# Kinetic mechanism and sub-ns measurements of the thermal spark in air

Nicolas Q Minesi, Pierre B Mariotto, Erwan Pannier, Axel Vincent-Randonnier, Gabi Daniel Stancu, Christophe O Laux

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Nicolas Q Minesi, Pierre B Mariotto, Erwan Pannier, Axel Vincent-Randonnier, Gabi Daniel Stancu, et al. Kinetic mechanism and sub-ns measurements of the thermal spark in air. Plasma Sources Science and Technology, 2023, 10.1088/1361-6595/acc9da. hal-04061759v2

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Submitted on 13 May 2023

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# Kinetic mechanism and sub-ns measurements of the thermal spark in air

Nicolas Q. Minesi<fnref n="1" />,<fnref n="2" />, Pierre B. Mariotto<fnref n="1" />, Erwan Pannier<fnref n="1" />, Axel Vincent-Randonnier<fnref n="3" />, Gabi Daniel Stancu<fnref n="1" />, and Christophe O. Laux<fnref n="1" />

E-mail: nicolas.minesi@centralesupelec.fr

## Abstract

This experimental and numerical study is focused on the formation of fully ionized plasmas in ambient air by nanosecond pulsed discharges, namely the thermal spark. The first contribution of this article is the experimental characterization of the electron number density during the pulse. An increase of the electron number density up to $10^{19} \text{ cm}^{-3}$ was measured with sub-nanosecond resolution via three techniques based on optical emission spectroscopy (OES): Stark broadening of $H_\alpha$, Stark broadening of $N^+/O^+$, and the continuum emission of electrons. The discharge diameter is measured with sub-nanosecond resolution using calibrated OES of the $N^+$ and $O^+$ lines. All measurements indicate a transition to a micrometric-size filament of fully ionized plasma in approximately 0.5 ns. The second main contribution of this work is the development of a 0-D kinetic mechanism to explain this observation. The mechanism includes 100 reactions, 12 species, and 12 excited electronic states. Particular attention is paid to modeling the $N_2$, $N_{2}^{+}$, N, and O electronic state kinetics using the electronic states as additional pseudo-species. Our results show that including the electron-impact ionization of the excited electronic states of N and O, in addition to those of $N_2$, is necessary to explain the experimental results, emphasizing the key role of excited state kinetics in the thermal spark formation.

Keywords: thermal spark, optical emission spectroscopy, ionization cross-section, electronic states, Z-pinch, filamentary discharge, Stark broadening

## 1. Introduction

In a book that has become a reference in plasma physics, Raizer classified the different types of plasma that can be produced in laboratories and nature. Among the various plasma sources in this ranking, “*the spark discharge stands out sharply*” [1]. In laboratories, a discharge is typically generated between two electrodes when an electric field above the breakdown threshold is applied. If the current is sustained for a time sufficiently long, “*the spark current produces a cathode spot and the spark transforms into [a continuous] arc*” [1]. This transition between the transient spark phase (or “breakdown”) and the arc phase was experimentally observed in millisecond discharges employed in the automotive industry for the ignition of combustible mixtures [2]. While the spark phase is typically a non-equilibrium plasma, the arc is at local thermal and chemical equilibrium. For decades, the short duration of nanosecond discharges

<footnote n="1">¹ EM2C, CNRS, CentraleSupélec, Université Paris Saclay, 3 Rue Joliot-Curie, 91190 Gif-sur-Yvette, France</footnote>

<footnote n="2">² Department of Mechanical and Aerospace Engineering, University of California, Los Angeles (UCLA), CA 90095, USA</footnote>

<footnote n="3">³ ONERA – Multi-Physics Department for Energy, Chemin de la Hunière, 91123 Palaiseau, France</footnote>
has been leveraged to generate non-equilibrium plasmas before their transitions to the arc at thermal equilibrium [3]. These non-equilibrium plasmas can be sustained at a quasi-steady state by repeating the nanosecond discharge at kHz frequencies. Nanosecond repetitively pulsed (NRP) discharges feature rich non-thermal chemistry and therefore present a broad interest in several fields of engineering [4], such as plasma-assisted combustion [5,6] and plasma medicine [7]. They have been studied in various gas mixtures with the electrodes facing open gaps [8–16] or separated by dielectric barriers [17–19].

In air, Pai et al. distinguished three regimes of NRP discharges: (i) NRP-corona, (ii) NRP-glow, and (iii) NRP-spark [9]. A summary of the typical characteristics of these discharges is provided in Table 1. Typical emission spectra of these NRP-corona, glow, and spark discharges in air are dominated by N₂(B) and N₂(C) emissions. The emission of O atoms and N₂⁺(B) is also present in the glow and spark regimes. Criteria for the corona-glow and glow-spark transitions were proposed in [9]. The corona and glow discharges are characterized by high voltages, low conduction currents (< 1 A), and low deposited energies (less than 100 µJ per pulse) [9]. These two non-equilibrium discharges produce moderate gas heating (less than 200 K) and relatively low electron number densities (less than 10¹³ cm⁻³ in air at atmospheric pressure) [10]. Corona discharges are confined near the electrodes, whereas the glow and the spark fill the entire interelectrode space. The NRP-spark is characterized by a much higher current (> tens of A) and higher energy deposition (> 1 mJ per mm per pulse) [9,11,20]. The gas temperature in the nanosecond spark was found to increase by thousands of kelvins in a few tens of nanoseconds due to the ultrafast heating mechanism [11,21–24]. Although the electron number density reaches 10¹⁵-10¹⁶ cm⁻³ in the nanosecond non-thermal spark [21], the plasma does not reach equilibrium.

<table><caption>**Table 1** Typical parameters of NRP corona, glow [10], spark [11,21], and thermal spark [25,26] [27] discharges in air near ambient pressure. The deposited electrical energy is estimated for discharge dimensions of approximately 1 mm.</caption><thead><tr><th rowspan="2"></th><th colspan="3">Non-equilibrium</th><th>Equilibrium</th></tr><tr><th>Corona</th><th>Glow</th><th>Spark</th><th>Thermal spark</th></tr></thead><tbody><tr><th>Regime</th><td colspan="3"></td><td></td></tr><tr><th>Deposited Energy</th><td>&lt;10 µJ</td><td>~10 µJ</td><td>~1 mJ</td><td>~1 mJ</td></tr><tr><th>Dominant Emission</th><td>N<sub>2</sub></td><td>N<sub>2</sub></td><td>N<sub>2</sub>, O</td><td>N, O, N<sup>+</sup>, O<sup>+</sup></td></tr><tr><th>Maximum electron Number Density</th><td>&lt;10<sup>13</sup> cm<sup>-3</sup></td><td>10<sup>13</sup> cm<sup>-3</sup></td><td>10<sup>15</sup>-10<sup>16</sup> cm<sup>-3</sup></td><td>10<sup>19</sup> cm<sup>-3</sup></td></tr><tr><th>Temperature Increase</th><td>Negligible</td><td>&lt;200 K</td><td>~1000 K</td><td>~40,000 K</td></tr></tbody></table>

However, recent observations of nanosecond discharges do not fit in the three regimes mentioned above. In a study of single microsecond pulses in ambient air, Lo et al. [28] observed a complete ionization of the gas with high electron number densities (10¹⁸ – 10¹⁹ cm⁻³) that do not correspond to any of the discharge regimes described above. The transition from the partially to fully ionized plasma, called “streamer-to-arc” was shorter than 5 ns and accompanied by a dramatic temperature rise, from 1,200 K to 36,000 K, in less than 5 ns. Observations of fully ionized plasmas under nanosecond discharges were also reported in water vapor [29,30] and air [31,32]. Observations of N⁺ emission for nanosecond discharges in pin-to-pin configuration were also presented in [33,34]. These authors showed that the “constriction” was accompanied by the emission of “spots” at the electrodes. In a series of papers, Parkevich et al. used laser interferometry and low astigmatism lenses to show that the fully ionized discharge propagates in the form of more than ten fully ionized filaments, 10-50 µm in diameter, whereas the discharge looks homogeneous if conventional lenses are used [35–38]. These publications in the 2010s extended the earlier studies performed decades ago for longer discharge durations. In 1977, Stritzke et al. [39] mapped the electron temperature during and after the full ionization using lines of N⁺, N⁺⁺, N⁺⁺⁺, and N⁺⁺⁺⁺. For the conditions of Stritzke et al., the emission of N⁺ appeared at electron number densities above 10¹⁶ cm⁻³, and the electron temperature reached 50,000 K. Similar ionization fractions were also measured in 1977 by Albrecht et al. [40] using Stark broadening. In the literature on transient fully ionized plasmas, several denominations were employed by different authors. We recently called this regime “the thermal spark” to distinguish it from the previously observed corona, glow, and non-thermal spark regimes
characterized by Pai et al. [9]. This naming emphasizes the thermal equilibrium state of the plasma (thermal) but also employs the denomination of Raizer for transient discharges (spark) [1].

A counterpart of the thermal spark was concomitantly observed in Surface Dielectric Barrier Discharge (SDBD). Houpt and Leonov [41] showed that the “contraction” of the channel of a 50-µs SDBD in ambient air is associated with the appearance of a continuum spectrum and lines of N+, N, O, and Hα. Using a 20-ns SDBD, Shcherbanev et al. [42] observed a transition from molecular emission to a continuum-dominated spectrum accompanied by an increased Stark broadening of Hα (20-30 nm) and O<sub>777nm</sub> (5 nm) lines. They named it the “streamer-to-filament” transition [43,44]. The same group measured electron number densities n<sub>e</sub> = 10<sup>18</sup> – 10<sup>19</sup> cm<sup>-3</sup> in similar configurations and carefully documented the propagation of the surface fully ionized filaments [42]. More recently, they measured surface filament diameters of 10 µm [45]. These values are comparable to those of Parkevich et al. [35–38,46] observed in pin-to-pin configuration and mentioned earlier. This flagrant similarity in temperature, composition, and spatiotemporal scales suggests that SDBD filaments and thermal sparks share similar kinetics.

It should be noted that the typical timescales of the transition to full ionization are sub-nanosecond, which, added to the micrometric dimensions, make the thermal sparks very challenging to characterize. Nevertheless, our group recently performed sub-nanosecond imaging and optical emission spectroscopy [25] and showed that thermal sparks are generated in five steps: (i) Formation of a partially ionized plasma across the inter-electrodes gap; (ii) Formation of a fully ionized filament at the cathode in less than 0.5 ns. The filament is at thermal equilibrium; (iii) Formation of a similar filament at the anode, delayed compared to the cathode filament; (iv) Propagation of these two fully ionized filaments toward each other; and (v) Merging of the filaments to form the thermal spark. This five-step process matched<fnref n="1" /> the observation also reported in [47, p.82]. After the energy deposition ends, the fast electron number density decay was proven to be due to an isentropic expansion during which the plasma remains at equilibrium [25,28,32]. Although the thermal spark formation was experimentally characterized, the reasons behind the abrupt formation of the fully ionized filaments remained unclear, especially the short (0.5 ns) transition from a partially to a fully ionized plasma. To answer these questions, numerical simulations of the thermal spark had to be performed.

Kinetic mechanisms for non-equilibrium discharges already exist in the literature [23,48–52], typically based on the work of Kossyi *et al.* [53] or Capitelli *et al.* [54]. For the streamer-to-filament transition (in SDBD), Shcherbanev *et al.* [42] performed kinetic simulations (230 Td in N₂ at 6 bar) to explain the fast increase of *n*<sub>e</sub> up to values of about 10<sup>19</sup> cm<sup>-3</sup>. The authors reported in their simulation an increase of the ionization degree by three orders of magnitude in 0.25 ns, due mainly to the dissociation and ionization of N₂(B) and N₂(C). However, this mechanism cannot produce ionization fractions above 10%. Therefore, a key mechanism is missing to explain the formation of a fully (100%) ionized plasma in a fraction of a nanosecond. To address this issue, we proposed in Ref. [26] to include the excitation and ionization of the N and O electronic states by electron impact. With this additional set of reactions, the full ionization in air was obtained. However, experimental confirmation of the model’s predictions was still needed.

This paper investigates the formation of thermal sparks generated in air by NRP discharges in a pin-to-pin configuration. First, in Section 2, the experimental setup is described. In Section 3, the temporal evolution of the electron number density and the discharge diameter are measured. In Section 4, a kinetic mechanism to explain the phenomenon of full ionization by nanosecond pulses is proposed. Finally, in Section 5, the results of these simulations are validated by comparison with the measurements.

<footnote n="1">¹ However, some experimental results demonstrated that the filaments formation (step 2. and 3.) were synchronized within 0.5 ns [46,116]. As underlined by [36,46,116], the electrode curvature and surface quality plays a role in the filament formation. The presence of a strong ionic current near the electrode could foster beneficial conditions, such as surface topology or temperature, that could lead to earlier filament formation.</footnote>
## 2. Experimental setup

The experimental setup, presented in Figure 1, is extensively described in our previous study [25]. Briefly, a nanosecond discharge is generated (FID FPG 30-100MC4K) at 1 kHz between two pin electrodes separated by 1.2 mm. The experiments are performed in ambient air at 0.1 MPa (humidity below saturation) and within 30 minutes before any erosion of the cathode is noticeable. In specific experiments, we use a mixture of ambient air and $H_2$ (3.7%) to measure the electron number density from the $H_α$ Stark broadening). The discharge is placed at the focal point of an off-axis parabolic mirror ($f = 10$ cm), as shown in Figure 1. The collimated light from the plasma is then refocused by a second parabolic mirror ($f = 20$ cm) onto the monochromator (Acton 500i) entrance slit. The monochromator is equipped with two gratings of 300 and 1800 gr·mm$^{-1}$, both blazed at 500 nm. An ICCD camera (Pi-MAX4) coupled to the monochromator is used to record the spectra with a gate of 450 ps. The pixel size of the ICCD is 13 µm. A HeNe laser (Thorlabs) is used to determine the instrumental broadening of the optical system. Absolute intensity calibration of the optical system is performed using an integration lamp (OL455, Gooch and Housego).

<figure><img src="image_2.png" /><figcaption>Figure 1 Experimental setup (same as in previous work [25] except for a change in the length of the coaxial cables).</figcaption></figure>

The discharge is generated by a positive 10-ns pulse (sent to the anode) and a negative 10-ns pulse (sent to the cathode). Hence, the energy deposition is monitored by two high-voltage probes (LeCroy PPE20kV) and two current transformers (Pearson 6585) connected to an oscilloscope (Lecroy HDO 6104). The measured voltage and current are represented in Figure 2. The voltage and current measurements are performed in the coaxial circuit, 4 meters away from the electrodes. Thus, the incident 10-ns voltage pulse can be separated from the reflected pulse arising due to impedance mismatch.
<figure><img src="image_3.png" /><figcaption>Figure 2 Measured voltage and current on a) the anode and b) the cathode sides. The amplitude difference between the incident pulses at the anode and the cathode is only due to a technical peculiarity of the pulser.</figcaption></figure>

The instantaneous power is the product of the voltage, U, and current, I. The total energy, E, is the time integration of the power in Eq. 1, where the indices ‘cat’ and ‘an’ denote the cathode and the anode, respectively.

$$E(t) = E_{cat}(t) + E_{an}(t) = \int_{0}^{t} I_{cat}(t)U_{cat}(t)dt + \int_{0}^{t} I_{an}(t)U_{an}(t)dt \quad \text{Eq. 1}$$

In Figure 2, $t = 0$ ns is defined as the moment the nanosecond pulse reaches the electrode, accounting for the 15-ns delay between the measurement of the voltage rise and its actual arrival at the electrode. As shown in Figure 3, the incident anode and cathode pulses carry 3.9 mJ to the interelectrode gap. Due to an impedance mismatch between the coaxial cable and the plasma, a part of the energy (1.5 mJ) is reflected towards the generator, defined as the upstream direction. Another impedance mismatch on the generator (switched OFF for $t > 10$ ns) leads to a new reflection. Thus, a 2nd pulse travels in the downstream direction and is detected at $t = 75$ ns. The second pulse carries ~ 0.8 mJ and deposits ~ 0.25 mJ in the interelectrode gap. For the same reasons, a 3rd pulse is detected at $t = 150$ ns and deposits another 0.25 mJ. This sequential energy deposition was already observed in previous work [20,25,55] and could explain the “ringing” observed in Ref. [56]. In the comparison of experiments and simulations in Section 5, we restrict the study of the electron number density increase for $t < 20$ ns, avoiding the complications due to secondary pulses. The instantaneous applied electric field and current, reconstructed from the sum of the incident and reflected pulses at both electrodes, are shown later in Figure 9.
<figure><img src="image_4.png" /><figcaption>Figure 3 Total transmitted energy according to Eq. 1. Approximately 2.5, 0.25, and 0.25 mJ are deposited in the 1st, 2nd, and 3rd pulses, respectively.</figcaption></figure>

<table><caption>Table 2 Summary of the experimental parameters of this work, compared to those of our previous study [25].</caption><thead><tr><th rowspan="2"></th><th colspan="2">Previous work</th><th rowspan="2">This work</th></tr><tr><th>Setup #1 in [25]</th><th>Setup #2 in [25]</th></tr></thead><tbody><tr><td>Electrode polarities</td><td>Cathode-Anode</td><td>Cathode-Ground</td><td>Cathode-Anode</td></tr><tr><td>Interelectrode gap</td><td>0.9 mm</td><td>2 mm</td><td>1.2 mm</td></tr><tr><td>Repetition frequency</td><td>10 Hz (single pulse)</td><td>10 Hz (single pulse)</td><td>1 kHz</td></tr><tr><td>Deposited energy</td><td>2.5 - 3.5 mJ per pulse</td><td>2.5 mJ per pulse</td><td>2.5 mJ (0 - 20 ns)<br>3.0 mJ (0 - 100 ns)</td></tr><tr><td>Maximum ΔV</td><td>8.5 kV</td><td>5.5 kV</td><td>8 kV</td></tr><tr><td>Pressure</td><td>5.8 kPa - 0.1 MPa</td><td>0.1 MPa</td><td>0.1 MPa</td></tr><tr><td>Gas</td><td>Ambient air</td><td>Ambient air<br>(with 1.6% of H<sub>2</sub>)</td><td>Ambient air<br>(with 3.7% of H<sub>2</sub>)</td></tr></tbody></table>

The experimental parameters used in the present study are summarized in Table 2. The experimental parameters of our previous work, performed in a single pulse regime, are also compared.

### 3. Experimental results

In Section 3.a, the measurements of the electron number density using the Stark broadening of $H_α$, at $λ = 656$ nm, and $N^+$, at $λ = 500$ nm, are described. Each line covers a different range of ionization degrees and their overlap serves as a validation of our procedure. In Section 3.b, the discharge radius is measured using the calibrated Optical Emission Spectroscopy (OES) of $N^+$ and $O^+$ recorded near $λ = 500$ nm. In Section 3.c, the electron number density is measured via the electron continuum emission. The measurements are spatially resolved along the interelectrode gap. In Section 3.d, we compare the results obtained with the different methods in the region near the cathode, which is the location where the transition to a fully ionized plasma is the fastest.
a. Stark broadening of H<sub>α</sub>

A review of line broadening for several atoms and ions, including N<sup>+</sup>, O<sup>+</sup>, and H<sub>α</sub>, was presented in our previous work [25]. In this Section, the primary sources of broadening of the H<sub>α</sub> transition are calculated, showing that Stark broadening is the dominant mechanism. For the N<sup>+</sup> lines, the methods employed in this study are the same as in [25] and are not described here.

The electron number density can be accurately measured using the Stark broadening of the H<sub>α</sub> line at 656 nm. The Half-Width at Half Maximum (HWHM) of H<sub>α</sub> results from several broadening mechanisms: Doppler, resonant, natural, Van der Waals, Stark, and instrumental [25,57,58]. In the following, we estimate the non-Stark contributions to the HWHM of the H<sub>α</sub> line and show that they are negligible. The broadening formulas used in the present work were initially developed by Griem [57] and simplified for H<sub>β</sub> in [58]. The work of Ref. [58] was recently improved and extended to H<sub>α</sub>, H<sub>β</sub>, O, N, O<sup>+</sup>, and N<sup>+</sup> [25]. In the following, the gas transformation is assumed to be isochoric with an initial ambient number density of 2.7×10<sup>19</sup> cm<sup>-3</sup>. The gas temperature varies from 300 K (initial condition) to ~30,000 K (fully ionized plasma), which induces moderate changes in the **Doppler** HWHM. The number density of atomic hydrogen is estimated to be below 10<sup>15</sup> cm<sup>-3</sup>, resulting in negligible **resonant broadening** (HWHM < 1 pm). **Natural broadening** is also negligible. **Van der Waals broadening** results from the perturbation of the H<sub>α</sub> transition by non-hydrogenic particles (N<sub>2</sub>, O<sub>2</sub>, O, N, O<sup>+</sup>, and N<sup>+</sup>). The sum of all the contributions is given in Eq. 2, where N is the total gas density, T<sub>gas</sub> the gas temperature, and x<sub>pert</sub> the perturber mole fraction. The coefficients γ<sub>pert</sub> are specific to each transition-perturber couple and are calculated in Ref. [25].

$$ \Delta\lambda_{\text{Van der Waals}}[\text{nm}] = \frac{N}{2.7 \times 10^{19} [\text{cm}^{-3}]} \left(\frac{T_{\text{gas}}}{273 \text{ [K]}}\right)^{0.3} \sum_{\text{pert}} \left(\frac{\gamma_{\text{pert}}}{100} x_{\text{pert}}\right) \quad \text{Eq. 2} $$

The sum in Eq. 2 can be simplified using γ<sub>mean</sub> defined in Eq. 3.

$$ \gamma_{\text{mean}} = \sum_{\text{pert}} (x_{\text{pert}}\gamma_{\text{pert}}) \qquad \text{Eq. 3} $$

In Table 3, the value of γ<sub>mean</sub> is calculated in air. During the thermal spark formation, the gas composition varies from N<sub>2</sub>:O<sub>2</sub> at 300 K, to N:O at 20,000 K, and finally N<sup>+</sup>:O<sup>+</sup>:e<sup>-</sup> at 40,000 K. For these compositions, the Van der Waals HWHM varies from 34 to 267 pm. For comparison, when the plasma is partially ionized at 20,000 K, a Stark HWHM of 34 pm corresponds to n<sub>e</sub> ~ 10<sup>14</sup> cm<sup>-3</sup>, which is well below our sensitivity limit. When the gas is fully ionized and T = 40,000 K, a Stark broadening of 34 pm would correspond to an electron number density of 3×10<sup>16</sup> cm<sup>-3</sup>, which is three orders of magnitude lower than the measured electron number density. The Van der Waals broadening is therefore negligible compared to the Stark broadening.

<table><caption>**Table 3** Estimated γ<sub>mean</sub> coefficient and Van der Waals broadening HWHM of the H<sub>α</sub> line in ambient air at 300 K, dissociated air at 20,000 K, and fully ionized air at 40,000 K.</caption><thead><tr><th>Composition</th><th>γ<sub>mean</sub></th><th>T (K)</th><th>N (cm<sup>-3</sup>)</th><th>Δλ<sub>VdW</sub> (pm)</th></tr></thead><tbody><tr><td>N<sub>2</sub> / O<sub>2</sub></td><td>3.36</td><td>300</td><td>2.7×10<sup>19</sup></td><td>34</td></tr><tr><td>N / O</td><td>3.64</td><td>20,000</td><td>5.4×10<sup>19</sup></td><td>150</td></tr><tr><td>N<sup>+</sup> / O<sup>+</sup></td><td>3.26</td><td>40,000</td><td>5.4×10<sup>19</sup></td><td>267</td></tr></tbody></table>

The instrumental broadening for a given spectrometer focal length mainly depends on the spectrometer slit opening and its grating. We used two gratings in this study to follow the variations of the H<sub>α</sub> HWHM from 0.1 nm to 20 nm. The corresponding instrumental HWHM is given in Table 4. A typical example of the H<sub>α</sub> line fit is shown in Figure 4. The H<sub>α</sub> width uncertainty is about ±20%, which leads to a 30% uncertainty on the n<sub>e</sub> measured with this technique.
<figure><img src="image_5.png" /><figcaption>Figure 4 Fit of the H<sub>α</sub> line at t = 2.25 ns (1800-gr/mm grating).</figcaption></figure>

The Stark broadening HWHM can be calculated<fnref n="2" /> with Eq. 4 [59,60].

$$ \frac{n_e}{10^{17} \text{ cm}^{-3}} = \left( \frac{\Delta\lambda_{\text{Stark,H}\alpha}}{0.549 \text{ nm}} \right)^{1.47134} \qquad \text{Eq. 4} $$

In this work, Stark broadening measurements are performed during the high-voltage pulse. In Ref. [61], it was shown that the applied electric field could have an impact on the Stark broadening. In that case, the width of the spectral line would be related to the current density instead of the electron number density. In Ref. [61], expressions are derived for the Stark broadening of H<sub>α</sub> and H<sub>β</sub> generated by the sum of two fields:

* A microscopic field, arising from the presence of ions and electrons in plasma. This is the typical Stark broadening effect.

* A macroscopic field applied in one direction during the experiment. This field can interfere with the Stark broadening.

According to Bastien & Marode [61], the H<sub>β</sub> Stark broadening is mainly due to the shifting of intense external lines caused by the static ions' microfield. Thus, the addition of an external field has an impact on the H<sub>β</sub> Stark broadening. On the other hand, Bastien & Marode showed that the addition of an external field has a weak impact on the H<sub>α</sub> lineshape. For instance, at n<sub>e</sub> = 10<sup>17</sup> cm<sup>-3</sup> and T<sub>e</sub> = 10,000 K, the Stark HWHM of H<sub>α</sub> remains independent of the applied field up to at least 12 kV·mm<sup>-1</sup> (four times the Holtsmark field, see Table 1 of [61]). In our experiments, the maximum applied field is 6.7 kV·mm<sup>-1</sup> when n<sub>e</sub> ~ 10<sup>17</sup> cm<sup>-3</sup>. Therefore, the applied field does not influence our Stark broadening measurements of H<sub>α</sub> during the pulse.

To summarize, the Doppler, resonant, and Van der Waals HWHM are negligible or smaller than the Stark broadening width, see Table 4. Their contribution is however included in the post-processing of the OES spectra.

<table><caption>Table 4 Estimation of the H<sub>α</sub> line HWHM (in pm) for the present conditions based on the formula given in Ref. [25]. See also Table 3 for the estimation of the Van der Waals HWHM.</caption><thead><tr><th>Time (ns)</th><th>T<sub>gas</sub> (K)</th><th>Doppler (pm)</th><th>Resonant (pm)</th><th>Van der Waals (pm)</th><th>Instrumental broadening (pm)</th><th>Stark (pm)</th></tr></thead><tbody><tr><td>0 – 4</td><td>300 – 3000</td><td>4 – 12</td><td>&lt;1</td><td>&lt; 34</td><td>48 (1800 gr·mm<sup>-1</sup>)</td><td>100 – 600</td></tr></tbody></table>

<footnote n="2">² The relation given initially by Gigosos *et al.* [59] must be corrected by a factor 2 as noted by Konjević *et al.* [60].</footnote>
<table><tr><td>3 – 20</td><td>3000 – 30,000</td><td>12 - 38</td><td>&lt;1</td><td>&lt; 267</td><td>342<br/>(300 gr·mm<sup>-1</sup>)</td><td>600 – 23,000</td></tr></table>

## b. Discharge radius measurement using N⁺ and O⁺ path-integrated number density

In this subsection, the results of the intensity-calibrated OES near 500 nm are presented and employed to measure the discharge optical diameter. In the work of Shcherbanev et al. [42], it was reported that the emission of filamentary SDBD was the sum of two concentric channels. In their case, the intense N⁺ and continuum spectral features were emitted from the central core, whereas a weaker N₂ emission originated from the outer shell. Shcherbanev et al. reported a central core Full Width at Half Maximum (FWHM) of 18 ± 2 µm. In a configuration similar to ours, Orrière measured the FWHM of a thermal spark [47]. He found an FWHM of 50 µm for a 6-kV discharge of 20 ns applied across a 1-mm gap between the electrodes<fnref n="3" />. However, no spectral filter was used; therefore, the measurement of Orrière et al. is an upper bound of their thermal spark diameter. In a 25-kV discharge of 40 ns at ambient conditions, Parkevich et al. [36] used interferometry to show that their discharge was composed of several fully ionized filaments, 10 – 50 µm in diameter. A recent study by Parkevitch and Khirianova [62] suggests that the formation of several smaller propagating filaments (instead of a single one) occurs above a current threshold of 40 – 100 A. Finally, Lo et al. clearly observed a constriction of the discharge channel during the transition to the thermal spark in a microsecond pulsed discharge in air [28].

In the present study, the plasma column diameter is estimated by focusing the discharge image on the CCD with a magnification factor of 2 and employing a 10-nm bandpass filter centered at 480 nm (Hard Coated OD 4.0 10-nm Bandpass Filter, Edmund Optics). An accurate measurement is challenging with our present setup because of the shot-to-shot spatial displacement of the discharge (~10 µm) and because the column width is comparable with the pixel width, which could result in pixel bleeding. Nevertheless, the CCD images provide a first-order estimate of the plasma diameter of the order or less than the pixel resolution, i.e. $\Delta l = 15$ µm. This value is similar to the measurements of Shcherbanev et al., Orrière, and Parkevich et al.

At T ~ 30,000 K, the plasma reaches complete ionization and hence thermal and chemical equilibrium [25]. At these temperatures, after an isochoric transformation, the plasma is mainly composed of electrons (50%), and ions N⁺ (40%) and O⁺ (10%) [63]. The electron number density was determined as described in Section 3.a. Thus, because the plasma is neutral, the number density of the ions (N⁺ and O⁺) is known and equal to $n_e$. The spectral emissivity of a uniform emission spectrum due to a given transition at $\lambda$ [nm] is a function of the upper state ion number density, $n_{ion}$[cm⁻<fnref n="3" />], the photon energy ($hc/\lambda$), the transition upper state energy and degeneracy, $\epsilon_u$ [J] and $g_u$, the temperature, T [K], the partition function, $Q(T)$, the transition Einstein coefficient, A [s⁻¹], the plasma optical diameter $\Delta l$ [m⁻¹], the absorption coefficient k [m⁻¹], and the broadening function, $\phi$ [nm⁻¹], see Eq. 5 [64].

$$ \varepsilon[\mathrm{W}\cdot\mathrm{cm}^{-3}\cdot\mathrm{sr}^{-1}\cdot\mathrm{nm}^{-1}] = n_{\mathrm{ion}} \frac{g_u \exp(-\varepsilon_u/kT)}{Q(T)} \frac{A}{4\pi} \frac{hc}{\lambda} \phi \frac{k}{1 - \exp(-k\Delta l)} \quad \text{Eq. 5} $$

Thus, the knowledge of the calibrated intensity spectrum of a given line of N⁺ or O⁺ and the electron density can provide a direct measurement of the discharge diameter, $\Delta l$.

<footnote n="3">³ For a 2-kV pulse across 200-µm gap, Orriere measured a FWHM of 40 µm.</footnote>
<figure><img src="image_6.png" /><figcaption>Figure 5 Fit of the N+ and O+ emission near the cathode at t = 9.75 ns. The continuum emission is approximated by a linear function (in blue) and the N+ and O+ emission is calculated by Specair (in red). The N+ and O+ lines are calculated, accounting for self-absorption.</figcaption></figure>

A typical spectrum is shown in Figure 5. The spectrum is fitted using an in-house extension of Specair [58,65] and accounting for self-absorption. For O+ and N+ lines, the Stark line broadening dominates and the lineshape fit provides another measurement of $n_e$. The uncertainty on the Stark broadening fitting (and $n_e$ determination) is below 20%. The continuum radiation (assumed to be linear) is subtracted and studied in Section 3.c. All electronic states of the ions are assumed to follow a Boltzmann distribution at $T_e$. The measured electronic temperature ($T_e$ = 35,000 K) corresponds to values usually reported in the thermal spark literature [25,28,32]. The accuracy of the $T_e$ measurements is ± 5%, implying that the path-integrated radiance is at best 30% accurate.

Because the plasma is globally neutral, we have $n_e = n_{N+} + n_{O+}$, where $n_e$ is taken from the N+ Stark broadening. The inferred plasma diameter evolution using this assumption and Eq. 5 is shown in Figure 6. This method theoretically offers a micrometer resolution and allows tracking the growth of the filament with sub-nanosecond resolution during its development. Initially, the thermal spark diameter is only about 1 µm, then grows to an asymptotic value of about 10 µm after about 20 ns. These filament diameters (about 10 µm at the end of the discharge pulse) are consistent with those reported for SDBD, i.e. 18 µm, during the transition to a fully ionized plasma [42]. Also, the estimate based on CCD imaging (15 µm) is within the error bars. This time-resolved optical diameter is used in the next section to process the electron continuum measurements.

We now discuss the potential effects arising due to such small diameters where the magnetic pressure could play a role. Assuming an isochoric heating from 300 to 30,000 K, complete dissociation and ionization, then the kinetic pressure is $P_k \sim 400$ atm. The magnetic pressure, $P_m$, can be estimated from Eq. 6, where $\mu_0 = 4\pi \cdot 10^{-7} \text{ H}\cdot\text{m}^{-1}$ is the magnetic permeability and B the tangential magnetic field.

$$P_m = \frac{B^2}{2\mu_0} \qquad \text{Eq. 6}$$

The tangential magnetic field on the filament boundary can be approximated by the magnetic field generated by an infinitely thin straight wire. Using $B = \mu_0 I / 2\pi R_f$, where I is the electrical current, and $R_f$ is the filament radius, the magnetic pressure is given by Eq. 7 [42].
$$P_m = 0.157 \left( \frac{I[A]}{R_f[\mu m]} \right)^2 \text{[atm]} \qquad \text{Eq. 7}$$

Using typical values corresponding to the present case, $I = 200$ A and $R_f = 1$ µm, we obtain $P_m = 6300$ atm ~ 10 $P_k$. Thus, a z-pinch effect is likely to reduce the discharge diameter and prevent its expansion until the end of the current pulse. For $t > 7$ ns, the current decreases (see later in Figure 9), which decreases the magnetic pressure and allows the diameter to increase.

No explanation is currently provided in the literature for the origin of micrometer filaments. The theory of Raizer [1] on leader propagation does not explain this phenomenon. As $n_e$ increases near the cathode, the cathode sheath size and the electron collision cross-section decrease. As a consequence, the cathode sheath becomes collisionless and runaway electrons can be formed [66–68]. These highly energetic electrons enter the partially ionized zone and could leave a trail of seed electrons behind, which could trigger the transition to a fully ionized region. This potential explanation needs validation and could be simulated in air by combining the approach of Levko and Raja [66,67] with the kinetic mechanism presented in the next section.

<figure><img src="image_7.png" /><figcaption>Figure 6 Discharge optical diameter evolution deduced from the ionic line radiation, and comparison with the diameter estimated using CCD imaging (dashed blue line).</figcaption></figure>

### c. Continuum radiation of electrons

This Section describes another method for measuring the electron number density, this time using the electron continuum emission. As shown in several works, continuum radiation can be significant in laser-induced plasmas [69], DBD discharges [42], or pin-to-pin nanosecond discharges [70]. Recently, laser absorption by the electron continuum was employed to measure the ionization degree of a fully ionized plasma [71].

The electron continuum emission results from the free-free radiation (also called Bremsstrahlung) and the free-bound radiation (also called recombination continuum). The continuum emission coefficient is given in Eq. 8, where $n_e$ and $n_i$ are the electron and ion number densities, $Z_i$ the ion charge, $T_e$ the electron temperature, $\xi$ the free-bound Gaunt factor, and $G$ the free-free Gaunt factor [72].

$$\varepsilon_c(\lambda, T_e, n_e) = C_8 Z_i \frac{n_e n_i}{\lambda^2 \sqrt{T_e}} [\xi(1 - e^{-u}) + G e^{-u}] \qquad \text{Eq. 8}$$

If several types of ions are present in the mixture, the total continuum radiation will be a sum over $Z_i$, $i$ being the ion species. In the present case, we consider that $N^+$ is the dominant ion and $Z_i = 1$. If the plasma is mainly composed of
neutral atoms, another formulation must be used [73,74]. The $C_8$ constant can be determined with Eq. 9, where e is the electron charge, $k_b$ the Boltzmann constant, c the speed of light, and $m_e$ the electron mass.

$$C_8 = \frac{16\pi e^6}{3c^2\sqrt{6\pi m_e^3 k_b}} = 1.63 \times 10^{-34} [\text{W}\cdot\text{m}^3\cdot\text{nm}\cdot\text{K}^{0.5}\cdot\text{sr}^{-1}] \qquad \text{Eq. 9}$$

The dimensionless variable u is defined in Eq. 10, where h is the Planck constant, $k_b$ the Boltzmann constant, c the speed of light and $\lambda$ the wavelength.

$$u = \frac{hc}{\lambda k_b T_e} \qquad \text{Eq. 10}$$

The free-free Gaunt factor of N⁺ (and O⁺) at $\lambda = 500$ nm varies from $G = 1.08$ to $G = 1.1$ for T in the range 32,000 – 42,000 K, interpolating<fnref n="4" /> the results of [75]. The free-bound Gaunt factor should be averaged on the atom's final states (i.e. after the electron has recombined with the ion). The free-bound Gaunt factor at $\lambda = 500$ nm (= 0.18 Ry) ranges between $\xi = 0.8$ and $\xi = 1.2$ depending on the final bound state (see the values calculated for any ions in Table 1 of Ref. [76]). Here we assume $\xi = 1$ and $G = 1.09$. These coefficients are considered to be accurate<fnref n="5" /> within 10%. Assuming $n_e = n_i$, we obtain $n_e$ [m⁻³] as a function of $\lambda$ [nm], the volumetric emission coefficient $\varepsilon_c$ [W·m⁻³·sr⁻¹·nm⁻¹], and $T_e$ [K], see [72]:

$$n_e^2 = \frac{\lambda^2 \sqrt{T_e}}{C_8 \xi (1 - e^{-u}) + Ge^{-u}} \varepsilon_c \qquad \text{Eq. 11}$$

Solving the radiative transport equation [64], the spectral emissivity $\varepsilon_c$ is related to the continuum intensity $I_c$ [W·m⁻²·sr⁻¹·nm⁻¹], the absorption coefficient k [m⁻¹], and the plasma width $\Delta l$ [m] as follows:

$$I_c = \varepsilon_c \frac{1 - \exp(-k\Delta l)}{k} \qquad \text{Eq. 12}$$

The absorption coefficient, k, is the sum of the continuum and bound-bound absorption, $k_{bb}$ and $k_c$, respectively. In a worst-case scenario, we assume that the absorption coefficient across the entire spectrum is equal to the bound-bound contribution, i.e. $k \sim k_{bb}$. For a plasma diameter of 15 µm and $n_e = n_{N+} = 10^{19}$ cm⁻³, we calculated with Speair that $k_{bb}\Delta l < 0.5$ between 460 and 530 nm. Based on this estimation, the plasma is optically thin, i.e. $I_c \approx \varepsilon_c\Delta l$. This approximation leads to an overestimation of $\varepsilon_c$ by 40% at most, which is acceptable, as shown in the next paragraph. Finally, Eq. 11 can be further simplified as given by Eq. 13, where $n_e$ [m⁻³] is a function of $\Delta l$ [m] the plasma width, $I_c$ [W·m⁻²·sr⁻¹·nm⁻¹] the continuum intensity, $T_e$ [K] the electron temperature, and $\lambda$ [nm] the wavelength of observation.

$$n_e = \left( \frac{\lambda^2 \sqrt{T_e}}{C_8 \xi (1 - e^{-u}) + Ge^{-u}} \frac{I_c}{\Delta l} \right)^{\frac{1}{2}} \qquad \text{Eq. 13}$$

Thanks to the power 1/2 on the right-hand side of Eq. 13, the impact on the electron number density of the uncertainty on the measured intensity $I_c$ is halved. The uncertainty on $T_e$ is about 10%. The free-free and free-bound coefficients vary slowly with $T_e$ and, in this case, are known within ±10%. The intensity calibration of the absolute spectrum is accurate within about ±20%. Considering the thin plasma assumption, the maximum uncertainty on $\varepsilon_c$ is +20% and -60%. The plasma thickness, $\Delta l$, is estimated to be accurate within 30%. Therefore, after propagation of the uncertainties through quadrature summation, the $n_e$ deduced from the continuum emission is found to be accurate within -70% and +40%.

<footnote n="4">⁴ We used the ChiantiPy Python package [118,119] at https://github.com/chianti-atomic/ChiantiPy.</footnote>

<footnote n="5">⁵ If needed, a more precise value could be obtained using the summation performed by ChiantiPy with the values of [120]. As described later, the dominant uncertainty comes from the plasma thickness and the Gaunt factors uncertainty is unimportant.</footnote>
d. Comparison of the electron density measurements

Electron number densities measured with the three techniques described above are compared in Figure 7. The Hα line measurements are performed using two gratings, with an overlap between the acquisitions to ensure data reproducibility. From *t* = 1.25 ns to *t* = 8 ns, *n*<sub>e</sub> increases from 10<sup>16</sup> to 3×10<sup>19</sup> cm<sup>-3</sup>. The steep increase of the electron number density (over a decade) at *t* = 4 ns corresponds to the filament formation. For *t* = 4 – 6 ns, the continuum emission measurements (triangles) are slightly above those obtained from Hα Stark broadening (circles), whereas the Stark broadening of N<sup>+</sup>/O<sup>+</sup> (squares) slightly overestimates the Hα Stark broadening measurements. For *t* > 6 ns, the measurements from the three techniques are in excellent agreement within the error bars. For *t* > 10 ns, the electron density slowly decays to 5×10<sup>18</sup> cm<sup>-3</sup> at 23 ns due to the isentropic expansion of the plasma channel [25,28]. These spectroscopic measurements are compared to those performed by Orrière and Pai [32] in a nanosecond discharge generated in a 200-μm gap via the estimation of the gap conductivity.

<figure><img src="image_8.png" /></figure>

Figure 7 Evolution of the electron number density near the cathode measured with Stark broadening of Hα (red circles), Stark broadening of N⁺ (blue squares), and continuum radiation (purple triangles). We assumed an optically thin plasma for continuum radiation interpretation only (uncertainty propagated in the error bars). Our measurements are compared to those of Orrière and Pai, see Figure 6 of [32].

4. Kinetic model of the thermal sparks

a. Introduction and numerical setup

In this Section, 0-D numerical simulations of plasma kinetics as a function of the reduced electric field are
performed using ZDPlasKin [78]. We assume an isochoric transformation of the plasma, which is typically the case
for nanosecond discharges. The heat capacity ratio is considered constant, c_p/c_v = 1.4, because it has a negligible effect
on the speed of ionization calculated in this work. The Electron Energy Distribution Function (EEDF) is calculated by
solving the Boltzmann equation accounting for the mixture composition at each timestep with BOLSIG+ [79,80], using
cross-sections of electron reactions taken mostly from the LXCat database [81–85]. BOLSIG+ was not tested by its
developers above an ionization fraction of 1% and we did not modify the solver. Nevertheless, it was shown in our
previous work that, for an ionization fraction above 1%, BOLSIG+ gives the same results as a kinetic simulation
assuming a Boltzmann EEDF, see Figure 11 of [26].
Several kinetic mechanisms for low-temperature plasmas are available in the literature [50,53,54] and could have been used as a baseline. For instance, the reaction set of Capitelli et al. was employed in our previous work on the thermal spark [86]. However, these mechanisms include hundreds of reactions, such as the formation of negative ions and N₂Oₓ with (x, y) ≥ 2, which are unnecessary to describe the thermal spark formation. Instead, in this work, we employ a simplified 38-reaction set from Laux et al. [8], supplemented with reactions taken from Popov [23] and Shcherbanev et al. [42]. The reaction set of Laux et al. is adequate because only the essential reactions and species driving the ionization and recombination of plasma discharges in air are included. The selected reactions taken from Popov include the ultrafast heating mechanism, whereas those from Shcherbanev et al. [42] account for the dissociation and ionization of N₂ excited states [87]. The originality of the present mechanism is the addition of the excitation and ionization of N and O electronically excited states [88–90]. This new mechanism includes N₂, O₂, NO, N, O, N₂⁺, O₂⁺, NO⁺, N⁺, O⁺, N⁺⁺, and O⁺⁺. A simplified representation of the kinetics and the main sources employed in this work is given in Figure 8.

<figure><img src="image_9.png" /><figcaption>Figure 8 Overview of the reactions included in the present mechanism [8,23,42,87–90].</figcaption></figure>

b. Detailed balance

The reverse rates are computed in these simulations via detailed balance [91,92]. Let us define Reac. 1 involving four species: A, B, C, and D.

$$A + B \leftrightarrow C + D \qquad \text{Reac. 1}$$

If the forward rate coefficient of Reac. 1, k_f, is known, then the reverse rate coefficient, k_r, can be computed using Eq. 14, where Zₓ is the partition function of species or pseudo-species X.

$$\frac{k_f}{k_r} = \frac{Z_C Z_D}{Z_A Z_B} (T) \qquad \text{Eq. 14}$$

If the collision partners are heavy particles, T = T<sub>gas</sub> in Eq. 14. However, if one electron is involved<sup>6</sup>, then T = T<sub>e</sub>. For instance, in the case of ionization/recombination of atomic nitrogen N(<sup>4</sup>S), see (R76), the detailed balance is equivalent to the Saha formula (Eq. 15), where m<sub>e</sub> is the electron mass, k<sub>b</sub> the Boltzmann constant, T<sub>e</sub> the electron temperature, h the Planck constant, g<sub>x</sub> the degeneracy of the species or pseudo-species X, I the ionization energy of N(<sup>4</sup>S), and k<sub>f,i</sub> the ionization rate computed with BOLSIG+.

<footnote n="6"><sup>6</sup> A particular treatment must be applied for three-body reactions involving two heavies and one electron but does not apply to this work [121].</footnote>
$$k_{r,i} = \frac{1}{2} \left( \frac{2\pi m_e k_b T_e}{h^2} \right)^{-\frac{3}{2}} \frac{g_{N(^4S)}}{g_{N^+}} \exp\left(\frac{I}{k_b T_e}\right) k_{f,i} \qquad \text{Eq. 15}$$

The first term in Eq. 15 is the inverse of the electron partition function and the factor 1/2 accounts for the electron spin. The Saha formula is exact if the EEDF is a Maxwellian (the use of $T_e$ requires to have a Maxwellian EEDF).

### c. List of reactions

This subsection provides the list of reactions and the motivation behind their selection.

<table><caption>Table 5 The baseline kinetic mechanism used in this work is mainly taken from [8]. The forward reaction coefficients are given, with their dependence on either the gas temperature or the electron temperature. The backward reaction rates are always calculated by detailed balance. The star symbol (*) refers to the forward reaction rates identical to those reported in Table 1 of [8].</caption><thead><tr><th colspan="2">Reaction & Temperature dependence</th><th colspan="3">k<sub>f</sub>=A T<sup>b</sup> exp(-E/RT)</th><th rowspan="2">Ref.</th></tr><tr><th></th><th></th><th>k<sub>f</sub></th><th>k<sub>r</sub></th><th>A (cm<sup>3</sup> s<sup>-1</sup>)</th><th>E/R (K)</th></tr></thead><tbody><tr><td colspan="6"><strong>O<sub>2</sub>, NO, N<sub>2</sub> Dissociation / Recombination [8]</strong></td></tr><tr><td>(R1)</td><td>O<sub>2</sub> + N<sub>2</sub> ↔ 2O + N<sub>2</sub></td><td>T<sub>g</sub> T<sub>g</sub></td><td>3.32 10<sup>-3</sup></td><td>-1.5</td><td>59,500</td><td>[93] *</td></tr><tr><td>(R2)</td><td>O<sub>2</sub> + O<sub>2</sub> ↔ 2O + O<sub>2</sub></td><td>T<sub>g</sub> T<sub>g</sub></td><td>3.32 10<sup>-3</sup></td><td>-1.5</td><td>59,500</td><td>[93] *</td></tr><tr><td>(R3)</td><td>O<sub>2</sub> + NO ↔ 2O + NO</td><td>T<sub>g</sub> T<sub>g</sub></td><td>3.32 10<sup>-3</sup></td><td>-1.5</td><td>59,500</td><td>[93] *</td></tr><tr><td>(R4)</td><td>O<sub>2</sub> + N ↔ 2O + N</td><td>T<sub>g</sub> T<sub>g</sub></td><td>1.66 10<sup>-2</sup></td><td>-1.5</td><td>59,500</td><td>[93] *</td></tr><tr><td>(R5)</td><td>O<sub>2</sub> + O ↔ 2O + O</td><td>T<sub>g</sub> T<sub>g</sub></td><td>1.66 10<sup>-2</sup></td><td>-1.5</td><td>59,500</td><td>[93] *</td></tr><tr><td>(R6)</td><td>O<sub>2</sub> + e ↔ O + O(<sup>3</sup>P) + e</td><td>T<sub>e</sub> T<sub>e</sub></td><td>BOLSIG+</td><td></td><td></td><td>[94]</td></tr><tr><td>(R7)</td><td>O<sub>2</sub> + e ↔ O + O(<sup>1</sup>S) + e</td><td>T<sub>e</sub> T<sub>e</sub></td><td>BOLSIG+</td><td></td><td></td><td>[94]</td></tr><tr><td>(R8)</td><td>NO + N<sub>2</sub> ↔ N + O + N<sub>2</sub></td><td>T<sub>g</sub> T<sub>g</sub></td><td>8.30 10<sup>-9</sup></td><td>0.0</td><td>75,500</td><td>[93] *</td></tr><tr><td>(R9)</td><td>NO + O<sub>2</sub> ↔ N + O + O<sub>2</sub></td><td>T<sub>g</sub> T<sub>g</sub></td><td>8.30 10<sup>-9</sup></td><td>0.0</td><td>75,500</td><td>[93] *</td></tr><tr><td>(R10)</td><td>NO + NO ↔ N + O + NO</td><td>T<sub>g</sub> T<sub>g</sub></td><td>1.83 10<sup>-7</sup></td><td>0.0</td><td>75,500</td><td>[93] *</td></tr><tr><td>(R11)</td><td>NO + N ↔ N + O + N</td><td>T<sub>g</sub> T<sub>g</sub></td><td>1.83 10<sup>-7</sup></td><td>0.0</td><td>75,500</td><td>[93] *</td></tr><tr><td>(R12)</td><td>NO + O ↔ N + O + O</td><td>T<sub>g</sub> T<sub>g</sub></td><td>1.83 10<sup>-7</sup></td><td>0.0</td><td>75,500</td><td>[93] *</td></tr><tr><td>(R13)</td><td>N<sub>2</sub> + N<sub>2</sub> ↔ 2N + N<sub>2</sub></td><td>T<sub>g</sub> T<sub>g</sub></td><td>1.16 10<sup>-2</sup></td><td>-1.6</td><td>113,200</td><td>[93] *</td></tr><tr><td>(R14)</td><td>N<sub>2</sub> + O<sub>2</sub> ↔ 2N + O<sub>2</sub></td><td>T<sub>g</sub> T<sub>g</sub></td><td>1.16 10<sup>-2</sup></td><td>-1.6</td><td>113,200</td><td>[93] *</td></tr><tr><td>(R15)</td><td>N<sub>2</sub> + NO ↔ 2N + NO</td><td>T<sub>g</sub> T<sub>g</sub></td><td>1.16 10<sup>-2</sup></td><td>-1.6</td><td>113,200</td><td>[93] *</td></tr><tr><td>(R16)</td><td>N<sub>2</sub> + N ↔ 2N + N</td><td>T<sub>g</sub> T<sub>g</sub></td><td>4.98 10<sup>-2</sup></td><td>-1.6</td><td>113,200</td><td>[93] *</td></tr><tr><td>(R17)</td><td>N<sub>2</sub> + O ↔ 2N + O</td><td>T<sub>g</sub> T<sub>g</sub></td><td>4.98 10<sup>-2</sup></td><td>-1.6</td><td>113,200</td><td>[93] *</td></tr><tr><td colspan="6"><strong>Zeldovich reactions [8]</strong></td></tr><tr><td>(R18)</td><td>N<sub>2</sub> + O ↔ NO + N</td><td>T<sub>g</sub> T<sub>g</sub></td><td>1.06 10<sup>-6</sup></td><td>-1.0</td><td>38,400</td><td>[93] *</td></tr><tr><td>(R19)</td><td>NO + O ↔ O<sub>2</sub> + N</td><td>T<sub>g</sub> T<sub>g</sub></td><td>1.39 10<sup>-11</sup></td><td>0.0</td><td>19,400</td><td>[93] *</td></tr><tr><td colspan="6"><strong>Associative ionization / Dissociative recombination [8]</strong></td></tr><tr><td>(R20)</td><td>N + O ↔ NO<sup>+</sup> + e</td><td>T<sub>g</sub> T<sub>e</sub></td><td>1.46 10<sup>-15</sup></td><td>1.0</td><td>31,900</td><td>[93]</td></tr><tr><td>(R21)</td><td>N + N ↔ N<sub>2</sub><sup>+</sup> + e</td><td>T<sub>g</sub> T<sub>e</sub></td><td>9.97 10<sup>-17</sup></td><td>1.5</td><td>67,500</td><td>[93]</td></tr><tr><td>(R22)</td><td>O + O ↔ O<sub>2</sub><sup>+</sup> + e</td><td>T<sub>g</sub> T<sub>e</sub></td><td>1.18 10<sup>-21</sup></td><td>2.7</td><td>80,600</td><td>[93]</td></tr><tr><td colspan="6"><strong>Electron Impact Ionization of O<sub>2</sub> and NO [8]</strong></td></tr><tr><td>(R23)</td><td>O<sub>2</sub> + e ↔ O<sub>2</sub><sup>+</sup> + 2e</td><td>T<sub>e</sub> T<sub>e</sub></td><td>BOLSIG+</td><td></td><td></td><td>[95]</td></tr><tr><td>(R24)</td><td>O<sub>2</sub> + e ↔ O<sup>+</sup> + O + 2e</td><td>T<sub>e</sub> T<sub>e</sub></td><td>BOLSIG+</td><td></td><td></td><td>[95]</td></tr><tr><td>(R25)</td><td>NO + e ↔ NO<sup>+</sup> + 2e</td><td>T<sub>e</sub> T<sub>e</sub></td><td>BOLSIG+</td><td></td><td></td><td>[96]</td></tr><tr><td>(R26)</td><td>NO + e ↔ N + O<sup>+</sup> + 2e</td><td>T<sub>e</sub> T<sub>e</sub></td><td>BOLSIG+</td><td></td><td></td><td>[96]</td></tr><tr><td>(R27)</td><td>NO + e ↔ O + N<sup>+</sup> + 2e</td><td>T<sub>e</sub> T<sub>e</sub></td><td>BOLSIG+</td><td></td><td></td><td>[96]</td></tr></tbody></table>
<table><thead><tr><th colspan="7">Charge Exchange [8]</th></tr></thead><tbody><tr><td>(R28)</td><td>N<sup>+</sup> + N<sub>2</sub> ↔ N + N<sub>2</sub><sup>+</sup></td><td>T<sub>g</sub> T<sub>g</sub></td><td>1.66 10<sup>-12</sup></td><td>0.5</td><td>12,000</td><td>[93]</td></tr><tr><td>(R29)</td><td>NO<sup>+</sup> + O ↔ N<sup>+</sup> + O<sub>2</sub></td><td>T<sub>g</sub> T<sub>g</sub></td><td>1.66 10<sup>-12</sup></td><td>0.5</td><td>77,200</td><td>[93] *</td></tr><tr><td>(R30)</td><td>NO + O<sup>+</sup> ↔ N<sup>+</sup> + O<sub>2</sub></td><td>T<sub>g</sub> T<sub>g</sub></td><td>2.32 10<sup>-19</sup></td><td>1.9</td><td>26,600</td><td>[93]</td></tr><tr><td>(R31)</td><td>NO<sup>+</sup> + N ↔ N<sub>2</sub> + O<sup>+</sup></td><td>T<sub>g</sub> T<sub>g</sub></td><td>5.65 10<sup>-11</sup></td><td>-1.08</td><td>12,800</td><td>[93]</td></tr><tr><td>(R32)</td><td>O<sup>+</sup> + N<sub>2</sub> ↔ N<sub>2</sub><sup>+</sup> + O</td><td>T<sub>g</sub> T<sub>g</sub></td><td>1.49 10<sup>-12</sup></td><td>0.36</td><td>22,800</td><td>[93]</td></tr><tr><td>(R33)</td><td>NO<sup>+</sup> + N ↔ N<sub>2</sub><sup>+</sup> + O</td><td>T<sub>g</sub> T<sub>g</sub></td><td>1.20 10<sup>-10</sup></td><td>0.0</td><td>35,500</td><td>[93] *</td></tr><tr><td>(R34)</td><td>O<sub>2</sub><sup>+</sup> + N ↔ N<sup>+</sup> + O<sub>2</sub></td><td>T<sub>g</sub> T<sub>g</sub></td><td>1.44 10<sup>-10</sup></td><td>0.14</td><td>28,600</td><td>[93]</td></tr><tr><td>(R35)</td><td>O<sub>2</sub><sup>+</sup> + N<sub>2</sub> ↔ N<sub>2</sub><sup>+</sup> + O<sub>2</sub></td><td>T<sub>g</sub> T<sub>g</sub></td><td>1.64 10<sup>-11</sup></td><td>0.0</td><td>40,700</td><td>[93]</td></tr><tr><td>(R36)</td><td>NO<sup>+</sup> + O<sub>2</sub> ↔ NO + O<sub>2</sub><sup>+</sup></td><td>T<sub>g</sub> T<sub>g</sub></td><td>3.99 10<sup>-11</sup></td><td>0.41</td><td>32,600</td><td>[93]</td></tr><tr><td>(R37)</td><td>NO<sup>+</sup> + O ↔ N + O<sub>2</sub><sup>+</sup></td><td>T<sub>g</sub> T<sub>g</sub></td><td>1.20 10<sup>-11</sup></td><td>0.29</td><td>48,600</td><td>[93]</td></tr></tbody></table>

Most of the reactions used by Laux *et al.* are taken from [93,97]. However, some reaction rate coefficients in [97] are calculated for $T_{gas} = 2000$ K, which is unsuitable for the present work. Therefore, the reaction rates calculated in [93] are replaced with calculations performed using BOLSIG+ with the cross-sections specified in Table 5. In the case of (R28), $N^+ + N_2 \leftrightarrow N + N_2^+$, the rate constant is taken from [93]. Note that some of the rate coefficients of charge exchange reactions were rounded off in Table 1 of [8], but the exact values were used in their simulation and ours.

The ionization of NO by electron impact was initially limited to (R25), NO + e $\leftrightarrow$ NO$^+$ + 2e, in [8]. We added here the formation of O$^+$ and N$^+$, (R26) and (R27) [96]. To our knowledge, the cross-section of NO dissociation by electron impact is not available in the literature [96]. BOLSIG+ computes the rate of O$_2$ dissociation by electron impact based on the cross-sections<fnref n="7" /> provided in [94]. In this model, the N$_2$ dissociation and ionization by electron-impact are state-specific and more details are given in Table 6. The heating due to NO$^+$, N$_2^+$, and O$_2^+$ dissociative recombination was found to have a minor impact on the temperature increase (less than 50 K) [42].

For electron-heavy reactions, the rate constant is a function of the electron temperature. For heavy-heavy reactions, the rate constant is a function of the gas temperature. As discussed in Section 4.b this is critical for associative ionization and dissociative recombination, whose rate constants depend on $T_g$ and $T_e$, respectively. Finally, the dissociative recombination of N$_2^+$, (R21), is calculated with detailed balance using Park's rate of associative ionization. An alternative method, not employed in this study, would have been to take the cross-sections calculated by [87] or measured by [98].

<table><caption>**Table 6 Additional set of reactions calculated with BOLSIG+**</caption><thead><tr><th>Reaction</th><th></th><th>Cross-section</th></tr></thead><tbody><tr><td colspan="3" style="text-align: center; background-color: #E0E0E0;">Electronic Excitation of N<sub>2</sub></td></tr><tr><td>(R37)</td><td>N<sub>2</sub> + e $\leftrightarrow$ N<sub>2</sub>(Y) + e<br/>Y = A, B, C, a'</td><td>[99]</td></tr><tr><td>(R38)</td><td>N<sub>2</sub>(Z) + e $\leftrightarrow$ N<sub>2</sub>(Y) + e<br/>Z = A, B, C, a'<br/>Y = B, C, a'</td><td>[87]</td></tr><tr><td colspan="3" style="text-align: center; background-color: #E0E0E0;">Ionization and dissociation of N<sub>2</sub></td></tr><tr><td>(R39)</td><td>N<sub>2</sub>(X) + e $\leftrightarrow$ N<sub>2</sub><sup>+</sup>(Y) + 2e<br/>Y = X, B</td><td>[100]</td></tr><tr><td>(R40)</td><td>N<sub>2</sub>(X) + e $\leftrightarrow$ N<sub>2</sub><sup>+</sup>(Y) + 2e<br/>Y = A, C</td><td>[87]</td></tr><tr><td>(R41)</td><td>N<sub>2</sub>(Z) + e $\leftrightarrow$ N<sub>2</sub><sup>+</sup>(Y) + 2e</td><td>[87]</td></tr></tbody></table>

<footnote n="7">⁷ Available in the LXCat *Phelps* database or the LXCat *LISBON-IST* database</footnote>
<table><tr><td></td><td>Z = A, B, C, a'</td><td></td></tr><tr><td></td><td>Y = X, A, B, C</td><td></td></tr><tr><td>(R42)</td><td>N<sub>2</sub>(X) + e &harr; N + N(<sup>2</sup>D) + e</td><td>[83,101]</td></tr><tr><td>(R43)</td><td>N<sub>2</sub>(Z) + e &harr; N + N(<sup>2</sup>D) + e</td><td>[87]</td></tr><tr><td></td><td>Z = A, B, a', C</td><td></td></tr><tr><td colspan="3" style="background-color: #E0E0E0; text-align: center; font-weight: bold;">Electronic transitions of N<sub>2</sub><sup>+</sup></td></tr><tr><td>(R44)</td><td>N<sub>2</sub><sup>+</sup>(Z) + e &harr; N<sub>2</sub><sup>+</sup>(Y) + e</td><td></td></tr><tr><td></td><td>Z = X, A, B, C</td><td>[87]</td></tr><tr><td></td><td>Y = X, A, B, C</td><td></td></tr></table>

The cross-sections of the electronic excitation of N₂(X) to higher excited states were measured by Cartwright *et al.* [102] and later renormalized by Trajmar *et al.* [103]. Usually, these cross-sections are linked to the *LXCaT Phelps Database* [104]. In our kinetic simulations, the cross-section of electron-impact excitation of N₂ to N₂(A³Σ⁺) is the sum of the cross-sections to the N₂(A, v') levels taken from [99]. The N₂(B³Πg, B³Σ⁻, W³Δu) states are lumped together in the N₂(B) state, using the renormalized cross-sections of [102]. This is also the case for N₂(C), which is the sum of N₂(C³Πu, E³Σ⁺) states. The N₂(a'¹Σ⁻, a'¹Πg, w'¹Δu) singlet states are lumped together in the N₂(a') state. This grouping was initially proposed in [53] and used in the kinetic mechanism<fnref n="8" /> of [51]. If further refinements of these reactions were needed, one should focus on distinguishing N₂(a'¹Σ⁻) and N₂(a'¹Πg) because their dissociation cross-sections differ by one order of magnitude.

The electronic excitation of N₂(A, B, C, a') by electron impact as well as the transitions between the N₂⁺(X, A, B, C) states by electron impact are taken from [87]. The ionization cross-sections of N₂(X) to N₂⁺(X, B), (R39), were estimated in [100] based on the (normalized) measurement of Cartwright *et al.* and the recommendations of Itikawa<fnref n="9" />. The ionization cross-sections of N₂(A, B, a', C), (R41), are taken from [87]. The dissociation cross-section of N₂ ground state, (R42), is taken from [83,101], whereas the dissociation cross-sections of the electronically excited states, (R43), are from [87]. In electron-impact dissociation of any electronic state of N₂, the fragments are N(⁴S) and N(²D) [83,101]. Reactions involving the excited electronic states of NO and O₂ are neglected in this work because these species are not dominant in these air plasmas<fnref n="10" />.

<footnote n="8">⁸ Available online: www.zdplaskin.laplace.univ-tlse.fr/author/sergey-pancheshnyi/index.html</footnote>

<footnote n="9">⁹ We note however that these ionization cross-sections include the dissociative ionization cross-sections. The impact of this approximation is limited in the present case because N₂(X) ionization only drives the partial ionization phase.</footnote>

<footnote n="10">¹⁰ If O₂ excited states were to be included in a simulation, we recommend using the cross-sections of the LXCat *Phelps database*. see [51]. It was argued, however, that these states have a minor impact on the chemistry [50]. For NO electronic excitation, note that non-investigated discrepancies exist between the LXCat *Itikawa database* and the LXCat *Hayashi database*.</footnote>
<table><caption>Table 7 Electronic excitation and ionization of atomic species by electron impact. The electron configuration and term are provided for each state. Rate coefficients are rounded in the present table and non-rounded values can be found in [90].</caption><thead><tr><th rowspan="2">Reaction</th><th colspan="3">k<sub>f</sub>=A T<sub>e</sub><sup>b</sup> exp(-E/RT)</th><th rowspan="2">Ref.</th></tr><tr><th>A (cm<sup>3</sup>s<sup>-1</sup>)</th><th>b</th><th>E/R (K)</th></tr></thead><tbody><tr><td colspan="5"><strong>Atomic electronic state transitions</strong></td></tr><tr><td>(R45) N(<sup>4</sup>S<sup>0</sup>) + e &harr; N(Y) + e<br/>Y=<sup>2</sup>D<sup>0</sup>, <sup>2</sup>P<sup>0</sup>, <sup>4</sup>P, <sup>2</sup>P, <sup>4</sup>P, <sup>2</sup>S<sup>0</sup>, <sup>4</sup>D<sup>0</sup></td><td></td><td></td><td>BOLSIG+</td><td>[88]</td></tr><tr><td>(R52) N(<sup>2</sup>D<sup>0</sup>) + e &harr; N(Y) + e<br/>Y = <sup>2</sup>P<sup>0</sup>, <sup>4</sup>P, <sup>2</sup>P, <sup>4</sup>P, <sup>2</sup>S<sup>0</sup>, <sup>4</sup>D<sup>0</sup></td><td></td><td></td><td>BOLSIG+</td><td>[88]</td></tr><tr><td>(R58) N(<sup>2</sup>P<sup>0</sup>) + e &harr; N(Y) + e<br/>Y = <sup>4</sup>P, <sup>2</sup>P, <sup>4</sup>P, <sup>2</sup>S<sup>0</sup>, <sup>4</sup>D<sup>0</sup></td><td></td><td></td><td>BOLSIG+</td><td>[88]</td></tr><tr><td>(R63) N(<sup>4</sup>P, 3s) + e &harr; N(Y) + e<br/>Y = <sup>2</sup>P, <sup>4</sup>P, <sup>2</sup>S<sup>0</sup>, <sup>4</sup>D<sup>0</sup></td><td></td><td></td><td>BOLSIG+</td><td>[88]</td></tr><tr><td>(R67) N(<sup>2</sup>P, 3s) + e &harr; N(Y) + e<br/>Y = <sup>4</sup>P, <sup>2</sup>S<sup>0</sup>, <sup>4</sup>D<sup>0</sup></td><td></td><td></td><td>BOLSIG+</td><td>[88]</td></tr><tr><td>(R70) N(<sup>4</sup>P, 2s<sup>2</sup>2p<sup>4</sup>) + e &harr; N(Y) + e<br/>Y = <sup>2</sup>S<sup>0</sup>, <sup>4</sup>D<sup>0</sup></td><td></td><td></td><td>BOLSIG+</td><td>[88]</td></tr><tr><td>(R72) N(<sup>2</sup>S<sup>0</sup>, 3p) + e &harr; N(<sup>4</sup>D<sup>0</sup>) + e</td><td></td><td></td><td>BOLSIG+</td><td>[88]</td></tr><tr><td>(R73) O(<sup>3</sup>P) + e &harr; O(<sup>1</sup>D, <sup>1</sup>S) + e</td><td></td><td></td><td>BOLSIG+</td><td>[89]</td></tr><tr><td>(R75) O(<sup>1</sup>D) + e &harr; O(<sup>1</sup>S) + e</td><td></td><td></td><td>BOLSIG+</td><td>[89]</td></tr><tr><td colspan="5"><strong>Atomic electronic state ionization</strong></td></tr><tr><td>(R76) N(<sup>4</sup>S<sup>0</sup>) + e &harr; N<sup>+</sup> + 2e</td><td></td><td></td><td>BOLSIG+</td><td>[88]</td></tr><tr><td>(R77) N(<sup>2</sup>D<sup>0</sup>) + e &harr; N<sup>+</sup> + 2e</td><td></td><td></td><td>BOLSIG+</td><td>[88]</td></tr><tr><td>(R78) N(<sup>2</sup>P<sup>0</sup>) + e &harr; N<sup>+</sup> + 2e</td><td></td><td></td><td>BOLSIG+</td><td>[88]</td></tr><tr><td>(R79) N(<sup>4</sup>P, 3s) + e &harr; N<sup>+</sup> + 2e</td><td>9.33 10<sup>-10</sup></td><td>0.49</td><td>49,120</td><td>[90]</td></tr><tr><td>(R80) N(<sup>2</sup>P, 3s) + e &harr; N<sup>+</sup> + 2e</td><td>1.67 10<sup>-9</sup></td><td>0.45</td><td>45,090</td><td>[90]</td></tr><tr><td>(R81) N(<sup>4</sup>P, 2s<sup>2</sup>2p<sup>4</sup>) + e &harr; N<sup>+</sup> + 2e</td><td>6.57 10<sup>-11</sup></td><td>0.74</td><td>41,660</td><td>[90]</td></tr><tr><td>(R82) N(<sup>2</sup>S<sup>0</sup>, 3p) + e &harr; N<sup>+</sup> + 2e</td><td>3.74 10<sup>-9</sup></td><td>0.42</td><td>34,580</td><td>[90]</td></tr><tr><td>(R83) N(<sup>4</sup>D<sup>0</sup>, 3p) + e &harr; N<sup>+</sup> + 2e</td><td>4.82 10<sup>-9</sup></td><td>0.41</td><td>32,820</td><td>[90]</td></tr><tr><td>(R84) O(<sup>3</sup>P) + e &harr; O<sup>+</sup> + 2e</td><td></td><td></td><td>BOLSIG+</td><td>[89]</td></tr><tr><td>(R85) O(<sup>1</sup>D) + e &harr; O<sup>+</sup> + 2e</td><td></td><td></td><td>BOLSIG+</td><td>[89]</td></tr><tr><td>(R86) O(<sup>1</sup>S) + e &harr; O<sup>+</sup> + 2e</td><td></td><td></td><td>BOLSIG+</td><td>[89]</td></tr></tbody></table>

In the present work, we neglect<fnref n="11" /> the formation of excited electronic states of N⁺ and O⁺. The forward rate coefficients of electron-impact ionization of the first three levels of N and O, (R76-R78) and (R84-86), are calculated with BOLSIG+ using the ionization cross-sections obtained by the B-spline R-matrix (BSR) approach [88,89]. For states above N(<sup>2</sup>D) and up to N(<sup>4</sup>D<sup>0</sup>), the ionization rate coefficients calculated in [90] are used. We will show in Section 5 that the reactions listed in Table 7 drive the final stage of thermal spark ionization. A summary of the electronic levels of N₂, N, and O modeled in the present mechanism is given in Table 8.

$$A + M \leftrightarrow A^{+} + e + M$$

<footnote n="11">¹¹ We also neglected the three-body recombination with heavy colliders, and therefore did not consider the following reaction, where A = N, O, and M is any heavy collider.</footnote>
<table><caption>Table 8 Electronic levels of N₂, N, and O included in the present work. The energy is given relative to the ground state [105,106] and *g* is the level statistical weight.</caption><thead><tr><th>State</th><th>Configuration</th><th>g</th><th>E<sub>up</sub> (eV)</th><th>State</th><th>g</th><th>E<sub>up</sub> (eV)</th></tr></thead><tbody><tr><td>N(<sup>4</sup>S<sup>0</sup>)</td><td>2s<sup>2</sup>2p<sup>3</sup></td><td>4</td><td>0</td><td>N<sub>2</sub>(X <sup>1</sup>Σ<sub>g</sub><sup>+</sup>)</td><td>1</td><td>0</td></tr><tr><td>N(<sup>2</sup>D<sup>0</sup>)</td><td>2s<sup>2</sup>2p<sup>3</sup></td><td>10</td><td>2.38</td><td>N<sub>2</sub>(A <sup>3</sup>Σ<sub>u</sub><sup>+</sup>)</td><td>3</td><td>6.23</td></tr><tr><td>N(<sup>2</sup>P<sup>0</sup>)</td><td>2s<sup>2</sup>2p<sup>3</sup></td><td>6</td><td>3.575</td><td>N<sub>2</sub>(B <sup>3</sup>Π<sub>g</sub>)</td><td>6</td><td>7.40</td></tr><tr><td>N(<sup>4</sup>P)</td><td>2s<sup>2</sup>2p<sup>2</sup>(<sup>3</sup>P)3s</td><td>12</td><td>10.33</td><td>N<sub>2</sub>(a' <sup>1</sup>Σ<sub>u</sub><sup>-</sup>)</td><td>1</td><td>8.45</td></tr><tr><td>N(<sup>2</sup>P)</td><td>2s<sup>2</sup>2p<sup>2</sup>(<sup>3</sup>P)3s</td><td>6</td><td>10.68</td><td>N<sub>2</sub>(C <sup>3</sup>Π<sub>u</sub>)</td><td>6</td><td>11.06</td></tr><tr><td>N(<sup>4</sup>P)</td><td>2s<sup>2</sup>2p<sup>4</sup></td><td>12</td><td>10.92</td><td>N<sub>2</sub><sup>+</sup>(X <sup>2</sup>Σ<sub>g</sub><sup>+</sup>)</td><td>2</td><td>0</td></tr><tr><td>N(<sup>2</sup>S<sup>0</sup>)</td><td>2s<sup>2</sup>2p<sup>2</sup>(<sup>3</sup>P)3p</td><td>2</td><td>11.60</td><td>N<sub>2</sub><sup>+</sup>(A <sup>2</sup>Π<sub>u</sub>)</td><td>4</td><td>1.13</td></tr><tr><td>N(<sup>4</sup>D<sup>0</sup>)</td><td>2s<sup>2</sup>2p<sup>2</sup>(<sup>3</sup>P)3p</td><td>20</td><td>11.75</td><td>N<sub>2</sub><sup>+</sup>(B <sup>2</sup>Σ<sub>u</sub><sup>+</sup>)</td><td>2</td><td>3.16</td></tr><tr><td>O(<sup>3</sup>P)</td><td>2s<sup>2</sup>2p<sup>4</sup></td><td>9</td><td>0</td><td>N<sub>2</sub><sup>+</sup>(C <sup>2</sup>Σ<sub>u</sub><sup>+</sup>)</td><td>2</td><td>8.02</td></tr><tr><td>O(<sup>1</sup>D)</td><td>2s<sup>2</sup>2p<sup>4</sup></td><td>5</td><td>1.97</td><td></td><td></td><td></td></tr><tr><td>O(<sup>1</sup>S)</td><td>2s<sup>2</sup>2p<sup>4</sup></td><td>1</td><td>4.19</td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>Table 9 Supplementary reactions for the thermal spark transition. Quenching reactions are taken from [23] and the ionization rate coefficients of N⁺ and O⁺ are calculated in this work.</caption><thead><tr><th></th><th>Reaction & Temp. dependence</th><th>A (cm<sup>3</sup> s<sup>-1</sup>)</th><th>b</th><th>E/R (K)</th><th>Ref.</th></tr></thead><tbody><tr><td colspan="6">Quenching of N<sub>2</sub> electronic excited states [23]</td></tr><tr><td>(R87)</td><td>N<sub>2</sub>(A) + O<sub>2</sub> ↔ N<sub>2</sub>(X) + 2O</td><td>T<sub>g</sub></td><td>9.81 10<sup>-14</sup></td><td>0.5</td><td>0</td><td>[53]</td></tr><tr><td>(R88)</td><td>N<sub>2</sub>(A) + O<sub>2</sub> ↔ N<sub>2</sub>(X) + O<sub>2</sub></td><td>T<sub>g</sub></td><td>4.33 10<sup>-14</sup></td><td>0.5</td><td>0</td><td>[53]</td></tr><tr><td>(R89)</td><td>N<sub>2</sub>(A) + O ↔ NO + N(<sup>2</sup>D)</td><td>T<sub>g</sub></td><td>4.00 10<sup>-11</sup></td><td>0.0</td><td>0</td><td>[107]</td></tr><tr><td>(R90)</td><td>N<sub>2</sub>(B) + N<sub>2</sub> ↔ N<sub>2</sub>(A) + N<sub>2</sub></td><td>T<sub>g</sub></td><td>1.00 10<sup>-11</sup></td><td>0.0</td><td>0</td><td>[53]</td></tr><tr><td>(R91)</td><td>N<sub>2</sub>(B) + O<sub>2</sub> ↔ N<sub>2</sub>(X) + 2O</td><td>T<sub>g</sub></td><td>3.00 10<sup>-10</sup></td><td>0.0</td><td>0</td><td>[53]</td></tr><tr><td>(R92)</td><td>N<sub>2</sub>(B) + O ↔ NO + N(<sup>2</sup>D)</td><td>T<sub>g</sub></td><td>3.00 10<sup>-10</sup></td><td>0.0</td><td>0</td><td>[108]</td></tr><tr><td>(R93)</td><td>N<sub>2</sub>(C) + N<sub>2</sub> ↔ N<sub>2</sub>(B) + N<sub>2</sub></td><td>T<sub>g</sub></td><td>1.00 10<sup>-11</sup></td><td>0.0</td><td>0</td><td>[53]</td></tr><tr><td>(R94)</td><td>N<sub>2</sub>(C) + O<sub>2</sub> ↔ N<sub>2</sub>(X) + O + O*</td><td>T<sub>g</sub></td><td>1.73 10<sup>-11</sup></td><td>0.3</td><td>0</td><td>[21]</td></tr><tr><td>(R95)</td><td>N<sub>2</sub>(C) + O ↔ NO + N(<sup>2</sup>D)</td><td>T<sub>g</sub></td><td>3.00 10<sup>-10</sup></td><td>0.0</td><td>0</td><td>[108]</td></tr><tr><td>(R96)</td><td>N<sub>2</sub>(a') + N<sub>2</sub> ↔ N<sub>2</sub>(B) + N<sub>2</sub></td><td>T<sub>g</sub></td><td>2.00 10<sup>-13</sup></td><td>0.0</td><td>0</td><td>[109]</td></tr><tr><td>(R97)</td><td>N<sub>2</sub>(a') + O<sub>2</sub> ↔ N<sub>2</sub>(X) + O + O*</td><td>T<sub>g</sub></td><td>2.80 10<sup>-11</sup></td><td>0.0</td><td>0</td><td>[109]</td></tr><tr><td>(R98)</td><td>N<sub>2</sub>(a') + O ↔ NO + N(<sup>2</sup>D)</td><td>T<sub>g</sub></td><td>3.00 10<sup>-10</sup></td><td>0.0</td><td>0</td><td>[108]</td></tr><tr><td colspan="6">Formation of doubly ionized ions</td></tr><tr><td>(R99)</td><td>N<sup>+</sup> + e ↔ N<sup>++</sup> + 2e</td><td>T<sub>e</sub></td><td>5.10 10<sup>-12</sup></td><td>0.68</td><td>349,690</td><td>This work</td></tr><tr><td>(R100)</td><td>O<sup>+</sup> + e ↔ O<sup>++</sup> + 2e</td><td>T<sub>e</sub></td><td>9.42 10<sup>-14</sup></td><td>0.98</td><td>381,240</td><td>This work</td></tr></tbody></table>

The quenching of N₂ excited states is mainly taken from the work of Popov [23], which is, to our knowledge, the most complete set of reactions for the simulation of non-thermal nanosecond discharges. It was shown in [23] that this set reproduces the ultrafast heating observed in [21] and [16,110] with good accuracy. The quenching rate coefficients of N₂(A) given in Table 9 are slightly modified compared to those of Popov because our model is not vibrational specific and does not include O₂ excited states. Therefore, it was assumed in (R88) that O₂ is not excited to the O₂(b<sup>1</sup>Σ<sub>g</sub><sup>+</sup>) state, which is different from the (R2) reaction in [23].

Also, we discarded the (R3), (R4) and (R6) reactions of [23] because they involve vibrationally excited molecules. The formation of NO by N₂(A) quenching, (R92), was neglected in the early work of Popov [111]. However, it was
shown in [107] and later in [108,112] that the formation of NO may have a higher impact than originally expected, which motivates its addition to this mechanism.

For the conditions of Rusterholtz et al. [21], Popov showed that the dissociative quenching of N₂(A, B, C, a') represent 63% of the total oxygen dissociation [21,23]. Regarding the production of metastable oxygen by N₂(C) quenching, Popov used the branching ratio (46% / 52% / 2%) of metastable argon quenching by O₂ [22,113]. The quenching of metastable xenon, which has a similar energy as N₂(B), does not produce excited atomic oxygen [113]. Therefore, we assume that only O(³P) is produced in N₂(B) quenching. Note that Popov used a rate coefficient constant with temperature for N₂(C) quenching, although it was shown that the rate varies with Tg⁰.³ [21]. The product of N₂(a') quenching is assumed to be O(³P) + O(¹D) with a probability of one [23]. We neglect as well the quenching of N(²D) and O(¹S, ¹D), reactions (R16) to (R34) in [23], because these reactions induce vibrational excitation, which is not included in the present mechanism. The quenching of N(²P) by N₂ was also neglected [114]. Using the rates from [50,53,54], we found that the formation and recombination of N₃⁺ induce negligible variations to the formation of the thermal spark. Therefore, the N₃⁺ chemistry is not included in this work.

Finally, the formation of N⁺⁺ and O⁺⁺ by electron-impact of N⁺ and O⁺ is included, (R99) and (R100). We integrated the cross-sections reported in [115] with a Maxwellian EEDF and fitted the results with Arrhenius laws. These Arrhenius rate coefficients fit the integrated results within 1% in the 40,000 – 73,000 K range, see Table 10, which corresponds to the range of temperature where these reactions have a significant impact.

<table><caption>**Table 10 Deviation of the Arrhenius fits compared to rates calculated from the cross-sections of N⁺ and O⁺ ionization by electron impact, (R99) and (R100).**</caption><thead><tr><th></th><th>23,000 K</th><th>30,000 K</th><th>40,000 K</th><th>73,000 K</th></tr></thead><tbody><tr><td>N⁺ → N⁺⁺</td><td>~ 1%</td><td>&lt; 1%</td><td>&lt; 1%</td><td>&lt; 1%</td></tr><tr><td>O⁺ → O⁺⁺</td><td>~ 30%</td><td>~ 10%</td><td>~ 1%</td><td>&lt; 1%</td></tr></tbody></table>

Note that this set of reactions is incomplete if one is interested in the heating generated by *non-thermal* sparks. For that purpose, the quenching of atomic metastable states may have to be included [49] along with other species such as O, O₃, N₃⁺, and N₄⁺ [50].

## 5. Comparison of the numerical kinetics with the experimental findings

As briefly mentioned before, we previously performed the simulation of canonical discharges in air to explain the formation of a fully ionized plasma in less than 10 ns [26]. In the present study, using a lighter baseline mechanism allows for the inclusion of the reverse rates of the reactions presented in Section 4.c. The chemical mechanism also gained clarity. Another novelty compared to [26] is the inclusion of N⁺⁺ and O⁺⁺ formation. Moreover, in the present work, an experimental voltage waveform is employed to calculate the reduced electric field. The applied voltage is calculated by summing the incident and reflected waveforms. We will show in this section that the sub-nanosecond transition to a fully ionized plasma is still driven by the formation of the excited electronic states of N and O, noted N* and O* here, and follows the same path as extensively described in [26]. The simulation is computationally expensive when accounting for electron-electron collisions in the EEDF calculation. The reader interested in conducting these simulations at a reduced CPU cost could assume a Boltzmann EEDF for an ionization fraction above 1%, which provides very similar results – see Figure 11 of [26].

<table><caption>**Table 11 Initial conditions and numerical parameters.** The initial N₂ and O₂ number densities correspond to air at 0.1 MPa (1 atm) and 300 K, and the initial electron and N₂⁺ number densities to typical ionization in streamer discharges.</caption><thead><tr><th>Initial gas state</th><th>Numerical parameters</th></tr></thead></table>
<table><thead><tr><th>N<sub>2</sub> (cm<sup>-3</sup>)</th><th>O<sub>2</sub> (cm<sup>-3</sup>)</th><th>N<sub>2</sub><sup>+</sup> (cm<sup>-3</sup>)</th><th>e- (cm<sup>-3</sup>)</th><th>T<sub>gas</sub> (K)</th><th>T<sub>e</sub> (K)</th><th>Time step (ps)</th></tr></thead><tbody><tr><td>1.90×10<sup>19</sup></td><td>5.06×10<sup>18</sup></td><td>10<sup>14</sup></td><td>10<sup>14</sup></td><td>300</td><td>1000</td><td>1.0</td></tr></tbody></table>

a. Comparison to electrical measurements

<figure><img src="image_10.png" /><figcaption>**Figure 9** *Experimental: Applied voltage (black) and current (green), both reconstructed from the difference between the incident and reflected pulses. Numerical: reduced field (red line), current density (purple), and electron number density (in blue).*</figcaption></figure>

The simulation uses the initial conditions and numerical parameters listed in Table 11. The time $t = 0$ ns is arbitrarily defined as the first measurement of the electron number density by Stark broadening of $H_\alpha$ with enough signal-to-noise ratio. This instant corresponds approximately to the top of the voltage ramp and the rise of the current. The applied voltage on the 1.2-mm gap is represented in Figure 9 and was reconstructed from the anode and cathode voltage pulse shapes. The reduced field, E/N, is calculated in the simulation at each time step, assuming a Laplacian field between the electrodes and accounting for the density change due to $N_2$ and $O_2$ dissociation. The reduced field is arbitrarily reduced by 10% to account for the cathode fall (900 V at maximal applied voltage). In BOLSIG+, the reduced field E/N is usually defined as the ratio of the electric field to the neutral particle density, $N_{\text{neutral}}$. However, the neutral particle number density can drop to negligible values. Thus, the reduced field tends toward infinity and the electron temperature increases above reasonable values. Therefore, we define $N$ as the heavy particle density, i.e. $N = N_{\text{ion}} + N_{\text{neutral}}$ [26,86]. This assumption, initially proposed in [26,86], was also successfully employed in [116].

In Figure 9, the experimental current going through the discharge gap has been reconstructed by synchronizing and summing the incident and reflected current pulses. For $t < 4$ ns, the typical diameter of the partially ionized plasma is 200 – 300 µm [25], which corresponds to current densities of $j = 10^4 – 10^5$ A·cm⁻² typically found in non-thermal sparks [21,23]. Similar values are found in the simulation up to $t = 4$ ns. Near $t = 4 – 8$ ns, the optical diameter measured with the ionic lines is ~1 µm (Figure 6) and the measured current reaches 150 – 200 A, which corresponds to $j \sim 5·10^9$ A·cm⁻² when the simulation predicts a lower value ($10^8$ A·cm⁻²). Several phenomena could explain this difference:

1. The current density measurement is rather sensitive to the discharge diameter measurement, because $j \propto 1/d^2$, and this difference could be reduced if the diameter were a few µm larger.

2. As observed in [36], there could be a multi-filament structure partially out of the spectrometer field of view.
3. Finally, parallel to the fully ionized core (~ 1 µm), a portion of the current might flow through an outer shell of the plasma channel (~ 100 µm) and have a non-negligible contribution. For instance, assuming the current flowing through the outer shell is the same after the filament formation (i.e. $I_{out} \sim 100$ A) would reduce the experimental current density in the core down to $j \sim 2 \cdot 10^9$ A·cm⁻².

Finally, for $t \sim 10$ ns, the current is still ~100 A but the diameter increases to 10 µm, which decreases the measured current density down to $j \sim 10^7 - 10^8$ A·cm⁻², close to what the simulation indicates.

<figure><img src="image_11.png" /></figure>

**Figure 10** Simplified discharge diameter evolution (red) used to compute the deposited energy (black) in the 0-D approximation from the simulated power (blue). The numerical value of the deposited energy (3.7 mJ) is close to the experimental one (2.5 mJ).

From our measurements in Section 3.b, a simplified evolution of the discharge diameter is used to calculate the deposited power, see Figure 10. The discharge power, *P* [W], is calculated from the power density, *E* × *j* [W·cm⁻³], multiplied by the discharge volume. The deposited energy, shown in Figure 10 from the time integration of *P*, starts increasing from *t* = 0 ns, which corresponds to our first measurement of the electron number density by Stark broadening. From 0 ns to 4 ns, the discharge diameter is assumed to be constant (*d* = 200 µm), as documented in our previous work [25]. The deposited energy sharply increases up to ~ 3.6 mJ, close to the experimental value of 2.5 mJ. From this time step, the discharge diameter drops, as shown by sub-nanosecond imaging in [25], down to *d* = 1 µm. For *t* > 4 ns, the diameter is assumed to increase linearly, see Section 3.b. This small diameter compensates for a high power density of 10¹¹ W·cm⁻³ and the deposited energy slowly varies for *t* > 4 ns. The presence of an outer shell of a partially ionized plasma (*n*ₑ ~ 10¹⁶-10¹⁷ cm⁻³ and *j* = 10⁵ A·cm²) would only slightly increase the total deposited energy by a few mJ.

Based on the current and energy comparison performed in this subsection, we consider the 0-D formalism sufficient to reproduce the main trends of the thermal spark formation. In the following subsection, the channels of electron formation are identified and compared to the experimental trends.
b. Comparison to optical measurements

<figure><img src="image_12.png" /><figcaption>Figure 11 Evolution of the species number densities and temperature during the thermal spark transition in air. Atomic O and O⁺ are omitted for clarity, but they follow the same trends as N and N⁺.</figcaption></figure>

In this section, we compare the optical measurements near the cathode and the simulation. The evolution of the plasma composition is shown in Figure 11 and follows the same behavior as the one reported in our previous work [26]. It is convenient to describe the number density evolution with, in Figure 12, the normalized electron number density variations, $v_e$ [s⁻¹], and the ionization frequency, $v_{ion,k}$ [s⁻¹], of reaction k where $Q_{ion,k}$ [cm⁻³·s⁻¹] is the electron production rate of reaction k and $Q_{rev,k}$ [cm⁻³·s⁻¹] the recombination rate in the reverse direction:

$$v_e = \frac{1}{n_e} \frac{dn_e}{dt} \qquad \text{Eq. 16}$$

$$v_{ion,k} = \frac{Q_{ion,k} - Q_{rev,k}}{n_e} \qquad \text{Eq. 17}$$

For $t < 1$ ns and according to the data in Figure 11 and Figure 12, the plasma remains partially ionized and is mostly composed of N₂ and O₂. At $t = 1$ ns, atomic nitrogen is quickly formed via the dissociation of electronic states of N₂ [26,42]. Ionization is driven by electron-impact on N₂ and O₂, while the ionization of N₂ excited electronic states gets stronger than that of O₂. The ionization fraction reaches ~1% and ~10% of N₂ is dissociated into N. The ionization frequencies by electron impact on N₂(X) and N₂⁺ are approximately equal but, for $t > 1$ ns, are both decreasing with time due to molecular dissociation, see Figure 12 (left). This fall of the ionization frequency from N₂ and N₂⁺ for $t > 1$ ns is compensated by an increase of the ionization frequency of N. The dissociative recombination of NO⁺, N₂⁺, and O₂⁺ induces a non-negligible drop in the total ionization frequency remaining near 1 GHz. The ionization frequency by N, $v_{ion,N}$, is first dominated by N(²D⁰), see Figure 12 (right). From $t > 1$ ns, the ionization frequencies from states above N(²D⁰) start dominating the ionization process. In practice, the addition of states higher than N(⁴P) at 10.92 eV does not significantly alter the kinetics of electron formation (comparison not shown). Near $t = 3$ ns, the dissociative recombination of molecules is negligible, and the reverse rate of N and O ionization is the dominant recombination channel. At this instant and in an interval of $\Delta t = 0.5$ ns, $n_e$ gains almost three orders of magnitude. Consequently, the ion-electron thermalization time, inversely proportional to $n_e$, significantly decreases below the nanosecond scale [1,26]. This sharp increase of $n_e$ drives the translational temperature thermalization with $T_e$ via electron-ion collisions, see also Figure 8 of [26]. The translational temperature, in Figure 11, reaches values near 10,000 K, below the
~30,000 K simulated in our previous work [26] and Shcherbanev et al. [42] because E/N quickly drops below 50 Td.
Note that if these simulations included the magnetic compression effect, the translational temperature would be higher,
but the inclusion of this effect falls outside of the presented 0-D formalism. After t = 4 ns, the plasma is fully ionized
and composed mainly of electrons, N+ and O+.

<figure><img src="image_13.png" /><figcaption>**Figure 12** Evolution of the effective ionization frequency, $v_e$ in dashed black, compared to the ionization frequency through the channels of ionization, $v_{ion,k}$ in full lines. (left) Sum of the ionization channels by electron impact of N$_2$, O$_2$, O and N. Only the major channels are shown and the recombination channels are hidden for clarity. (right) Detailed contributions of each electronic state of N.</figcaption></figure>

The sharp increase of the ionization frequency at $t = 3.5$ ns is also influenced by the following mechanism.
The increase of $n_e$ induces a faster ionization rate (by electron impact) and a faster formation of N* and O* (by electron
impact). The ionization cross-sections of these electronic states present higher amplitudes and lower ionization
thresholds than those of the electronic ground state. In parallel, the increase of $n_e$ shifts the underpopulated high-energy
tail of the EEDF closer to a Boltzmann distribution, see for instance Figure 9 of [26]. Therefore, the ionization rate is
significantly increased. All these retroactive loops lead to a dramatic increase in the electron number density.
<figure><img src="image_14.png" /><figcaption>**Figure 13** Comparison of the electron number density obtained by simulation (black solid line) with measurements performed by Stark broadening and continuum emission. Full ionization is not reached if the simulation is run without the ionization of O and N excited states (red solid line).</figcaption></figure>

Figure 13 compares the evolution of the electron number density measured with Stark broadening of N+, Stark broadening of Hα, and the free-free/free-bound continuum, with the electron number density calculated with our kinetic mechanism. The “1-level model” corresponds to the mechanism of Section 4.c with all reactions except the formation and ionization of N* and O*. In this case, $n_e$ reaches at most $3 \cdot 10^{16}$ cm⁻³ at the end of the 10-ns pulse. With the “Full mechanism”, the electron-impact ionization of the first eight electronic states of N and three electronic states of O is added. The very abrupt ionization phase matches the one observed in the experiments. Note that the two simulations agree for $t < 1$ ns as expected when, below an ionization fraction of 1%, the kinetics of atomic excited electronic states is not primarily driving the chemistry.

The decay of $n_e$ at later times is due to an isentropic expansion lasting more than 100 ns [25,28]. In these simulations, it was assumed that the discharge volume is constant, which is usually correct in non-thermal nanosecond discharges. We believe that the agreement would be improved if the model included the plasma channel expansion, which could begin as early as $t = 6$ ns, see also Figure 6. The consideration of the radiative losses, which could be non-negligible but not dominant (see [25,26]), would also represent an improvement [117]. Although this 0-D approximation is suitable for capturing the main trends of the thermal spark formation, a 1-D formalism would also be helpful to quantitatively assess the second-order influence of the magnetic compression.

## 6. Conclusions

In this work, we study the formation of thermal sparks generated at ambient conditions by an 8-kV pulse of 10 ns across a 1.2-mm gap. In the first part, the increase of the electron number density near the cathode is measured using several optical emission spectroscopy methods. First, the Stark broadening of Hα, at 656 nm, and N+, at 500 nm, are used. Then, after intensity calibration of the emission spectrum near $\lambda = 500$ nm, the optical diameter of the discharge is obtained with sub-nanosecond resolution from atomic ion lines by assuming plasma neutrality. This diameter indicates that magnetic compression is likely to play a role during a few nanoseconds and is later used to determine the electron number density via the continuum radiation near $\lambda = 500$ nm. The three methods used to measure $n_e$ agree with one another and show that the thermal spark’s full ionization occurs on a timescale of approximately 0.5 ns.
In the second part of this article, a 0-D kinetic mechanism is developed to explain the observed abrupt increase of the ionization degree of the nanosecond thermal spark. The mechanism includes N₂, O₂, NO, N, O, N₂⁺, O₂⁺, NO⁺, N⁺, O⁺, N⁺⁺, and O⁺⁺ species. The novelty of the present kinetic mechanism is the addition of the ionization of the excited electronic states of N and O by electron impact. These reactions are found to be essential to reproduce the experimental data obtained in the first part of the article. The agreement between simulations and experiments strongly supports the key role of the atomic excited states in the ionization of the thermal spark filaments. However, the topology and the propagation of the fully ionized filaments remain unclear and require further investigation. Including the electron-impact ionization of N and O excited states in 1-D or 2-D simulations, together with incorporating plasma-electrode interactions, runaway electrons, magnetic compression, radiation, and ionization wave propagation would help to further describe the nanosecond thermal spark.

## 7. Acknowledgments

This work has been supported by the French National Research Agency (ANR-16-CE22-0005 PASTEC, ANR-16-CE30-0004 ASPEN). Dr Nicolas Minesi has been supported by an IDEX Ph.D. fellowship (ANR-11-IDEX-0003-02). The authors express their gratitude to all the contributors of LXCat, BOLSIG+, and ZDPlasKin. We thank the anonymous reviewers for their valuable comments and feedback.

## 8. References

[1] Raizer, Y. P. *Gas Discharge Physics*. Springer Verlag. Berlin., 1991.

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