4, includ-
ing the region corresponding to the low vibrational levels, is
much higher than the ones related to the other test cases and
this is evidence of the considerable NO formation (see the next
figure 7).
<figure><img src="image_10.png" /><figcaption>Figure 6. N₂, O₂ and NO vibrational energies. The e-V processes and e-D reactions are included as in the test cases TC1–TC4 of table 2 (see the different curves). T = 500 K.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 7. N, O and NO number densities. The e-V processes and e-D reactions are included as in the test cases TC1–TC4 of table 2 (see the different curves). T = 500 K.</figcaption></figure>

Recalling that during the post-discharge the electron-heavy particle collisions are not included in the solution system, it is reasonable that, precisely in post-discharge, after a certain time the molecules distributions are no more affected by the discharge conditions, indeed at $t = 5 \times 10^{-3}$ s the distributions relating to test cases TC1–TC3 (curves 1–3) are almost the same; on the other side the distributions corresponding to the test case TC4 (pink curves 4) are still a bit different from the others. The explanation is that the vibrational equilibrium is not yet reached at $t = 5 \times 10^{-3}$ s.

As already stressed, in figure 5 only the N₂, O₂ and NO vibrational distributions at the end of the discharge and at the end of the post-discharge are shown, even if it would be interesting to follow the evolution of the vibrational distributions over time. Alternatively, the investigation of the trends of the molecules vibrational energies can give at least an idea of how the vibrational distributions move away from the initial Boltzmann distributions in the discharge and slowly approach new equilibrium distributions in the post-discharge. The molecules vibrational energies are calculated in the usual way as

$$E_{\text{vib,AB}} = \sum_{v=0}^{v_{\text{last}}} \frac{n_{v,AB}}{n_{\text{tot}}} E_{v,AB}, \quad (27)$$

where AB is either N₂ or O₂ or NO, and they are reported in figure 6 for the test cases TC1–TC4. Please note that in the plots of this figure, as well as in other following figures, a vertical line at $t = 2 \times 10^{-7}$ s separates the discharge and the post-discharge regions.

The discussion of the evolution over time of the vibrational energies of the molecules in the different test cases is in line with the discussion of the vibrational distribution dependence on the e-V and e-D processes considered both at the end of the discharge and of the post-discharge (figure 5). Moreover, by studying the figure a couple of observations deserve to be done. The first one is that the main contribution to the total vibrational energy $E_{\text{vib}}$, i.e. the sum of the three molecules vibrational energies, is given by the nitrogen vibrational energy. The second one is that $E_{\text{vib,N2}}$ and $E_{\text{vib,O2}}$ slightly increase, then they stay more or less constant and finally they slightly decrease, everything in no more than one order of magnitude. On the other side, $E_{\text{vib,NO}}$ increases by more than six orders of magnitude and then it starts to decrease.

Coming back to the vibrational distributions, they are also linked to the species number densities to the extent that an increasing in a molecule number density reflects in a higher vibrational distribution in all the vibrational range, as observed for the NO distribution at the end of the discharge when all
<figure><img src="image_12.png" /><figcaption>**Figure 8.** N₂, O₂ and NO vibrational distributions at the end of the post-discharge. The simulation temperature values are 500 K (TC4 of table 2), 380 K (TC5) and 300 K (TC7).</figcaption></figure>

the e-V and e-D processes are considered (see pink curve 5 of figure 5, test case TC4).

Figure 7 reports the evolution over time of the N, O and NO number densities in the same conditions of the figures 5 and 6.

The increasing of atomic nitrogen and atomic oxygen densities when a large number of e-V collisions are considered (curves 1, 2, 3) is clear, showing how the pumping of the higher vibrational levels due to the e-V collisions helps the dissociation by heavy particle collisions. Anyway, the atomic densities stay many orders of magnitude lower than the N₂ and O₂ molecular densities, being the last ones not reported in the figure. If the e-D collisions are included (pink curve 4, test case TC4), the atoms increasing in the discharge is, as expected, much more evident and the same happens in the first part of the post-discharge.

Regarding the NO number density, like the ones of the atoms, it increases as a greater number of e-V collisions (curves 1–3) are considered. When also the e-D reactions (curve 4) are taken into account, the NO number density increasing in the first 1.8 × 10⁻⁵ s is also higher with respect to the previous test cases.

In any case the NO number density increases, reaches a maximum, in the range $t = 4.2 \times 10^{-5} - 1.4 \times 10^{-4}$ s depending on the test case, and then it starts to decrease. This increasing-decreasing, together with the e-D reactions, is due to the mutual effect of the two Zeldovich reactions and it will be better investigated later on, during the discussion of the pseudo-first order formation and decomposition constants obtained in the test case TC5 (figure 10), constants that show the same trend as those obtained (and not reported in this work) in TC4.

Please note that the NO number densities in the test cases TC1-TC4 behaves as the NO vibrational energies in the same cases (figure 6).

## 3.2. The role of temperature

At this stage the role of the different electron-heavy particle collisions (15)–(20) has been understood and it is time to investigate the role of the working temperature, to this aim the test cases TC4, TC5 and TC7 are compared. They are run

including all the kinetic processes (2)–(20), considering the three different temperatures 500 K, 380 K and 300 K and, which is important, without considering the gas recirculation in the model, so that the discussion of the test case TC6 is postponed.

Firstly, it has been verified that all the analyzed quantities, during the discharge and in the temperature range 300–500 K, are minimally or not affected at all by the temperature.

In the post-discharge the increasing of the temperature accelerates the approach of the molecules to their vibrational equilibrium, as clear in figure 8 where the N₂, O₂ and NO vibrational distributions at the end of the post-discharge are shown. In the case of NO the vibrational distribution is lower at 380 K than at 300 K and it is lower at 500 K than at 380 K in all the vibrational energy range and this is fully in line with the NO density trend reported in figure 9. In the same figure 9 also the atomic densities are plotted and, as reasonable, the N and O densities in the post discharge increase with the temperature.

Concerning the nitrogen monoxide, as anticipated in the discussion of the NO density in the test cases TC1–TC4 (figure 7), its trend is due to the mutual effect of the two Zeldovich reactions ((10) and (11)) and this effect can be understood through the analysis of the pseudo-first order formation and decomposition constants.

For this discussion the focus is on the case TC5, related to the temperature $T = 380$ K.

In figure 10 both the different terms of the NO formation and decomposition constants (left panel) and the global NO formation and decomposition constants (right panel) are plotted. More precisely, both the decomposition constants due to only heavy particle collisions (blue curve 1) and those ones due to all the implemented reactions (black curve 3) are plotted in the right panel of the figure.

The NO formation and decomposition description is quite tricky. These quantities need to be analyzed by considering not only the species number densities but also the molecular vibrational distributions, indeed the rate coefficients of the Zeldovich reactions, both forward and backward, depend on the vibrational levels of the reactants and of the products.

The first process that gives its contribution to the NO formation is the second Zeldovich forward reaction (11) (green curve
<figure><img src="image_13.png" /><figcaption>**Figure 9.** N, O and NO number densities. The simulation temperature is, time by time, 500 K (TC4 of table 2), 380 K (TC5) and 300 K (TC7).</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Figure 10.** Different terms of the NO formation and decomposition constants (left panel) and global NO formation and decomposition constants (right panel). T = 380 K. TC5 in table 2.</figcaption></figure>

3 in the left panel of figure 10), even if the N density is lower than the O and NO densities (figure 9) and the O₂ density is lower than the N₂ density.

The reason can be found in the Zeldovich forward and backward rate coefficients at low temperature (see the reactions rates at T = 500 K in figures 1 and 2 of [30] and at T = 380 K in figures 1 and 2 of this work).

The StS rate coefficients of the first Zeldovich forward reaction (10) are negligible for the first N₂ vibrational levels, therefore, keeping in mind that the initial distributions of the three molecules are Boltzmann ones, so that only the first vibrational levels have a significant population density, it is straightforward that this reaction cannot be effective at the beginning of the discharge. Later on, when the N₂ vibrational distribution starts to be pumped, the first Zeldovich forward reaction (10) affects the NO production (blue curve 1 in the left panel of figure 10). On the opposite, the rates of the second Zeldovich forward reaction (11) are effective also for the lower O₂ vibrational levels v, allowing this reaction since the very beginning of the discharge (green curve 3).

Let us focus now on the two backward Zeldovich reactions. At the very beginning of the discharge, once more, only

the first NO vibrational levels w have a significant population density and for these levels the rate coefficients of the first backward Zeldovich reaction (10) are in a useful range, whereas the rates of the second backward Zeldovich reaction (11) are negligible: this explains the overcoming of the first mentioned reaction on the second one in the first $2.8 \times 10^{-11}$ s (red curve 2 and violet curve 4 in the left panel of the figure). After that, up to about $4.6 \times 10^{-6}$ s, the second backward Zeldovich reaction (11) prevails over the first one (10), indeed the NO vibrational distribution increases, allowing the use of higher rates in reaction (11), and moreover the O density is still higher than the N density (green curves 2 in figure 9). Afterwards the contribution of the second backward Zeldovich reaction to the NO deactivation decreases and then increases again, however staying below the first backward Zeldovich reaction contribution, showing more or less the same trend of the O atom density (green curve 2 in the central panel of figure 9).

Before concluding, it is mandatory to stress that the NO decomposition in the discharge region is mainly due to the e-D reactions (see black curve 6 in the left panel of figure 10), whose effect overcomes the effects of the two backward
<figure><img src="image_15.png" /><figcaption>**Figure 11.** Different terms of the N₂ (left panel) and O₂ (right panel) pseudo-first order decomposition constants. *T* = 380 K. TC5 in table 2.</figcaption></figure>

Zeldovich reactions (curves 2 and 4); this behavior is also deductible by observing that in the right panel of the same figure the curve 3, that includes the e-D reactions (20) in the NO decomposition constant (23), is many order of magnitude higher than curve 1.

Without distinguishing between the different e-D and exchange reactions (right panel of figure 10), the NO decomposition (black curve 3) prevails on the NO formation (red curve 2) in almost all the discharge zone up to *t* = 2.9 × 10⁻⁸ s, then the NO formation prevails between *t* = 2.9 × 10⁻⁸ s and *t* = 2.8 × 10⁻⁴ s and finally an equilibrium between formation (red curve 2) and decomposition (blue curve 1) is reached.

Beside the NO pseudo-first order formation and decomposition constants it is also useful to investigate the behavior of the N₂ and O₂ pseudo-first order decomposition constants, whose different terms are plotted in figure 11. It is extremely interesting to note that in the discharge, as well as it happens to NO, the greatest N₂ and O₂ decomposition is due to the e-D reactions (black curve 6), i.e. reactions (18) and (19), respectively.

Aside from the e-D reactions (18), the major contribution to the N₂ decomposition is given by the forward Zeldovich reaction (10) (violet curve 4 in the left panel of the figure), followed by the contributions of the direct dissociation caused either by O + N₂ collision (12) in the first 3.7 × 10⁻⁶ s (pink curve 5) or by N + N₂ collision (3) since *t* = 3.7 × 10⁻⁶ s (blue curve 1).

About the molecular oxygen O₂, except for the e-D reaction (19), the major contribution to its decomposition is given by the Zeldovich reaction (11) (violet curve 4 in the right panel of the figure) only in the first 5.9 × 10⁻¹² s and after *t* = 1.9 × 10⁻⁷ s and by the O + O₂ direct dissociation (7) (blue curve 1 in the right panel of the figure) in the range 5.9 × 10⁻¹² to −1.9 × 10⁻⁷ s.

At this stage it deserves to be pointed out that the differences of the N₂, O₂ and NO pseudo-first order decomposition constants as well as of the NO formation constants with respect to the corresponding quantities obtained in [30], are due to both the different initial mixture composition and the initial gas excitation that in this work is caused by an electric discharge including the electron-heavy particle collisions,

whereas in [30] it was caused by a much higher initial temperature suddenly decreased. On the other side, the working temperature, 380 K instead of 500 K, does not affect the trend of these constants, as verified by plotting them for the test case TC4 of table 2 (not reported in this paper for conciseness).

Before to conclude this section, it could be interesting to stress that the results are also affected by the electron density. At *T* = 380 K, the N, O and NO number densities obtained with an electron percentage equal to 5.2 × 10⁻⁵%, 5.2 × 10⁻⁴%, 5.2 × 10⁻³% and 5.2 × 10⁻²%, respectively, are calculated and compared. The corresponding plots are not reported because beyond the aim of this work, however some features deserve to be pointed out. The increasing of the electron density causes the atomic densities increasing during the discharge. In the post discharge the N densities, sooner or later, start to decrease and the atomic nitrogen density obtained at the electron percentage of 5.2 × 10⁻⁵% stays many orders of magnitude lower than the others. The atomic oxygen in the post-discharge, except for the electron percentage of =5.2 × 10⁻⁵% for which it stays constant, experiences a decreasing, a minimum and a further increasing. The higher is e⁻%, the earlier is the time at which the decreasing and the further increasing starts and the minimum occurs. Finally, the higher is the electron density, the higher the NO density is. In particular, the increasing and the decreasing of NO start earlier as the electron density increases.

### 3.3. The role of recirculation

The last feature that deserves a focus is the ability to simulate in a zero-dimensional numerical model the gas recirculation occurring in an experimental setup, as explained in section 2.2, and how this recirculation can affect the gas behavior. To this aim the results obtained at *T* = 380 K both for a stationary gas (test case TC5) and for a recirculating one (TC6) with ε = 1400 * Δ*t* are compared in the figures 12–16.

As explained in section 2.2, the input of new fresh gas during the discharge—post-discharge cycle pushes back the gas mixture towards its initial conditions because the new gas is
<figure><img src="image_16.png" /><figcaption>**Figure 12.** N₂, O₂ and NO vibrational distributions at the end of the post-discharge. The simulation of the gas flow in the reactor is either neglected (TC5 of table 2) or taken into account (TC6). T = 380 K.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Figure 13.** N₂, O₂ and NO vibrational energies. The simulation of the gas flow in the reactor is either neglected (TC5 of table 2) or taken into account (TC6). T = 380 K.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>**Figure 14.** N, O and NO species number densities. The simulation of the gas flow in the reactor is either neglected (TC5 of table 2) or taken into account (TC6). T = 380 K.</figcaption></figure>

entered exactly at the initial set conditions, in other words this fresh gas input delays the physical-chemical evolution of the gas itself.

However, the flow rate chosen in TC6 (i.e. 3.3 cm³s⁻¹) is not enough to affect the mixture properties up to about t = 2 × 10⁻⁵–5 × 10⁻⁴ s, when it causes lower values of the different calculated variables: vibrational distributions (in figure 12 only the final ones at t = 5 × 10⁻³ s are plotted),

molecular vibrational energies (figure 13), N, O and NO densities (figure 14), N₂ and O₂ pseudo-first order decomposition constants (figure 15) and, finally, NO formation and decomposition constants (figure 16).

The O and NO trends (figure 14) are in line with those of figure 2 of [67]. It is mandatory to stress that in [67] there is a typo [68]: the labels of the third and fourth curves are inverted, so that ‘NO 760 Torr’ is actually ‘O 760 Torr’ and vice versa.
<figure><img src="image_19.png" /><figcaption>**Figure 15.** Global N₂ (left panel) and O₂ (right panel) pseudo-first order decomposition constants. The simulation of the gas flow in the reactor is either neglected (TC5 of table 2) or taken into account (TC6). T = 380 K.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>**Figure 16.** Global NO formation and decomposition constants. The simulation of the gas flow in the reactor is either neglected (TC5 of table 2) or taken into account (TC6). T = 380 K.</figcaption></figure>

<caption>**Figure 17.** N₂, N, O₂, O and NO number densities. The conditions are those related to TC5 (table 2), except that 4 impulses are now considered: after t = 10⁻⁵ s from the beginning of each discharge—post-discharge cycle, a new electric pulse is given.</caption>

The other important feature, before to compare the curves of the two works, is that in [67] each density is normalized to its own maximum value. Both in the experimental (see figure 2 of [67]) and in the numerical (figure 14(b), green curve 2) case the atomic oxygen in the post-discharge decreases, reaches a minimum, increases and finally decreases again; however, the minimum in the numerical test cases is reached later: at about 2 × 10⁻⁵ s, against the 4 × 10⁻⁶ s of the experiment. The NO (figure 14(c), green curve 2) increases, then it decreases; the maximum is around 10⁻⁴ s both in the numerical test and in the experiment. The comparison is also in line with the results in [68]. This is a very preliminary comparison between experimental and theoretical results; a deeper combined work, both from the experimental side and from the theoretical side, is planned to investigate the behavior of the ns discharges.

The NO formation and decomposition constants (figure 16) behave in an unexpected way. Up to about t = 1.5 × 10⁻⁵ s both kNO,dec (red curve 1 and green curve 2) and kNO,form (violet curve 3 and pink curve 4) do not depend on the gas velocity, after that both kNO,dec and kNO,form split and at this time kNO,dec (red curve 1) and kNO,form (violet curve 3) for the stationary

gas become more or less equal one to each other and the same happens to kNO,dec (green curve 2) and kNO,form (pink curve 4) for the flowing gas. It seems that, for the adopted conditions, the effect of the gas recirculation starts when the equilibrium between NO decomposition and NO formation is reached.

### 3.4. The role of multiple pulses

In figure 17, reporting the species number densities, the kinetics is studied with application of multiple impulses of the same duration as the first one, at intervals of 10⁻⁵ s. The initial conditions are the ones of TC5 (see table 2). O and N atoms are produced in correspondence with the impulses, and immediately contribute to new NO production. The enhancement in the NO production is clear in the figure, in correspondence with each impulse, with an increasing of one order of magnitude after four impulses.

In the post-discharge phase there is a partial loss of product, but even considering this loss a final product one order of
<figure><img src="image_21.png" /><figcaption>**Figure 18.** NO number density. The conditions are those related to TC5 (table 2). Either a unique impulse (blue curve) or 3 pulses (red curve) or 4 pulses (green curve) are considered in the simulation.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>**Figure 19.** N, O and NO number densities time evolution with second pulse at $5 \times 10^{-3}$ s after the first one. Curve 1(blue): first pulse evolution; curve 2 (red): second pulse evolution.</figcaption></figure>

magnitude higher than in the one-impulse case is obtained, as shown in figure 18.

It is verified that by running a second pulse at $t = 5 \times 10^{-3}$ s with the same pulse duration of 200 ns, after further $5 \times 10^{-3}$ s (see section 2.3 Input parameters) the species number densities are almost the same as without the second pulse, as shown in figure 19, at variance with the previous test case. The results show the importance of a high repetition rate to get more efficient NO production.

This model can be the basis for a systematic study in strict collaboration with experimental researchers aimed at assessing the best strategy for maximizing the performance of nitric oxide production from air by non-equilibrium plasmas.

## 4. Conclusions

By considering a mixture composed of nitrogen and 5% oxygen, the N₂ fixation, the NO formation and the oxygen behavior are numerically investigated in an atmospheric pressure and low temperature discharge—post-discharge environment.

The discharge goes on $2 \times 10^{-7}$ s, whereas the post-discharge lasts $5 \times 10^{-3}$ s. The chosen reduced electric field during the discharge is $E/n_{tot} = 74 \times 10^{-21}$ Vm$^2$.

In this study various features are analyzed by calculating the species number densities, the molecular vibrational population distributions, the vibrational energies, the pseudo-first order decomposition and formation constant of the molecules.

Firstly, at a temperature equal to 500 K, the different electron-heavy particle e-V and e-D reactions are assessed and it is verified that all of them are essential in the simulations. In fact neglecting some of them underestimates the effect of the discharge.

Secondly, the role of the gas temperature is explored by replicating the calculations at 300 K, 380 K and 500 K. In this temperature range, the temperature almost does not affect the analyzed quantities during the discharge. On the other side, during the post-discharge the vibrational distributions of the medium-high levels of N₂ and O₂ and of all the vibrational levels of NO, as well as the NO number density, decrease as the temperature increases. As guessed, the opposite happens to the N and O number density.
At the same time, the study of the NO pseudo-first order formation and decomposition constants at T = 380 K helps in understanding the NO behavior and shows how its trend is due to the mutual effect of the two Zeldovich reactions. Moreover, it is realized that the NO decomposition in the discharge region is mainly due to the e-D reactions, whose effect overcomes the effects of the two backward Zeldovich reactions. In the same way, also the N₂ and O₂ decomposition in the discharge is essentially due to the e-D reactions.

At this stage, the simulation of the gas recirculation is implemented in the zero-dimensional code. Gas recirculation means that fresh gas is pumped in the apparatus at a certain rate. This fresh gas enters the device at the initial conditions, delaying the physical-chemical evolution of the gas itself, as expected.

Finally, a train of either three or four impulses is applied to the system, showing how multiple pulses, at the proper repetition rate, enhance the NO formation.

The plan for the next future is to use the model developed in the present study in synergy with the experimental researchers to assess the best configuration to maximize the nitrogen monoxide production from air by means of non-equilibrium plasmas.

## Data availability statement

The data cannot be made publicly available upon publication because they are not available in a format that is sufficiently accessible or reusable by other researchers. The data that support the findings of this study are available upon reasonable request from the authors.

## Acknowledgment

Work funded by the European Union under the National Recovery and Resilience Plan (NRRP)—NextGenerationEU, projects PRIN 2022 PLASMODD Prot. 2022J5NBBN.

V.L. thanks Region Bretagne for funding via Stratégie d'Attractivité Durable (SAD) program.

The authors are grateful to Dr Giorgio Dilecce (CNR ISTP, Italy), Dr Paolo F. Ambrico (CNR ISTP, Italy), Dr Domenico Bruno (CNR ISTP, Italy) and Dr Francesco Taccogna (CNR ISTP, Italy) for useful discussion.

## ORCID iDs

I Armenise ID 0000-0002-9440-0026

V Laporta ID 0000-0003-4251-407X

F Esposito ID 0000-0003-0586-5866

## References

[1] Šimek M and Bonaventura Z 2018 Non-equilibrium kinetics of the ground and excited states in N₂–O₂ under nanosecond discharge conditions: an extended scheme and comparison with available experimental observations *J. Phys. D* **51** 504004

[2] Guerra V, Tejero-del-Caz A, Pintassilgo C D and Alves L L 2019 Modelling N₂–O₂ plasmas: volume and surface kinetics *Plasma Sources Sci. Technol.* **28** 073001

[3] Taccogna F and Dilecce G 2016 Non-equilibrium in low-temperature plasmas *Eur. Phys. J. D* **70** 251

[4] Jans E R, Raskar S, Yang X and Adamovich I V 2021 Kinetics of metastable N₂(A³Σu⁺,ν) molecules in high-pressure nonequilibrium plasmas *Plasma Sources Sci. Technol.* **30** 025003

[5] Raskar S, Berry M, Ting L T, Jans E R and Adamovich I V 2024 MHz sampling rate measurements of N₂(A³Σu⁺) population in a Ns pulse discharge in a heated plasma flow reactor *Plasma Sources Sci. Technol.* **33** 105019

[6] Kunova O V and Nagnibeda E A 2014 State-to-state description of reacting air flows behind shock waves *Chem. Phys.* **441** 66–76

[7] Kustova E V, Savelev A S and Kunova O V 2018 Rate coefficients of exchange reactions accounting for vibrational excitation of reagents and products *AIP Conf. Proc.* **1959** 060010

[8] Su W, Bruno D and Babou Y 2018 State-specific modeling of vibrational relaxation and nitric oxide formation in shock-heated air *J. Thermophys. Heat Transfer* **32** 337–52

[9] Campoli L, Kunova O, Kustova E and Melnik M 2020 Models validation and code profiling in state-to-state simulations of shock heated air flows *Acta Astronaut.* **175** 493–509

[10] Kravchenko D, Kunova O, Kustova E and Melnik M 2024 Reflected shock waves in air components and their mixtures: validation of theoretical models *Acta Astronaut.* **218** 47–58

[11] Kravchenko D, Kunova O, Kustova E and Melnik M 2024 Effect of exchange reactions and NO vibrational excitation on shock-heated air component flows *Chem. Phys. Lett.* **847** 141331

[12] Ferreira C M, Gordiets B, Tatarova E, Henriques J and Dias F M 2012 Air–Water microwave plasma torch as a NO source for biomedical applications *Chem. Phys.* **398** 248–54

[13] Graves D B 2014 Reactive species from cold atmospheric plasma: implications for cancer therapy *Plasma Process. Polym.* **11** 1120–7

[14] Brun P, Bernabè G, Marchiori C, Scarpa M, Zuin M, Cavazzana R, Zaniol B and Martines E 2018 Antibacterial efficacy and mechanisms of action of low power atmospheric pressure cold plasma: membrane permeability, biofilm penetration and antimicrobial sensitization *J. Appl. Microbiol.* **125** 398–408

[15] Zampieri L, Ibba L, Agus R, Furno I, Zuin M, De Masi G, Cordaro L, Cavazzana R and Martines E 2025 Combined characterization of a plasma jet: FTIR, chemical surveys and antibacterial effect *Plasma Chem. Plasma Process.* **45** 485–501

[16] Chen H, Yuan D, Angjian W, Lin X and Xiaodong L 2021 Review of low-temperature plasma nitrogen fixation technology *Waste Dispos. Sustain. Energy* **3** 201–17

[17] Waskow A, Ibba L, Leftley M, Howling A, Ambrico P F and Furno I 2021 An *in situ* FTIR study of DBD plasma parameters for accelerated germination of *arabidopsis thaliana* seeds *Int. J. Mol. Sci.* **22** 11540

[18] Šimek M and Homola T 2021 Plasma-assisted agriculture: history, presence, and prospects—a review *Eur. Phys. J. D* **75** 210

[19] Ambrico P F, Zicca S, Ambrico M, Rotondo P R, De Stradis A, Dilecce G, Saponari M, Boscia D and Saldarelli P 2022 Low temperature plasma strategies for *xylella fastidiosa* inactivation *Appl. Sci.* **12** 4711

[20] Maliszewska I, Nowinski D, Fujera J, Arora G, Šimek M and Czapka T 2024 The efficacy of non-equilibrium dielectric barrier discharge in air on the elimination of
phytopathogenic fungi from wooden surface *Food Control* **162** 110438

[21] Aceto D, Ambrico P and Esposito F 2024 Air cold plasmas as a new tool for nitrogen fixation in agriculture: underlying mechanisms and current experimental insights *Front. Phys.* **12** 1455481

[22] Van Alphen S, Vermeiren V, Butterworth T, van den Bekerom D C M, van Rooij G J and Bogaerts A 2020 Power pulsing to maximize vibrational excitation efficiency in N₂ microwave plasma: a combined experimental and computational study *J. Phys. Chem. C* **124** 1765–79

[23] Vervloessem E, Aghaei M, Jardali F, Hafezkihiabani N and Bogaerts A 2020 Plasma-based N₂ fixation into NOₓ: insights from modeling toward optimum yields and energy costs in a gliding arc plasmatron *ACS Sustain. Chem. Eng.* **8** 9711–20

[24] Bahnamiri O S, Verheyen C, Snyders R, Bogaerts A and Britun N 2021 Nitrogen fixation in pulsed microwave discharge studied by infrared absorption combined with modelling *Plasma Sources Sci. Technol.* **30** 065007

[25] Vervloessem E, Gorbanev Y, Nikiforov A, De Geyter N and Bogaerts A 2022 Sustainable NOₓ production from air in pulsed plasma: elucidating the chemistry behind the low energy consumption *Green Chem.* **24** 916–29

[26] Fujera J, Hoffer P, Prukner V, Rotondo P R, Arora G, Jirásek V, Luke's P and Šimek M 2024 Surface DBD in moist air for nitrogen fixation: a comparative study of pulsed versus amplitude-modulated AC powered discharge *J. Appl. Phys.* **57** 405205

[27] Fujera J, Homola T, Jirásek V, Ondráček J, Tarabová B, Prukner V and Šimek M 2024 Aerosol-based multihollow surface DBD: a promising approach for nitrogen fixation *Plasma Sources Sci. Technol.* **33** 075002

[28] Yu S, Vervloedt S C L and von Keudell A 2024 Controlled synthesis of NO in an atmospheric pressure plasma by suppressing NO destruction channels by plasma catalysis *J. Phys. D* **57** 245203

[29] Esposito F 2022 On the relevance of accurate input data for vibrational kinetics in air cold plasmas: the case of nitrogen fixation *Plasma Sources Sci. Technol.* **31** 094010

[30] Armenise I 2023 Low temperature state-to-state vibrational kinetics of O + N₂(v) and N + O₂(v) collisions *Chem. Phys.* **571** 111937

[31] Armenise I and Esposito F 2021 N+O₂(v) collisions: reactive, inelastic and dissociation rates for state-to-state vibrational kinetic models *Chem. Phys.* **551** 111325

[32] Hong Q, Bartolomei M, Pirani F, Esposito F, Sun Q and Coletti C 2022 Vibrational deactivation in O(³P) + N₂ collisions: from an old problem towards its solution *Plasma Sources Sci. Technol.* **31** 084008

[33] Esposito F and Armenise I 2021 Reactive, inelastic, and dissociation processes in collisions of atomic nitrogen with molecular oxygen *J. Phys. Chem. A* **125** 3953–64

[34] Esposito F, Macdonald R, Boyd I D, Neitzel K and Andrienko D A 2019 Heavy-particle elementary processes in hypersonic flows In *Hypersonic Meteoroid Entry Physics* (IOP Publishing) pp 16–1 to 16–32

[35] Esposito F 2019 Reactivity, relaxation and dissociation of vibrationally excited molecules in low-temperature plasma modeling *Rend. Lincei Sci. Fis. Nat.* **30** 57–66

[36] Esposito F, Garcia E and Laganà A 2017 Comparisons and scaling rules between N+N₂ and N₂+N₂ collision induced dissociation cross sections from atomistic studies *Plasma Sources Sci. Technol.* **26** 045005

[37] Esposito F and Armenise I 2017 Reactive, inelastic, and dissociation processes in collisions of atomic oxygen with molecular nitrogen (https://doi.org/10.1021/acs.jSEA.7b04442)

Esposito F and Armenise I 2020 Correction to reactive, inelastic, and dissociation processes in collisions of atomic oxygen with molecular nitrogen *J. Phys. Chem. A* **124** 1465

[38] Celiberto R et al 2016 Atomic and molecular data for spacecraft re-entry plasmas *Plasma Sources Sci. Technol.* **25** 033004

[39] Capitelli M, Celiberto R, Colonna G, Esposito F, Gorse C, Hassouni K, Laricchiuta A and Longo S 2016 Reactivity and relaxation of vibrationally/rotationally excited molecules with open shell atoms *Fundamental Aspects of Plasma Chemical Physics; Springer Series on Atomic, Optical, and Plasma Physics* vol 85 (Springer) pp 31–56

[40] Laporta V, Little D A, Celiberto R and Tennyson J 2014 Electron-impact resonant vibrational excitation and dissociation processes involving vibrationally excited N₂ molecules *Plasma Sources Sci. Technol.* **23** 065002

[41] Laporta V, Celiberto R and Tennyson J 2013 Resonant vibrational-excitation cross sections and rate constants for low-energy electron scattering by molecular oxygen *Plasma Sources Sci. Technol.* **22** 025001

[42] Laporta V, Celiberto R and Tennyson J 2015 Dissociative electron attachment and electron-impact resonant dissociation of vibrationally excited O₂ molecules *Phys. Rev. A* **91** 012701

[43] Laporta V, Vialetto L and Guerra V 2022 Vibrational excitation cross sections for non-equilibrium nitric oxide-containing plasma *Plasma Sources Sci. Technol.* **31** 054001

[44] Laporta V, Tennyson J and Schneider I F 2020 Vibrationally resolved NO dissociative excitation cross sections by electron impact *Plasma Sources Sci. Technol.* **29** 05LT02

[45] Laporta V, Heritier K L and Panesi M 2016 Electron-vibration relaxation in oxygen plasmas *Chem. Phys.* **472** 44–49

[46] Armenise I, Capitelli M, Colonna G and Gorse C 1996 Nonequilibrium vibrational kinetics in the boundary layer of re-entering bodies *J. Thermophys. Heat Transfer* **10** 397–405

[47] Capitelli M, Armenise I and Gorse C 1997 State-to-state approach in the kinetics of air components under re-entry conditions *J. Thermophys. Heat Transfer* **11** 570–8

[48] Armenise I, Capitelli M and Gorse C 1998 Nitrogen nonequilibrium vibrational distributions and non-Arrhenius dissociation constants in hypersonic boundary layers *J. Thermophys. Heat Transfer* **12** 45–51

[49] Armenise I and Capitelli M 2005 State to state vibrational kinetics in the boundary layer of an entering body in earth atmosphere: particle distributions and chemical kinetics *Plasma Sources Sci. Technol.* **14** S9–S17

[50] Armenise I 2017 Excitation of the lowest CO₂ vibrational states by electrons in hypersonic boundary layers *Chem. Phys.* **491** 11–24

[51] Esposito F, Armenise I and Capitelli M 2006 N-N₂ state to state vibrational-relaxation and dissociation rates based on quasiclassical calculations *Chem. Phys.* **331** 1–8

[52] Esposito F, Armenise I, Capitta G and Capitelli M 2008 O-O₂ state-to-state vibrational relaxation and dissociation rates based on quasiclassical calculations *Chem. Phys.* **351** 91–98

[53] Billing G D and Fisher E R 1979 VV and VT rate coefficients in N₂ by a quantum-classical model *Chem. Phys.* **43** 395–401

[54] Billing G D 1986 Vibration-vibration and vibration-translation energy transfer, including multiquantum transitions in atom-diatom and diatom-diatom collisions *Nonequilibrium Vibrational Kinetics* ed M Capitelli (Springer) pp 85–112

[55] Billing G D and Kolesnick R E 1992 Vibrational relaxation of oxygen. State to state rate constants *Chem. Phys. Lett.* **200** 382–6

[56] Cacciatore M, Capitelli M and Dilonardo M 1978 Non-equilibrium vibrational population and dissociation rates of
oxygen in electrical discharges: the role of atoms and of the recombination process *Beitr. Plasmaphys. 18* 279–99

[57] Doroshenko V M, Koudriatsev N N, Novikov S S and Smetanin V V 1990 Dependence of heat transfer on the formation of vibrationally excited nitrogen molecules during the recombination of atoms in a boundary layer *High Temp. 28* 70–76

[58] Akpinar S, Armenise I, Defazio P, Esposito F, Gamallo P, Petrongolo C and Sayos R 2012 Quantum mechanical and quasiclassical born-Oppenheimer dynamics of the reaction $N_2(X^1\Sigma_g^+)+O(^3P)\rightarrow N(^4S)+NO(X^2\Pi)$ on the $N_2O\bar{a}^3A$" and $b^3A$' surfaces" *Chem. Phys. 398* 81–89

[59] Rockwood S D 1973 Elastic and inelastic cross sections for electron-Hg scattering from Hg transport data *Phys. Rev. A* **8** 2348–58

[60] Rockwood S D and Greene A E 1980 Numerical solutions of the Boltzmann transport equation *Comput. Phys. Commun.* **19** 377–93

[61] Capitelli M, Dilonardo M and Gorse C 1981 Coupled solutions of the collisional Boltzmann equation for electrons and the heavy particle master equation in nitrogen *Chem. Phys. 56* 29–42

[62] Bardsley J N and Mandl F 1968 *Rep. Prog. Phys.* **31** 471

[63] Domcke W 1991 Physics reports *Eur. J. Pharmacol.* **208** 97

[64] Werner H-J *et al* 2020 *J. Chem. Phys.* **152** 144107

[65] Tennyson J 2024 *J. Phys. B* **57** 233001

[66] Tejero-del-Caz A, Guerra V, Gonçalves D, Lino da Silva M, Marques L, Pinhão N, Pintassilgo C D and Alves L L 2019 The Lisbon kinetics Boltzmann solver *Plasma Sources Sci. Technol.* **28** 043001

[67] Ambrico P F, Mrkvickova M, Ambrico M, Dilecce G and Šimek M 2022 OPO-based measurements of atonic oxygen and nitric oxide in nanosecond discharges *FLTPD XIV Frontiers in Low Temperature Plasma Diagnostics* (Book of Abstract) (Levico Terme, Italy, 1–5 May 2022) p 31

[68] Ambrico P F and Dilecce G private communication

[69] Gordiets B F, Ferreira C M, Guerra V L, Loureiro J M A H, Nahorny J, Pagnon D, Touzeau M and Vialle M 1995 Kinetic model of a low-pressure $N_2-O_2$ flowing glow discharge *IEEE Trans. Plasma Sci.* **23** 750–67

[70] Youfan H *et al* 2021 Zero-dimensional and pseudo-one-dimensional models of atmospheric-pressure plasma jets in binary and ternary mixtures of oxygen and nitrogen with helium background *Plasma Sources Sci. Technol.* **30** 105017

[71] Arora G, Hoffer P, Prukner V, Bilek P and Šimek M 2024 Properties and characteristics of the nanosecond discharge developing at the water-air interface: tracking evolution from a diffused streamer to a spark filament *Plasma Sources Sci. Technol.* **33** 025025