# Study of CO2 recombination kinetics

Corentin H C Grimaldi, Sean Mcguire, Christophe O Laux

► To cite this version:

Corentin H C Grimaldi, Sean Mcguire, Christophe O Laux. Study of CO2 recombination kinetics. Plasma Sources Science and Technology, 2025, 34 (11), pp.115005. 10.1088/1361-6595/ae1058. hal-05379065

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Submitted on 24 Nov 2025

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# Study of CO₂ recombination kinetics

To cite this article: Corentin H C Grimaldi et al 2025 *Plasma Sources Sci. Technol.* **34** 115005

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

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# Study of CO₂ recombination kinetics

Corentin H C Grimaldi<sup>*</sup>, Sean McGuire* and Christophe O Laux<sup>†</sup>

Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire EM2C, 91190 Gif-sur-Yvette, France

E-mail: sean.mc-guire@centralesupelec.fr

Received 9 April 2025, revised 25 July 2025

Accepted for publication 7 October 2025

Published 12 November 2025

<figure><img src="image_2.png" /></figure>

CrossMark

## Abstract

This paper presents an analysis of thermochemical data obtained in a high temperature recombining CO₂ mixture at atmospheric pressure. For the analysis, three kinetic models taken from the literature are used. All models appear to capture the measured CO density evolution. However, they all underpredict the measured CO₂ density evolution by a multiplicative factor of roughly five. Modifications to the baseline models are considered. A Monte-Carlo analysis is used to simultaneously vary several rates, and as part of an effort to modify the baseline models in as minimal a way as possible. Furthermore, the Monte-Carlo analysis allows us to identify different modifications to the baseline models that result in good agreement with experiment. This offers a level flexibility in the choice of modifications made, in order to account for prior knowledge of certain reaction rates that may be well known. In several cases, the suggested modifications require an increase of the rate of CO₂ recombination by a factor of approximately 10. We argue that such an increase could be explained by recombination through excited states of CO₂.

Keywords: recombining plasma, CO₂ kinetics, chemical nonequilibrium

## 1. Introduction

Recombining CO₂ plasmas occur during planetary entry into the atmosphere of Mars/Venus [1, 2] and in CO₂ conversion applications seeking to produce CO using electric discharges [3]. While a lot of work has focused on CO₂ dissociation kinetics (see, for example, [4–6]), comparatively little work has focused on recombination kinetics. Of note are the works of Huczko and Szymafiski [7], Yang et al [8, 9], Brabbs and Belles [10], Hardy et al [11], Kondratiev and Intezarova [12], and Slanger et al [13]. One major difficulty in studying recombination processes is achieving a well-characterized initial state at high temperature. The inductively coupled plasma (ICP) torch at CentraleSupélec used for the study analyzed in the present article operates at atmospheric pressure. The high

collision frequency at this pressure ensures that the plasma is close to local thermal equilibrium at the exit of the torch [14]. Therefore, the initial conditions for the recombination studies can be characterized.

The experiment presented in the companion article [15] and the thesis of Grimaldi [16] studied the evolution of a recombining CO₂/Ar mixture produced by the ICP torch. Argon was mixed with CO₂ to stabilize the plasma. The injected mixture consisted of 90% Argon and 10% CO₂ by mass. This was done by mounting a water-cooled tube at the exit of the plasma torch (figure 1). The tube forces chemical recombination as the high temperature mixture transits the tube and cools down. The tube, made of brass, has an inner diameter of 1 cm. Measurements were made at the exit of tubes of various lengths. Infrared emission spectroscopy was used to measure the temperature, CO density, and CO₂ density. The emission was calibrated in absolute intensity, collected perpendicular to the jet axis, and measured as a function of jet radius. Absorption from ambient CO₂ and H₂O in the optical path was minimized by purging a portion of the optical path with N₂. The remaining absorption was accounted for in the spectral analysis by simulating the absorption from H₂O and CO₂ and by varying the mole fractions of these species along

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

<footnote>* Author to whom any correspondence should be addressed.</footnote>
<figure><img src="image_4.png" /><figcaption>**Figure 1.** The recombination tube is water-cooled (indicated in blue) and mounted above the torch to rapidly cool the plasma and trigger chemical recombination. The text/lines at the entrance/exit of the tube indicate the measured temperature and primary chemical components.</figcaption></figure>

the optical path to produce the best fit of the experimental spectra. Spatially resolved measurements were achieved by Abel inverting the line-of-sight integrated spectra. Figure 2 shows the measured evolution in these parameters at the center of the recombining flow ($r = 0$ mm) as it transits the recombination tube. Note that there is no electric field in the recombination tube and, given that the plasma is at atmospheric pressure, thermal equilibrium is expected (same translational temperature of electrons and heavy species). The goal of this article is to compare these experimental results with 0D kinetic simulations.

Section 2 introduces the kinetic mechanisms used for the simulations. Sections 3 and 4 discuss the chemical composition and temperature at the recombination tube inlet and the velocity profile of the flow. The conditions at the recombination tube inlet serve as the initial conditions for the kinetic simulations. The velocity profile is used to convert the evolution with distance seen in figure 2 to an evolution in time, which is necessary for comparison with the 0D kinetic simulations. Section 5 presents chemical kinetic simulations using the estimated time-temperature profile. While the measured CO density is found to be correctly predicted by the model, there is a significant discrepancy between the predicted and measured CO₂ densities. Prior studies have shown that uncertainties in reaction rates can lead to large uncertainties in the simulation results [17, 18]. In section 6, a parametric study is undertaken which examines what rates could be modified in the baseline kinetic models to improve agreement.

## 2. Kinetic mechanism

Many CO₂ kinetic mechanisms have been published in the literature [18–30]. Three kinetic mechanisms are used in this work—those of Park [19], Johnston and Brandis [22], and Cruden et al [26]. These will be referred to in this paper as the

<table><caption>**Table 1.** Argon charge exchange and electron impact ionization rates. A is in cm³ mole⁻¹ s⁻¹ and Eₐ is in K.</caption><thead><tr><th>Reactions</th><th>T<sub>x</sub></th><th>A</th><th>n</th><th>E<sub>a</sub></th></tr></thead><tbody><tr><td colspan="5">Charge exchange reactions</td></tr><tr><td>Ar<sup>+</sup> + O ⇌ Ar + O<sup>+</sup></td><td>T<sub>tr</sub></td><td>3.85 × 10<sup>12</sup></td><td>0</td><td>0</td></tr><tr><td>Ar<sup>+</sup> + C ⇌ Ar + C<sup>+</sup></td><td>T<sub>tr</sub></td><td>2.86 × 10<sup>13</sup></td><td>0</td><td>0</td></tr><tr><td>Ar<sup>+</sup> + O<sub>2</sub> ⇌ Ar + O<sub>2</sub><sup>+</sup></td><td>T<sub>tr</sub></td><td>2.77 × 10<sup>13</sup></td><td>0</td><td>0</td></tr><tr><td>Ar<sup>+</sup> + CO ⇌ Ar + CO<sup>+</sup></td><td>T<sub>tr</sub></td><td>5.42 × 10<sup>13</sup></td><td>0</td><td>0</td></tr><tr><td colspan="5">Electron-impact ionization</td></tr><tr><td>Ar<sup>+</sup> + e<sup>-</sup> + e<sup>-</sup> ⇌ Ar + e<sup>-</sup></td><td>T<sub>ve</sub></td><td>1.75 × 10<sup>21</sup></td><td>-1.0</td><td>-47800</td></tr></tbody></table>

Park, Johnston, and Cruden models, respectively. These kinetic models describe the high temperature conditions targeted in this paper and have been used to model atmospheric entry into the Mars atmosphere [31, 32]. The Johnston and Cruden models both use the Park model as a baseline, but involve modifications of certain rates. The appendix provides the list of reactions used as well as the associated rate coefficients. The Cantera software was used for the kinetic simulations [33]. Reverse reaction rates were calculated using detailed balance. The enthalpy and entropy of the various species were calculated using the NASA 9-coefficient polynomials [34].

Due to the small amount of argon in the Martian atmosphere (1.93% by volume), Ar⁺ only forms in low concentrations in the shock layer of the entry capsule. This species is therefore neglected in the Park, Johnston and Cruden models. However, in our experiments, 90% of the mixture is argon. For these conditions, Ar⁺ represents 5% of all ionized species at the recombination tube inlet (see table 2). This concentration is no longer negligible. Therefore, we added charge-exchange reactions involving Ar⁺ and electron-impact ionization of argon to the models. Table 1 shows the reactions added and the rate constants used. The electron-impact ionization rate constant is taken from Owano et al [35]. The rate constants of charge-exchange with O, O₂, and CO are taken from Beuthe and Chang [36]. For lack of data in the literature for the charge-exchange reaction with atomic carbon, the mean value of the previous charge-exchange rate constants is taken. These reactions do not play a significant role but are important for the determination of the initial state as detailed in the next section.

## 3. Inlet chemical composition and temperature

Knowledge of the chemical composition of the plasma at the inlet of the tube is necessary to set the initial conditions for the recombination simulations. However, a close inspection of figure 2 shows that the measured CO density is slightly underpopulated with respect to its equilibrium value. Therefore, rather than assuming full equilibrium, we relax this assumption and assume only partial equilibrium. Under the assumption of partial equilibrium, the chemical reactions that determine the chemical composition of the mixture must first be identified. A subset of these reactions, together with the available
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Measured CO (black) and CO₂ (red) densities on centerline as a function of axial distance. The measured temperature is shown on the upper axis for reference, as is the equilibrium density of CO and CO₂ (solid line).</figcaption></figure>

experimental data, is then used to estimate the gas composition. This section details the approach and method used for this calculation, which sets the densities used as input conditions for the kinetic simulations that follow. For all calculations shown here, the Johnston kinetic model is used. However, the results do not change if the Park or Cruden models are used.

As a first step, kinetic simulations were performed to model the chemical evolution of the plasma between the induction region of the torch to the nozzle exit. The maximum temperature reached by the plasma in the induction region is estimated to be around 9000 K [37]. The temperature at the exit of the torch was measured to be about 6500 K (figure 2). Numerical simulations conducted by Candler *et al* showed that the pressure difference of the plasma from the inflow to the outflow reservoir of the ICP torch is about 3200 Pa in the nitrogen experiments of [38]. This is negligible in comparison to atmospheric pressure. Therefore, the pressure is taken constant at 1 atm for the simulations that follow.

Consider the case of an instantaneous drop in temperature from 9000 K to 6500 K. The plasma is initially assumed to be in equilibrium at 9000 K—the starting point of the simulation. Figure 3 shows the evolution of the chemical composition according to the Johnston kinetic model. Chemical equilibrium is reached in about 5 ms. This value is very close to the timescale for the flow to pass from the induction region to the exit of the torch, as estimated by Laux for an air mixture and a 5 cm diameter nozzle [39]. In other words, these simulations suggest that the flow may not have completely reached equilibrium at the exit of the torch. This is consistent with our observations that CO is slightly underpopulated.

Chemical equilibrium is not established due to incomplete reactions such as the recombination of CO (C + O + M ⇌ CO + M) and C₂ (C + C + M ⇌ C₂ + M). Despite this, other faster reactions are equilibrated at the exit of the torch. A reaction is considered equilibrated when the forward reaction rate R_f is equal to the reverse reaction rate R_r. This equality imposes that the ratio of the densities of the species involved in the reaction is equal to the ratio at equilibrium, i.e., equal to the equilibrium constant of the reaction Kₐₗ.

To find the equilibrated reactions at the exit of the torch, the forward and reverse rates of each reaction were evaluated. The instantaneous temperature drop from 9000 K to 6500 K represents a worst-case scenario from the standpoint of achieving chemical equilibrium at the torch exit. Indeed, for this instantaneous temperature drop scenario, the rate constants that govern the evolution in the chemical composition are all calculated at the gas temperature of 6500 K. Meanwhile, between 9000 K and 6500 K, the rate constants are smallest at 6500 K. Therefore, this calculation uses the smallest (i.e. slowest) rate constants and takes the longest time to equilibrate given the temperature range considered. After only 1 ms—one order of magnitude below the estimated time for the flow to reach the torch exit—several reactions are already equilibrated in the instantaneous temperature drop scenario. The appendix contains tables that list the equilibrated and nonequilibrated reactions. In these tables, the reactions are sorted by the value of the relative difference between the forward and reverse reaction rates (4th column). A value close to 1 indicates a one-sided reaction (fully forward or fully backward) whereas a value close to 0 indicates that the reaction is equilibrated. The threshold for considering the reaction to be equilibrated was fixed to 0.05.
<figure><img src="image_6.png" /><figcaption>**Figure 3.** Evolution of the chemical composition of a plasma initially in equilibrium at 9000 K with an imposed instantaneous temperature drop to 6500 K using Johnston's kinetic mechanism. Neutral species are in solid lines, ionized species and electrons are in dashed lines.</figcaption></figure>

At the exit of the torch, on centerline, we have a direct measurement of CO density from the infrared emission measurements. At this location, we do not have an estimate of CO₂ density as the CO₂ is fully dissociated due to the high temperature. Of the 13 species included in the kinetic models, the density of 1 is therefore known but the densities of the other 12 species remain unknown: Ar, Ar⁺, C, O, C₂, O₂, CO₂, C⁺, O⁺, O₂⁺, CO⁺, and e⁻. These 12 species must be determined to fully characterize the conditions at the inlet of the recombination tube. As explained in the thesis of Mariotto for a similar test case using a N₂/Ar plasma [40], the missing densities can be inferred given the information already known (P_atm, T, [CO]) by applying the assumption of partial equilibrium and imposing additional constraints. Equation (1) lists the set of algebraic equations that must be solved. The individual equations arise from the following considerations:

* The gas at the exit of the torch is at atmospheric pressure P_atm. This condition is set in line (a).

* The fractions of carbon and oxygen nuclei in the mixture αC and αO, respectively, are conserved. These conditions are set in lines (b) and (c). Note that αC + αO + αAr = 1. Only two equations are independent. The conservation of argon nuclei is therefore redundant and is not used. In the denominator of lines (b) and (c), γM represents the number of nuclei in the species M.

* Charge neutrality of the gas. This condition is set in line (d).

* Reactions 1–5 and 7–9 in table 11 are equilibrated. These conditions are set in lines (e) to (l). Note that all the reactions listed in table 11 cannot be used since some are simply a linear combination of others and are therefore redundant for this calculation. For example, R6 = R3 – R4, R10 = R8 – R9, R11 = R5 + R10. In the case of R6 = R3 – R4, we can only use two of these reactions (since the third

follows immediately). The two reactions closest to partial equilibrium (as determined by the smallest values of $|R_f - R_r|/R_f + R_r|$) are retained.

$$ 
\left\{
\begin{aligned}
\frac{P_{\text{atm}}}{k_b T} &= \sum_{\text{species}} [\text{M}] && (a) \\
\frac{[\text{C}] + [\text{C}^+] + 2[\text{C}_2] + [\text{CO}] + [\text{CO}^+] + [\text{CO}_2]}{\sum_{\text{species}} \gamma_M [\text{M}]} &= \alpha_C && (b) \\
\frac{[\text{O}] + [\text{O}^+] + 2[\text{O}_2] + 2[\text{O}_2^+] + [\text{CO}] + [\text{CO}^+] + 2[\text{CO}_2]}{\sum_{\text{species}} \gamma_M [\text{M}]} &= \alpha_O && (c) \\
[\text{Ar}^+] + [\text{C}^+] + [\text{O}^+] + [\text{CO}^+] + [\text{O}_2^+] &= [\text{e}^-] && (d) \\
\frac{[\text{CO}][\text{O}]}{[\text{CO}_2]} &= K_{\text{eq}}^{R8} && (e) \\
\frac{[\text{C}_2][\text{O}]}{[\text{C}][\text{CO}]} &= K_{\text{eq}}^{R7} && (f) \\
\frac{[\text{CO}][\text{O}_2]}{[\text{CO}_2][\text{O}]} &= K_{\text{eq}}^{R9} && (g) \\
\frac{[\text{CO}^+][\text{e}^-]}{[\text{C}][\text{O}]} &= K_{\text{eq}}^{R2} && (h) \\
\frac{[\text{O}_2^+][\text{e}^-]}{[\text{O}_2][\text{O}]} &= K_{\text{eq}}^{R5} && (i) \\
\frac{[\text{Ar}][\text{O}^+]}{[\text{Ar}^+][\text{O}]} &= K_{\text{eq}}^{R1} && (j) \\
\frac{[\text{Ar}][\text{C}^+]}{[\text{Ar}^+][\text{C}]} &= K_{\text{eq}}^{R4} && (k) \\
\frac{[\text{Ar}][\text{CO}^+]}{[\text{Ar}^+][\text{CO}]} &= K_{\text{eq}}^{R3} && (l)
\end{aligned}
\right.
\quad (1)
$$

The above system of equations was carefully conditioned to be numerically solved using the Levenberg-Marquardt algorithm. The inputs of this system are the pressure P_atm, temperature T, and the CO density [CO]. The 12 independent equations allow us to solve for the 12 missing densities. As a consistency check, this set of equations was solved for several
<table><caption>**Table 2.** Chemical composition of the plasma at the exit of the torch on centerline. Comparison between equilibrium and non-equilibrium density values at 6650 K. The values are sorted according to their abundance.</caption><thead><tr><th>Species</th><th>Non-equilibrium density (m<sup>-3</sup>)</th><th>Equilibrium density (m<sup>-3</sup>)</th><th>Relative difference</th></tr></thead><tbody><tr><td>Ar</td><td>8.8 × 10<sup>23</sup></td><td>8.9 × 10<sup>23</sup></td><td>-1.1%</td></tr><tr><td>O</td><td>1.4 × 10<sup>23</sup></td><td>1.3 × 10<sup>23</sup></td><td>8.6%</td></tr><tr><td>CO</td><td>3.9 × 10<sup>22</sup></td><td>5.2 × 10<sup>22</sup></td><td>-25%</td></tr><tr><td>C</td><td>4.9 × 10<sup>22</sup></td><td>3.7 × 10<sup>22</sup></td><td>32%</td></tr><tr><td>e<sup>-</sup></td><td>7.1 × 10<sup>20</sup></td><td>4.6 × 10<sup>20</sup></td><td>56%</td></tr><tr><td>C<sup>+</sup></td><td>6.3 × 10<sup>20</sup></td><td>3.9 × 10<sup>20</sup></td><td>61%</td></tr><tr><td>Ar<sup>+</sup></td><td>3.9 × 10<sup>19</sup></td><td>3.2 × 10<sup>19</sup></td><td>21%</td></tr><tr><td>C<sub>2</sub></td><td>1.9 × 10<sup>19</sup></td><td>2.1 × 10<sup>19</sup></td><td>-8.7%</td></tr><tr><td>O<sup>+</sup></td><td>2.1 × 10<sup>19</sup></td><td>1.6 × 10<sup>19</sup></td><td>33%</td></tr><tr><td>CO<sup>+</sup></td><td>1.4 × 10<sup>19</sup></td><td>1.5 × 10<sup>19</sup></td><td>-7.9%</td></tr><tr><td>O<sub>2</sub></td><td>1.6 × 10<sup>19</sup></td><td>1.4 × 10<sup>19</sup></td><td>18%</td></tr><tr><td>CO<sub>2</sub></td><td>1.3 × 10<sup>18</sup></td><td>1.6 × 10<sup>18</sup></td><td>-18%</td></tr><tr><td>O<sub>2</sub><sup>+</sup></td><td>3.5 × 10<sup>16</sup></td><td>4.6 × 10<sup>16</sup></td><td>-24%</td></tr></tbody></table>

temperatures with the CO equilibrium density as an input. The calculation returned the equilibrium composition as expected. The missing densities can now be calculated to determine the chemical composition at the exit of the torch. Table 2 shows the comparison between the equilibrium densities of the species and those computed under the assumption of partial equilibrium by solving the above set of equations. These densities were computed at 6650 K, the temperature measured at the exit of the torch on centerline. As expected, a slight departure from equilibrium is observed. Among the neutral species, atomic carbon presents the largest relative difference of 32% between its equilibrium and non-equilibrium values. The highest relative difference is for the ion C⁺, with a value of 61%. The non-equilibrium density values, in the 2nd column of table 2, represent the initial chemical state for the recombination simulations developed in section 3.

## 4. Time-temperature profile

Figure 2 shows the relation between the temperature and the length of the tube. Formally, the temperature *T* can be expressed as a function of the axial position *z*:

$$T = f(z). \qquad (2)$$

However, to simulate the evolution of the chemical composition of the gas through the recombination tube, the time-temperature history profile is required. The link between the axial temperature profile and time-temperature history profile is the velocity of the gas. Unfortunately, we do not have a measurement of the gas velocity but we will see that this is not critical for the conclusions drawn in this paper. As discussed in more detail in [16], we consider three different possibilities:

* A constant velocity profile $U_{gas}(r) = \text{const}$, which corresponds to a plug flow in a tube and approximates the profile

of a fully turbulent flow where the laminar sublayer is very thin.

* A parabolic velocity profile $U_{gas}(r) = \alpha(r - R)(r + R)$, which corresponds to a fully developed laminar flow with temperature-independent properties in a tube.

* A velocity profile that is self-similar to the enthalpy profile $U_{gas}(r) = \beta(h_{gas}(r) - h_{wall})$, where $h_{wall}$ is the enthalpy at the wall. In viscous flows where the volumetric power source and pressure gradient can be neglected, the nondimensional momentum equation is mathematically self-similar to the nondimensional energy equation if the Prandtl number $Pr = 1$, which is approximately true for the mixture studied here where $Pr \sim 2/3$ [16].

Given that the pressure, temperature profile, and mass flow rate are known, the constants $\alpha$ and $\beta$ can be determined. Figure 4 shows the axial evolution of the gas velocity through the recombination tube that results from these three profiles. The gas velocity at the center of the plasma jet ($r = 0$ mm) given by each radial velocity profile assumption is shown. The axial velocity profiles were obtained by interpolating and smoothing the discrete values plotted in the figure. The fastest axial gas velocity profile is the parabolic profile, which begins at $925 \text{ m s}^{-1}$ at the tube inlet and drops to $455 \text{ m s}^{-1}$ at the exit of the 35 cm tube. The slowest velocity profile is the constant profile which begins at $385 \text{ m s}^{-1}$ and drops to $215 \text{ m s}^{-1}$. The self-similar profile lies between these two extremes, going from $730 \text{ m s}^{-1}$ at the tube inlet to $275 \text{ m s}^{-1}$ at the exit of the 35 cm tube.

The gas velocity $U_{gas}$ can now be expressed as a function of the axial position *z*:

$$U_{gas} = \frac{dz}{dt} = g(z). \qquad (3)$$

Solving this first-order differential equation gives the axial position as a function of the time:

$$z = h(t). \qquad (4)$$

The three differential equations—one for each velocity profile assumption—were solved numerically. Combining equations (2) and (4) gives the temperature as a function of time:

$$T = f(h(t)). \qquad (5)$$

Figure 5 shows the time-temperature history profile for each radial velocity profile. These time-temperature history profiles can now be used to perform chemical kinetic simulations. Although the time-temperature history profile is used for the simulations, all results are presented as a function of tube length.

## 5. Kinetic mechanisms and nominal simulation

Figure 6 shows the comparison between the CO density measurements and the predictions of the kinetic models. Figure 7
<figure><img src="image_7.png" /><figcaption>**Figure 4.** Axial velocity profiles at the center of the plasma jet given by the parabolic (green), self-similar (red), and constant (black) radial velocity profiles. The curves result from interpolation and smoothing of the individual calculations performed for various tube lengths (dots).</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Figure 5.** Axial velocity profiles at the center of the plasma jet given by the parabolic (green), self-similar (red), and constant (black) radial velocity profiles. The curves result from interpolation and smoothing of the individual calculations performed for various tube lengths (dots).</figcaption></figure>

shows the same comparison for CO₂. The solid lines represent the equilibrium densities. The predictions using the constant, self-similar, and parabolic velocity profiles are indicated by the dotted, dashed, and dash-dotted lines, respectively. The CO and CO₂ densities predicted using the constant velocity profile are always slightly higher than the predictions using the self-similar velocity profile. Conversely, the densities predicted using the parabolic profile are always slightly lower than the predictions using the self-similar velocity profile.

In figure 6, the CO density predictions using the Park model (top) globally agree with the measurements except at 20 cm where the density is underpredicted by a factor of 2, regardless of the velocity profile used. The predictions using the Johnston model (middle) improve the agreement between 5 cm and 20 cm. The CO density at 20 cm is correctly predicted. The

predictions using the Cruden model (bottom) are quite different. The Cruden model predicts a much faster return to equilibrium than the Park and Johnston models. Chemical equilibrium is reached around 17 cm, leading to an overprediction of the CO density between 10 cm and 15 cm. The three kinetic models predict the same final CO density at 35 cm. Overall the predictions using the Johnston model are the closest to our measurements.

The three models predict the same evolution of the CO₂ density through the recombination tube and fail to reproduce the measurements. As shown in figure 7, the three models underpredict the measurements by a factor 10 at 25 cm, a factor 7 at 30 cm, and a factor 5 at 35 cm, regardless of the velocity profile used. The total number of carbon atoms present in the plasma is determined by the initial mass flow rate injected in
<figure><img src="image_9.png" /><figcaption>**Figure 6.** CO density predictions using Park (top), Johnston (middle), and Cruden (bottom) kinetic models, with the constant, self-similar, and parabolic velocity profiles.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>**Figure 7.** CO₂ density predictions using Park (top), Johnston (middle), and Cruden (bottom) kinetic models, with the constant, self-similar, and parabolic velocity profiles.</figcaption></figure>
the torch, and by the mixture composition. After a tube length of 20 cm, the species accounting for the majority of carbon atoms are CO and CO₂. Therefore, the CO and CO₂ densities are not independent. At first, it might seem inconsistent that the models succeed in matching the CO density and not the CO₂ density. However, the CO₂ density measured at 35 cm is about ten times lower than the CO density. Therefore, an underprediction of a factor 5 in the CO₂ density only implies an overprediction of the CO density by 10%.

## 6. Parametric study

The previous section showed that the models cannot reproduce our measurements. This fact could be explained by three possible scenarios.

1. The plasma on tube centerline does not behave in a manner consistent with our assumption of a 0D flow. For example, turbulent mixing or diffusion not accounted for in our 0D model could alter the chemical composition of the jet.

2. The models are not sufficiently detailed to capture the CO₂ recombination process. Indeed, the models used for the chemical simulation are not state specific. A non-Boltzmann vibrational distribution function and any subsequent impact on the global rate is not considered. However, our spectroscopic measurements indicated that both CO and CO₂ follow a Boltzmann vibrational distribution.

3. The kinetic model rate constants are incorrect or important reactions are missing.

The first and third possibilities are, in our view, the most likely. The first point was found to be important in the recent work of Van Poyer et al [41] who performed detailed 2D modeling of CO₂ conversion using microwave plasmas. However, as shown in the companion paper and thesis [16], the measured radial profiles of the CO and CO₂ densities are quite flat for tube lengths at or above 10 cm, and therefore we do not expect the 0D assumption to be invalid. Nevertheless, it would be interesting in future work to apply the model of [41] to the present recombination tube experiments in order to fully verify this point. The third point is explored in this section by investigating possible modifications to the models that improve the agreement between the predictions and our measurements. In this work, we limit ourselves to the three mechanisms of Park, Johnston, and Cruden. The potential impact of other reactions not included in these models is left to future work<fnref n="1" />.

First, we begin with a global sensitivity analysis where a large number of modifications to all reaction rates are tested. The resulting spread in the CO₂ density predictions provides an estimate of the sensitivity of the kinetic models to these variations. This global sensitivity analysis shows that the

measurements cannot be reproduced by a small modification of the rates. Rather, a large modification is required to fit our data.

Second, a local sensitivity analysis is conducted where the reaction rates are separately modified one by one to identify which modifications have the most impact on the CO₂ density prediction. This local sensitivity analysis shows that 5 reactions determine the CO₂ density predictions. Other reactions play a minor role and are subsequently neglected for the following study.

Finally, a quasi Monte-Carlo (QMC) investigation is conducted to identify possible modifications that give an acceptable agreement between the predictions and the measurements. Here, simultaneous modifications to the rates of the five reactions previously identified are considered. This is done to account for any potential coupling effects amongst these five reactions.

For this entire section, the kinetic models will be tested using the self-similar velocity profile. As shown by figure 7, this choice has little impact on the CO₂ density predictions. Therefore, the conclusions of this section apply for all velocity profiles considered.

### 6.1. Global sensitivity analysis

A global sensitivity analysis was conducted to study the sensitivity of the CO₂ density predictions to the rate constants of the kinetic models. 10 000 simulations with modified rate constants were performed to observe the impact on the CO₂ prediction. This analysis is global since all reaction rates of the models are simultaneously modified in each simulation. Both forward and reverse rate constants of a given reaction were multiplied by the same factor since the reverse rate constant is computed using detailed balance. The forward and reverse rates of each reaction *i* were multiplied by independent random factors $x_i$:

$$k_{f,r}^i = x_i k_{f,r}^{\text{nominal},i} \qquad (6)$$

where $x_i$ is a random sample of the variable $X_i$ discussed below, $k_{f,r}^{\text{nominal},i}$ represents the nominal forward or reverse rate constants as proposed in the model, and $k_{f,r}^i$ represents the forward or reverse rate constants used in the altered simulation.

The choice of the distribution of the random variable $X_i$ has a strong impact on the results. A log-normal distribution was used, i.e. the distribution of $\log(X_i)$ follows a normal distribution, $\log(X_i) \sim N(\mu, \sigma^2)$, where $\mu$ is the average and $\sigma$ is the standard deviation. This means that a normal distribution was used to randomly sample the logarithm of the variable $X_i$. This choice was made to give the same importance to simulations with rate constants multiplied by *m* and simulations with rate constants divided by *m*. The log-normal distribution has the added advantage of suppressing extreme values located far from the mean. Therefore, extreme values, although sampled, are unlikely. The mean of the normal distribution $\mu$ was taken equal to 0 to center the simulations around the nominal case where $k_{f,r}^i = k_{f,r}^{\text{nominal},i}$. The uncertainty in the rate

<footnote n="1">¹ The impact of the Boudouard reaction CO₂ + C ↔ 2CO, often mentioned in the literature on CO₂ conversion, was estimated using the rate taken from reference [54].</footnote>
<figure><img src="image_11.png" /><figcaption>Figure 8. Log-normal distribution of the random variable $X_i: \log(X_i) \sim N(\mu, \sigma^2)$, with $\mu = 0$ and $\sigma = \log(2)/1.96$.</figcaption></figure>

constants was not specified in the Park, Johnston or Cruden models. However, other articles such as Ibragimova et al [42] suggest that this uncertainty is on the order of 2, at least for the reaction CO₂ + M ⇌ CO + O + M. With M = Ar, the ratio of the Park, Johnston and Cruden rates is between 2 and 3 for the 3500–8000 K temperature range and below 4 for the 3000–3500 K range. Therefore, the standard deviation of the normal distribution σ was fixed to log(2)/1.96 using the quantile function of the normal distribution. This implies that 95% of the samples $x_i$ are between 1/2 and 2. While this analysis is not an uncertainty analysis, this simple approach allows us to see the global sensitivity of the CO₂ density predictions to the rate constants used in the kinetic models.

Figure 8 shows the log-normal distribution used for all reactions. The log-normal distribution plotted was generated from a uniformly distributed random variable $U \sim U[0, 1]$ using the inverse cumulative distribution function of the log-normal distribution [43–45]:

$$F^{-1}(u) = \exp\left(\mu + \sigma\sqrt{2\operatorname{erf}^{-1}(2u-1)}\right) \quad (7)$$

The results of the 10 000 simulations using the Johnston model are presented in figure 9. This analysis was conducted for two log-normal distributions, using a standard deviation equal to log(2)/1.96 in the top portion of the figure, and a standard deviation equal to log(3)/1.96 in the bottom portion of the figure. The two black dashed lines indicate the 95% percentile—i.e. 95% of the altered simulations are comprised between these two lines. Each percentile is represented by a color indicated on the colormap. Figure 9 shows that the CO₂ density measurements cannot be reproduced using a simple multiplicative factor of 2 or 3 to modify the reaction rates. Indeed, the CO₂ density measurements are not in the 95% percentile area—the largest discrepancy is at 25 cm. The same results were observed using the Park model and the Cruden model. This indicates that stronger modifications to the kinetic models are required to match the CO₂ density measurements.

## 6.2. Local first order sensitivity analysis

A local sensitivity analysis was conducted to identify individual reactions that have the strongest impact on the CO₂ density predictions. To do this, simulations were performed with only one reaction rate modified while the others were fixed to their nominal value. The local first order sensitivity index $\eta_i$ of each reaction is defined as follows:

$$\eta_i = \frac{n_{\text{CO}_2}^{35\text{cm}}([1 + \varepsilon]k_{f,r}^{\text{nominal},i}) - n_{\text{CO}_2}^{35\text{cm}}(k_{f,r}^{\text{nominal},i})}{n_{\text{CO}_2}^{35\text{cm}}(k_{f,r}^{\text{nominal},i})} \quad (8)$$

where $n_{\text{CO}_2}^{35\text{cm}}$ is the CO₂ density predicted at the 35 cm tube exit, $\varepsilon$ is an infinitesimally small number used to vary the rate constants $k_{f,r}^{\text{nominal},i}$. The value of $\eta_i$ is positive when an increase of the rate constant leads to an increase of the final CO₂ density, and negative when it leads to a decrease. This index is local, meaning it depends on the nominal value of the rate constants $k_{f,r}^{\text{nominal},i}$ and is valid only for values close to this nominal value. Nonlinear effects associated with larger changes in $k_{f,r}^{\text{nominal},i}$ or due to simultaneous changes of several rate constants (cross-sensitivity), are not accounted for though these are expected. As before, only the Johnston model is used here. However, the results do not change significantly if the Park or Cruden models are used instead.

Figure 10 shows the local first order sensitivity results. Only the highest indexes are presented in figure 10. Indexes of reactions not shown were more than 1000 times lower than the highest value. The most important reaction that impacts the final CO₂ density is the three-body recombination reaction of CO₂. Its sensitivity index is 10 times larger than the next largest index. The three-body recombination of O₂ and the neutral exchange reaction CO₂ + O ⇌ O₂ + CO are ranked second and third, respectively. The three-body recombination of CO and the neutral exchange reaction CO + O ⇌ O₂ + O are ranked fourth and fifth, respectively. These 5 most important reactions are listed and numbered in table 3 and will be referred to for the remainder of this paper. The reactions are
<figure><img src="image_12.png" /><figcaption>**Figure 9.** Global sensitivity analysis of the CO₂ density predictions to the rate constants of the Johnston model. Spread patterns for 10 000 altered simulations in which each rate constant was modified by a random factor following a log-normal distribution centered on 0 and with a standard deviation of log (2) / 1.96 (top) and log (3) / 1.96 (bottom).</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Figure 10.** Local first order sensitivity indexes of the reactions influencing the final density of CO₂ at the 35 cm tube exit for the Johnston kinetic model.</figcaption></figure>
<table><caption>**Table 3.** Reactions playing the major role in the final CO₂ density predicted at the exit of the 35 cm tube. Numbered according to order of appearance in the kinetic model.</caption><thead><tr><th>Reaction</th><th>Number</th><th>Symbols</th></tr></thead><tbody><tr><td>O₂ + M ↔ O + O + M</td><td>1</td><td>k<sub>f,r</sub><sup>nominal,1</sup>, k<sub>f,r</sub><sup>1</sup>, x₁</td></tr><tr><td>CO + M ↔ C + O + M</td><td>2</td><td>k<sub>f,r</sub><sup>nominal,2</sup>, k<sub>f,r</sub><sup>2</sup>, x₂</td></tr><tr><td>CO₂ + M ↔ CO + O + M</td><td>3</td><td>k<sub>f,r</sub><sup>nominal,3</sup>, k<sub>f,r</sub><sup>3</sup>, x₃</td></tr><tr><td>CO + O ↔ C + O₂</td><td>4</td><td>k<sub>f,r</sub><sup>nominal,4</sup>, k<sub>f,r</sub><sup>4</sup>, x₄</td></tr><tr><td>CO₂ + O ↔ O₂ + CO</td><td>5</td><td>k<sub>f,r</sub><sup>nominal,5</sup>, k<sub>f,r</sub><sup>5</sup>, x₅</td></tr></tbody></table>

numbered according to their order of appearance in the kinetic models (not based upon their level of importance). Other reactions play a minor and negligible role in the recombination process of CO₂.

Figure 11 shows the impact of a modification to the rate constant of the 5 reactions listed in table 3 on the CO₂ density prediction along the recombination tube. A few trends are worth mentioning. First, all reactions except #3 show a saturation effect—increasing the rate constant to infinity has a finite limiting effect on the CO₂ density. Second, increasing the rate constant of reaction #3 by a factor of ~10 results in a much better agreement with the CO₂ density measurements.

### 6.3. QMC investigation

The local sensitivity analysis identified the five reactions that have a major impact on the CO₂ recombination process and CO₂ density predictions. These reactions are listed in table 3. This section identifies ways in which these five reactions may be modified in order to achieve an acceptable agreement between the CO₂ density predictions and the measurements. Unlike the local sensitivity analysis, wherein only one reaction rate was modified at a time, here we modify all five reaction rates simultaneously.

A QMC approach was used to efficiently probe the 5-dimensional parametric space of possible modifications to the reaction rates [46]. The vector $x = (x_1, x_2, x_3, x_4, x_5)$ uniquely determines a modified simulation (recall that the rate coefficient $k_{f,r}^i$ is linked to the nominal rate coefficient $k_{f,r}^{nominal,i}$ via $k_{f,r}^i = x_i k_{f,r}^{nominal,i}$). The difference between the Monte-Carlo and QMC methods is in the choice of the vector $x$. Contrary to the Monte-Carlo method, which is based on random numbers, a QMC method uses low-discrepancy predetermined sequences. The main advantage is a faster rate of convergence. In our case it is an efficient way to probe the parametric space of possible modifications. The QMC approach provides a method of sampling different values of the vector $x$ within a defined space. The approach used here can be divided into two primary steps:

1. The Sobol sequence was used to generate quasi-random numbers in the 5D unit cube [47]. The coordinates of these numbers are uniformly distributed along the segment (0, 1) in each dimension, thus providing five random distributions along the segment (0, 1) for each of the five variables $x_i$ used to modify the reaction rates.

2. Each of the five distributions produced in step 1 is then transformed using the inverse cumulative distribution function of the log-normal distribution presented in equation (7). This produces a log-normal distributed set. Different log-normal distributions were used for each dimension. All log-normal distributions were centered on 0 but had different standard deviations. Table 4 lists the standard deviations used in the analysis. The standard deviations were chosen to reflect the effective range identified by the local sensitivity analysis.

More than 500 000 simulations were made for each kinetic model, each corresponding to a separate vector $x$. The agreement between the CO₂ density prediction of a modified simulation and the measurements was computed using equation (9),

$$S = \sqrt{\sum_{z \in \{25,30,35\}} \left( \frac{n_{\text{CO}_2}^{\text{pred}}(z) - n_{\text{CO}_2}^{\text{meas}}(z)}{n_{\text{CO}_2}^{\text{meas}}(z)} \right)^2} \quad (9)$$

This value is referred as the score $S$ of the simulation and was compared to a threshold value to characterize whether a simulation gives an acceptable agreement or not. All simulations with a score higher than this threshold are rejected while simulations with a lower score (i.e. a better score) are selected. The threshold $S_{max}$ used here was computed as follows:

$$S_{\max} = \sqrt{\left(\frac{35}{100}\right)^2 + \left(\frac{35}{100}\right)^2 + \left(\frac{35}{100}\right)^2} \quad (10)$$

This threshold allows a maximum discrepancy of 35% between the CO₂ density prediction and the measurements made at the 25 cm, 30 cm, and 35 cm tube exits. A threshold of 30% eliminated many interesting possibilities for altering the rates. Conversely, a threshold of 40% did not—rather, it simply improved upon possibilities already identified with a threshold of 35%. Therefore, a threshold of 35% was chosen.
<figure><img src="image_14.png" /><figcaption>**Figure 11.** Impact of rate modifications to the five reactions playing the major role in the CO₂ density predicted at the exit of the 35-cm tube identified in the local sensitivity analysis.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>**Figure 11.** (Continued.)</figcaption></figure>

<table><caption>**Table 4.** Standard deviations of log-normally distributed sets used to explore the 5D parametric space of possible modifications to the kinetic models.</caption><thead><tr><th>Reactions</th><th>O<sub>2</sub> + M &harr; O + O + M</th><th>CO + M &harr; C + O + M</th><th>CO<sub>2</sub> + M &harr; CO + O + M</th><th>CO + O &harr; C + O<sub>2</sub></th><th>CO<sub>2</sub> + O &harr; CO + O<sub>2</sub></th></tr></thead><tbody><tr><th>σ</th><td>log(50)<br>1.96</td><td>log(20)<br>1.96</td><td>log(15)<br>1.96</td><td>log(25)<br>1.96</td><td>log(100)<br>1.96</td></tr></tbody></table>

<table><caption>**Table 5.** Reaction rate factors found to give the best agreement between the CO₂ density predictions and the measurements.</caption><thead><tr><th rowspan="2">Reaction</th><th colspan="2">Park</th><th colspan="2">Johnston</th><th colspan="2">Cruden</th></tr><tr><th>Mean</th><th>Min-max</th><th>Mean</th><th>Min-max</th><th>Mean</th><th>Min-max</th></tr></thead><tbody><tr><td>x<sub>O₂</sub>+M &harr; O+O+M</td><td>41</td><td>19-88</td><td>45</td><td>19-107</td><td>2614</td><td>1324-5160</td></tr><tr><td>x<sub>CO</sub>+M &harr; C+O+M</td><td>1</td><td>0.20-5.0</td><td>1</td><td>0.18-5.2</td><td>1</td><td>0.23-4.4</td></tr><tr><td>x<sub>CO₂</sub>+M &harr; CO+O+M</td><td>28</td><td>25-32</td><td>28</td><td>24-32</td><td>1<sup>2</sup></td><td>0.20-2.8</td></tr><tr><td>x<sub>CO</sub>+O &harr; C+O<sub>2</sub></td><td>1</td><td>0.24-3.9</td><td>1</td><td>0.24-4.3</td><td>1<sup>2</sup></td><td>0.14-2.8</td></tr><tr><td>x<sub>CO₂</sub>+O &harr; CO+O<sub>2</sub></td><td>0.14</td><td>0.03-0.63</td><td>0.14</td><td>0.03-0.74</td><td>4.1</td><td>3.6-4.7</td></tr></tbody></table>
<table><caption>**Table 6.** Closest modifications from the nominal simulation while preserving an acceptable agreement between the CO₂ density predictions and the measurement ($S < S_{max}$).</caption><thead><tr><th rowspan="2">Model</th><th colspan="2">Park</th><th colspan="2">Johnston</th><th colspan="2">Cruden</th></tr><tr><th>Mean</th><th>Min-max</th><th>Mean</th><th>Min-max</th><th>Mean</th><th>Min-max</th></tr></thead><tbody><tr><td>xO<sub>2</sub>+M ↔ O+O+M</td><td>2.7</td><td>2.0-3.7</td><td>2.1</td><td>1.4-3.0</td><td>77</td><td>68-88</td></tr><tr><td>xCO+M ↔ C+O+M</td><td>1</td><td>0.50-2.2</td><td>1</td><td>0.52-2.1</td><td>1</td><td>0.45-2.36</td></tr><tr><td>xCO<sub>2</sub>+M ↔ CO+O+M</td><td>9.7</td><td>8.4-11.2</td><td>9.8</td><td>7.5-12.9</td><td>10.1</td><td>9.1-11.1</td></tr><tr><td>xCO+O ↔ C+O<sub>2</sub></td><td>1</td><td>0.48-2.3</td><td>1</td><td>0.52-2.1</td><td>1</td><td>0.64-1.5</td></tr><tr><td>xCO<sub>2</sub>+O ↔ CO+O<sub>2</sub></td><td>1<sup>2</sup></td><td>0.34-1.4</td><td>1<sup>2</sup></td><td>0.32-1.23</td><td>1</td><td>0.62-1.6</td></tr></tbody></table>

This value is slightly above the uncertainty value of the measurements and was found to give an overall acceptable agreement between the modified simulations and the measurements. Note that, if a simulation prediction is really close to the measurement at 30 cm, a more distant prediction at 25 cm or 35 cm is allowed by this criterion.

Table 5 shows the vectors giving the best agreement with the CO₂ density measurements for each kinetic model. The mean, minimal and maximal values presented here were determined from all simulations that fell within 5% of the minimal S score achieved. This avoids specifying a precise value to a factor which, in reality, can fluctuate to some extent and produce the same agreement.

Table 5 shows that the best agreement is obtained with similar modification factors for the Park and Johnston models. The best agreement is achieved by multiplying the O₂ + M ↔ O+O+M reaction rate by a factor between 20 and 100, the CO₂ + M ↔ CO+O+M reaction rate by a factor between 24 and 32, and the CO₂ + O ↔ CO+O₂ reaction rate by a factor between 0.03 and 0.7. Modifications of the CO + M ↔ C+O+M or CO+O ↔ C+O₂ reaction rates have no impact on the agreement and therefore the nominal rates should be conserved.

The best agreement using the Cruden model is achieved for an increase of the O₂ + M ↔ O+O+M reaction rate by a factor between 1324 and 5160, and an increase of the CO₂ + O ↔ CO+O₂ reaction rate by a factor between 3.6 and 4.7. The rate constant of CO₂ + M ↔ CO+O+M does not need modification.

The modifications proposed in table 5 correspond to the best agreement achieved. However, the modifications proposed are quite far from the nominal values, as illustrated by the value of $x_1 = 2614$ proposed for the O₂ + M ↔ O+O+M reaction rate of the Cruden model. Modifications closer to the nominal simulation can be found while preserving an acceptable agreement between the CO₂ density prediction and the measurements ($S < S_{max}$). To do so, the ‘distance’ D between the modified simulation and the nominal simulation

is computed using equation (11),

$$D = \sqrt{\sum_{i=1}^{5} \left( \log \left( \frac{k_{f,r}^{i}}{k_{f,r}^{\text{nominal},i}} \right) \right)^{2}} \quad (11)$$

A smaller value of D indicates that the modified reaction rates are closer to the nominal rates. The selected simulations ($S < S_{max}$) are sorted according to D. All of these simulations represent acceptable agreement according to our criterion. Sorting them according to D allows us to see which of these acceptable modifications is closest to the nominal simulation. We distinguish between these closest modifications and the best modifications discussed previously. Table 6 shows the closest modifications for each kinetic model. The mean, the minimal and the maximal values are computed from simulations yielding a similar distance D (between 100 and 300 simulations depending on the model). As expected, the factors $x_i$ are much closer to one, meaning that the proposed modifications are closer to the nominal simulation. For the Park and Johnston models, the closest modification was obtained by altering the same reaction rates as for the best modification, but with a smaller modification of the rate. Indeed, the closest modification for the Park and Johnston models consists in an increase of the O₂ + M ↔ O+O+M reaction rate by a factor between 2 and 3 and an increase of the CO₂ + M ↔ CO+O+M reaction rate by a factor around 10. Once again, it appears that modifications of the CO + M ↔ C+O+M or CO+O ↔ C+O₂ reaction rates have no impact on the agreement and therefore the nominal rates should be conserved. Contrary to the modification achieving the best agreement, no modification of the CO₂ + O ↔ CO+O₂ reaction rate is required to achieve an acceptable agreement. For the Cruden model, a modification of the O₂ + M ↔ O+O+M reaction rate is still required but with a much lower factor between 68 and 88. However, a change of the CO₂ + M ↔ CO+O+M reaction rate by a factor between 9 and 11 is proposed instead of a change of the CO₂ + O ↔ CO+O₂ reaction rate. Other reaction rates appear to have no impact on the agreement and therefore the nominal rates should be conserved.

All models suggest that the minimal changes from the nominal simulation giving an acceptable agreement require an increase of the CO₂ three-body recombination rate by a factor of about 10. For each kinetic model, figure 12 shows the selected simulations (red area) with a score below the defined threshold ($S < S_{max}$) that give an acceptable agreement. Since the measured CO₂ density is about ten times lower than the

<footnote n="2">² The exact values of the means of these reaction rates are different from 1. However, it appears that the same agreement is achieved with whether an increase or a decrease of these rates, as shown by the extremal values. Thus, the nominal rates should be conserved. Furthermore, the simulation score of the updated simulations is still within 5% of the minimal score achieved. The exact value of the mean can still be retrieved by taking the geometric mean of the min and max values.</footnote>
<figure><img src="image_16.png" /><figcaption>**Figure 12.** CO₂ density predictions with the modified Park (top), Johnston (middle), and Cruden (bottom) kinetic models. The simulation minimizing the residual (best) is the dash-dotted line while the simulation minimizing the changes (closest) is the solid line.</figcaption></figure>
<table><caption>**Table 7.** Alternative modifications that preserve an acceptable agreement between the CO₂ density predictions and the measurements ($S < S_{max}$).</caption><thead><tr><th>Reaction</th><th>x<sub>1</sub></th><th>x<sub>2</sub></th><th>x<sub>3</sub></th><th>x<sub>4</sub></th><th>x<sub>5</sub></th><th>&alpha;<sub>i</sub></th></tr></thead><tbody><tr><td rowspan="2">Park</td><td>2.7</td><td>1</td><td>9.7</td><td>1</td><td>1</td><td>(1, 1, 1, 1, 1)</td></tr><tr><td>14</td><td>1</td><td>1</td><td>1</td><td>10.9</td><td>(1, 1, 5, 1, 1)</td></tr><tr><td rowspan="3">Johnston</td><td>2.1</td><td>1</td><td>9.8</td><td>1</td><td>1</td><td>(1, 1, 1, 1, 1)</td></tr><tr><td>10.5</td><td>1</td><td>1</td><td>1</td><td>6.2</td><td>(1, 1, 5, 1, 1)</td></tr><tr><td>1</td><td>2.5</td><td>9.3</td><td>1</td><td>0.36</td><td>(5, 1, 1, 1, 1)</td></tr><tr><td rowspan="3">Cruden</td><td>77</td><td>1</td><td>10.1</td><td>1</td><td>1</td><td>(1, 1, 1, 1, 1)</td></tr><tr><td>112</td><td>1</td><td>1</td><td>1</td><td>9.5</td><td>(1, 1, 5, 1, 1)</td></tr><tr><td>55</td><td>1</td><td>10.2</td><td>1</td><td>0.15</td><td>(9, 1, 1, 1, 1)</td></tr></tbody></table>

predicted CO density, changing the CO₂ density prediction has little impact on the CO density prediction. In other words, the CO density is still correctly matched with these changes. Since CO₂ density measurements are only available after 25 cm, the predictions are more spread out before 25 cm than after. Figure 12 also shows the comparison between the modification yielding the best agreement (dash-dotted line) and the closest modification from the nominal simulation (solid line). Regardless of the kinetic model, both simulations are similar before 20 cm and after 30 cm but differ in between. The maximum discrepancy is observed at 25 cm. Although both simulations underpredict the CO₂ density at 25 cm, better agreement is achieved for the best simulation, as expected.

Table 6 shows the closest modifications—determined according to the distance *D* presented in equation (11)—of the nominal simulation. In this ranking, modifications to all reactions are treated equally. However, modifications of reaction rates that are relatively uncertain should be given more weight than for other more certain rates. To find alternative modifications that respect such a weighting, and yet remain relatively close to the nominal simulation, the modifications are ranked using the non-isotropic distance $D_\alpha$ defined as follows:

$$D_{\alpha} = \sqrt{\sum_{i=1}^{5} \alpha_{i} \left( \frac{k_{f,r}^{i}}{k_{f,r}^{\text{nominal},i}} \right)^{2}} \quad (12)$$

The distance $D_\alpha$ depends upon the reaction weights $\alpha_i$ that must be specified. A high value of a reaction weight penalizes modifications to the corresponding reaction rate. Therefore, setting a coefficient of $\alpha_3 = 5$ penalizes changes to the rate of the CO₂ + M $\leftrightarrow$ CO + O + M reaction and allows us to find alternative modifications/solutions that give acceptable agreement while preserving the nominal reaction rate of reaction 3.

Table 7 lists several alternative modifications found using this method for each kinetic model. The last column shows the reaction weights used for the calculation of $D_\alpha$. It is hard to find all the possible modifications given the infinite combination of reaction weights $\alpha_i$. A thorough survey of the literature for different measurements of each of these reactions would allow for an assessment in the scatter in the

predicted reaction rate values. This might help assign reasonable values of $\alpha_i$ to find the most reasonable modification in the nominal rates and therefore is recommended for future work. However, it is interesting to note that an acceptable agreement can be achieved for all models without changing the rate of the CO₂ + M $\leftrightarrow$ CO + O + M reaction. For the three models, preserving the CO₂ three-body recombination rate CO₂ + M $\leftrightarrow$ CO + O + M requires increases of both O₂ + M $\leftrightarrow$ O + O + M and CO₂ + O $\leftrightarrow$ CO + O₂ reaction rates (second line of each model in table 7). Increasing these reaction rates quickens the return to equilibrium (remember that both forward and reverse rates are modified in this study). Therefore, the O₂ + M $\leftrightarrow$ O + O + M rate increase favors the reverse rate and produces more O₂ which is in turn converted into CO₂ with the enhanced exchange reaction CO₂ + O $\leftrightarrow$ CO + O₂. Indeed, the CO₂ + O $\leftrightarrow$ CO + O₂ rate increase favors the reverse rate since CO is overpopulated and CO₂ is under populated. Increasing these two reaction rates allows to match the CO₂ predictions with the measurements without modifying the CO₂ + M $\leftrightarrow$ CO + O + M rate constant.

Furthermore, for the Cruden model, no simulation with the nominal rate for the O₂ + M $\leftrightarrow$ O + O + M reaction was found to give an acceptable agreement. The minimal change required for this reaction rate is a factor of 55 (last line of table 7). Figure 13 shows how a few of the possible alternate modifications compare with the measured values. Given the wide spread in the predictions of the O and O₂ densities, measurements of these species densities could tell us which modification to choose.

The analysis presented in this section limited the possible modifications to the 5 reactions listed in table 3. These reactions were identified by the local first order sensitivity analysis as having the strongest impact on the CO₂ density predictions. The result of this local analysis depends on the values of the rate constants. As a consistency check, the same local first order local sensitivity analysis was conducted for all proposed modifications. The same 5 reactions were identified in all configurations to have the strongest impact on the CO₂ density predictions. While this does not ensure that other reactions have no impact, this gives confidence in the study limited to these 5 reactions.
<figure><img src="image_17.png" /><figcaption>Figure 13. Neutral species density predictions for two possible modifications of the Johnston model. Both predict similar CO and CO₂ densities.</figcaption></figure>

We conclude with a final remark concerning the Park and Johnston models. Dissociation reactions such as CO₂ + M ↔ CO + O + M have different rates depending on the collision partner. In this study, these original rates were conserved and thus also depend upon the collision partner. However, the same multiplying factor xᵢ was applied to the rate constants for all colliders. It should be noted that, for the reaction CO₂ + M ↔ CO + O + M, our analysis showed that argon colliders (M = Ar) are responsible for 35% of the total recombination. The higher density of argon atoms is compensated by the lower rate constant. CO, O and O₂ colliders are respectively responsible for 30%, 17% and 10% of the total recombination. The Cruden model is not concerned by this remark since all collision partners have the same rate—no rate efficiencies indicated. In this case, argon colliders are responsible for 87% of the total recombination, while CO and O₂ are only responsible for 7% and 3%, respectively. These percentages are the same for all dissociation reactions studied in this article, given our mixture is composed of 90% of argon by mass. Therefore, the modifications of the dissociation reactions proposed in this section concern mostly the reactions involving argon as a collider.

### 6.4. Recombination through excited states

The preceding sections used the Park, Johnston, and Cruden models to make predictions concerning the evolution of the chemical composition of a recombining CO₂ mixture. Until now, we have based our analysis on recombination rates

obtained through the detailed balance principle with dissociation rate constants from the literature. These dissociation rate constants were measured in shock tube experiments, usually by monitoring the evolution of the density of the ground state of the molecule undergoing dissociation. If the rate of dissociation is much faster than the rate at which ground states equilibrate with excited electronic states, the measured rate constants represent the rate of dissociation from the ground states of the molecules and not from excited intermediate states:

$$O_2 (X^3\Sigma_g^-) + M \leftrightarrow O + O + M \quad (13)$$

$$CO (X^1\Sigma^+) + M \leftrightarrow C + O + M \quad (14)$$

$$CO_2 (X^1\Sigma_g^+) + M \leftrightarrow CO + O + M . \quad (15)$$

Furthermore, the contribution of excited states to the partition functions is negligible for the temperature range of interest. For example, the contribution of O₂ excited states to the partition function is less than 11% below 6500 K. Therefore, the contribution of excited states to the equilibrium constant is also negligible: Kₐeq ≅ Kₐeq,x. Consequently, the recombination rate constants obtained by applying detailed balance to the dissociation rate constants account primarily for recombination into the ground states of the molecules.

However, recombination can also lead to excited states of the molecules, for example the first excited states:

$$O + O + M \leftrightarrow O_2 (a^1\Delta_g) + M \quad (16)$$

$$C + O + M \leftrightarrow CO (a^3\Pi) + M . \quad (17)$$
These excited states can then undergo radiative or collisional transitions to the ground state of the molecules. Such additional reactions necessarily increase the global effective recombination rate constants. For example, for N₂ recombination, the recombination rate constant $k_A^R$ to the first excited state N₂(A<fnref n="3" />Σ⁺ₐ) is almost twice as large as the recombination rate constant $k_X^R$ to the ground state N₂(X¹Σ⁺ₗ):

$$N + N + M \underset{k_X^R}{\rightarrow} N_2 (X^1\Sigma^+) + M \quad (18)$$

$$N + N + M \underset{k_A^R}{\rightarrow} N_2 (A^3\Sigma^+) + M \quad (19)$$

$$k_A^R = 1.7k_X^R, [47] \quad (20)$$

$$k_A^R = 1.8k_X^R, [48]. \quad (20)$$

This means that the global effective recombination rate constant of N atoms is 3 times higher when including recombination to N₂(A<fnref n="3" />Σ⁺ₐ). Colonna and Capitelli showed that the recombination to the excited states of N₂ can double the density of molecular nitrogen in a nitrogen expansion flow [50]. For the recombination of C and O atoms, Shui et al found a similar result with an effective recombination rate constant including the excited states (a<fnref n="3" />Π, a'<fnref n="3" />Σ⁺, d<fnref n="3" />Δ, e<fnref n="3" />Σ⁻, I¹Σ⁻, A¹Π) almost 10 times higher than the recombination rate constant to the ground state CO(X¹Σ⁺) alone [48].

Such alternative recombination pathways also exist for CO₂ recombination. For example, recombination to the ¹³B₂Σ⁺ₐ and ¹¹B₂Δₐ excited states of CO₂ followed by deexcitation through radiative or collisional transitions could increase the global effective recombination rate of CO₂. Slanger et al [13] studied the three-body recombination of O with CO over the temperature range 250–370 K. For CO as the third body (CO + O + CO = CO₂ + CO), they proposed the following recombination rate constant<fnref n="3" />:

$$k_{\text{rec}}^{\text{Slanger}} = 6.5 \times 10^{-33} \exp\left(-\frac{2185}{T}\right) (\text{cm}^6\text{molec}^{-2}\text{s}^{-1}). \quad (21)$$

This effective recombination rate, measured in a recombining CO₂ plasma, necessarily incorporates recombination through the excited states of CO₂. The extrapolation of the Slanger rate beyond 370 K should be taken with caution but it is interesting to compare it against the recombination rate constant of Johnston, computed using detailed balance from the dissociation rate constant given in [22]. For the 3 000–4000 K temperature range, the Slanger rate is between 3.1 × 10⁻³³ and 3.8 × 10⁻³³ cm⁶molec⁻² s⁻¹, whereas the Johnston rate is between 5 × 10⁻³⁴ and 4.5 × 10⁻³⁴ cm⁶molec⁻² s⁻¹. Thus the extrapolated Slanger rate is between 6 and 10 times higher than the Johnston rate in the temperature range of interest in our study.

For the recombination of O atoms, it can be shown using the rate constants of Park [51] that the effective rate constant of

recombination into the excited states O₂(a), O₂(b), and O₂(c) is comparable with the rate constant of recombination into the ground state (Reaction 7.22). Thus, this would double the global rate of recombination of O atoms.

It is interesting to note that the factor 10 enhancement of CO + O + M → CO₂ + M and the factor 2 enhancement of O + O + M → O₂ + M are consistent with the modifications proposed in table 6 for the mechanisms of Park and Johnston.

These alternative recombination pathways could justify the previously proposed ‘enhanced’ effective recombination rates without necessarily increasing the effective dissociation rates if the rate of dissociation is much faster than the rate of electronic excitation. Future work should include recombination into the excited states of CO₂, CO, and O₂. This will require an electronic (or even vibronic) specific kinetic model with state-specific reaction rates, as done by Mariotto [40] for nitrogen plasmas, Dubuet [52] for air plasmas, or Annaloro [20] for CO₂ plasmas.

## 7. Conclusions

The chemical kinetic simulations of the recombining CO₂ plasma flow showed that modifications to the kinetic models are needed to reproduce the measured CO₂ density. Several combinations of rate constant modifications exist that enable matching both the CO and CO₂ density measurements (see tables 6 and 7). Our data cannot discriminate between these proposed modifications since they both predict similar CO and CO₂ density evolutions. The increase by a factor of 10 in the rate of the recombination reaction CO + O + M ↔ CO₂ + M and by a factor of 2 the rate of the recombination reaction O + O + M ↔ O₂ + M seems to be the most promising modification. These increases could potentially be justified by recombination via excited states of CO₂ and O₂.

A database link is provided below. The database contains the measurements analyzed in this paper (CO density, CO₂ density, and temperature - all as a function of tube length). It also contains implementations of the Park, Johnston and Cruden kinetic models within Cantera, and an example python script showing how to use these models.

## Data availability statement

The data that support the findings of this study are openly available at the following URL/DOI: https://doi.org/10.57745/V00CSP [53].

## Acknowledgment

This work was financially supported by the Agence Nationale de la Recherche (ANR) under the CO2REC project (Grant Number: ANR-18-CE05-0003) and the European Research Council (ERC) under the European Union's Horizon 2020 research & innovation program (Grant Agreement Number: 101021538).

<footnote n="3">³ In the original article the activation energy is 4 340 cal mol⁻¹. Therefore, it has been multiplied by 4.186/8.314 to obtain 2 185 Kelvin.</footnote>
## Appendix. Kinetic mechanisms and partial equilibrium calculations

Tables 8–10 provide the mechanisms and rate constants used in this article. For every reaction, the forward reaction can be determined as follows:

$$k_f = AT_x^n \exp \left( -\frac{E_a}{T_x} \right).$$

Tables 11 and 12 list the results of the kinetic study presented in section 2, wherein an instantaneous temperature drop from 9000 K to 6500 K is imposed. Table 11 lists the reactions that are equilibrated after 1 ms, whereas table 12 lists the reactions that are not equilibrated.

<table><caption>**Table 8.** Park model without nitrogen containing species. A is in cm³ mole⁻¹ s⁻¹ and E_a is in K.</caption><thead><tr><th>Reactions</th><th>M</th><th>T<sub>x</sub></th><th>A</th><th>n</th><th>E<sub>a</sub></th></tr></thead><tbody><tr><td colspan="6" style="text-align: center; font-weight: bold;">Dissociation reactions</td></tr><tr><td>C<sub>2</sub> + M ⇌ C + C + M</td><td>All</td><td>T<sub>a</sub></td><td>3.7 × 10<sup>14</sup></td><td>0</td><td>69900</td></tr><tr><td rowspan="2">O<sub>2</sub> + M ⇌ O + O + M</td><td>C, O</td><td rowspan="2">T<sub>a</sub></td><td>1.0 × 10<sup>22</sup></td><td rowspan="2">-1.5</td><td rowspan="2">59750</td></tr><tr><td>Ar, C<sub>2</sub>, O<sub>2</sub>, CO, CO<sub>2</sub></td><td>2.0 × 10<sup>21</sup></td></tr><tr><td rowspan="3">CO + M ⇌ C + O + M</td><td>Ar</td><td rowspan="3">T<sub>a</sub></td><td>2.3 × 10<sup>19</sup></td><td rowspan="3">-1.0</td><td rowspan="3">129000</td></tr><tr><td>C, O</td><td>3.4 × 10<sup>20</sup></td></tr><tr><td>C<sub>2</sub>, O<sub>2</sub>, CO, CO<sub>2</sub></td><td>2.3 × 10<sup>20</sup></td></tr><tr><td rowspan="3">CO<sub>2</sub> + M ⇌ CO + O + M</td><td>Ar</td><td rowspan="3">T<sub>a</sub></td><td>6.9 × 10<sup>20</sup></td><td rowspan="3">-1.5</td><td rowspan="3">63275</td></tr><tr><td>C, O</td><td>1.4 × 10<sup>22</sup></td></tr><tr><td>C<sub>2</sub>, O<sub>2</sub>, CO, CO<sub>2</sub></td><td>6.9 × 10<sup>21</sup></td></tr><tr><td colspan="6" style="text-align: center; font-weight: bold;">Neutral exchange reactions</td></tr><tr><td>CO + O ⇌ C + O<sub>2</sub></td><td></td><td>T<sub>tr</sub></td><td>3.9 × 10<sup>13</sup></td><td>-0.18</td><td>69200</td></tr><tr><td>CO + C ⇌ C<sub>2</sub> + O</td><td></td><td>T<sub>tr</sub></td><td>2.0 × 10<sup>17</sup></td><td>-1.00</td><td>58000</td></tr><tr><td>CO<sub>2</sub> + O ⇌ O<sub>2</sub> + CO</td><td></td><td>T<sub>tr</sub></td><td>2.1 × 10<sup>13</sup></td><td>0</td><td>27800</td></tr><tr><td colspan="6" style="text-align: center; font-weight: bold;">Associative ionization reactions</td></tr><tr><td>O + O ⇌ O<sub>2</sub><sup>+</sup> + e<sup>-</sup></td><td></td><td>T<sub>tr</sub></td><td>7.1 × 10<sup>2</sup></td><td>2.70</td><td>80600</td></tr><tr><td>C + O ⇌ CO<sup>+</sup> + e<sup>-</sup></td><td></td><td>T<sub>tr</sub></td><td>8.8 × 10<sup>8</sup></td><td>1.00</td><td>33100</td></tr><tr><td colspan="6" style="text-align: center; font-weight: bold;">Charge exchange reactions</td></tr><tr><td>O<sub>2</sub><sup>+</sup> + O ⇌ O<sup>+</sup> + O<sub>2</sub></td><td></td><td>T<sub>tr</sub></td><td>4.0 × 10<sup>12</sup></td><td>-0.09</td><td>18000</td></tr><tr><td>C<sup>+</sup> + CO ⇌ CO<sup>+</sup> + C</td><td></td><td>T<sub>tr</sub></td><td>1.0 × 10<sup>13</sup></td><td>0</td><td>31400</td></tr><tr><td>C<sup>+</sup> + O<sub>2</sub> ⇌ O<sub>2</sub><sup>+</sup> + C</td><td></td><td>T<sub>tr</sub></td><td>1.0 × 10<sup>13</sup></td><td>0</td><td>9400</td></tr><tr><td colspan="6" style="text-align: center; font-weight: bold;">Electron-impact ionization</td></tr><tr><td>C + e<sup>-</sup> ⇌ C<sup>+</sup> + e<sup>-</sup> + e<sup>-</sup></td><td></td><td>T<sub>ve</sub></td><td>3.9 × 10<sup>33</sup></td><td>-3.78</td><td>130700</td></tr><tr><td>O + e<sup>-</sup> ⇌ O<sup>+</sup> + e<sup>-</sup> + e<sup>-</sup></td><td></td><td>T<sub>ve</sub></td><td>3.9 × 10<sup>33</sup></td><td>-3.78</td><td>158500</td></tr><tr><td colspan="6" style="text-align: center; font-weight: bold;">Radiative recombination reactions</td></tr><tr><td>O + e<sup>-</sup> ⇌ O + hν</td><td></td><td>T<sub>ve</sub></td><td>1.07 × 10<sup>11</sup></td><td>-0.52</td><td>0</td></tr><tr><td>C + e<sup>-</sup> ⇌ C + hν</td><td></td><td>T<sub>ve</sub></td><td>2.02 × 10<sup>11</sup></td><td>-0.46</td><td>0</td></tr></tbody></table>
<table><caption>**Table 9.** Johnston's model without *N* compounds. *A* is in cm³ mole⁻¹ s⁻¹ and *E*<sub>a</sub> is in *K*. Modifications from Park's model are in green, additions in red.</caption><thead><tr><th>Reactions</th><th><i>M</i></th><th><i>T</i><sub>x</sub></th><th><i>A</i></th><th><i>n</i></th><th><i>E</i><sub>a</sub></th></tr></thead><tbody><tr><td colspan="6">Dissociation reactions</td></tr><tr><td>C<sub>2</sub> + M ↔ C + C + M</td><td>All</td><td><i>T</i><sub>a</sub></td><td>4.50 × 10<sup>18</sup></td><td>-1.0</td><td>71 500</td></tr><tr><td rowspan="3">CO + M ↔ C + O + M</td><td>Ar</td><td rowspan="3"><i>T</i><sub>a</sub></td><td>1.20 × 10<sup>20</sup></td><td rowspan="3">-1.0</td><td rowspan="3">129 000</td></tr><tr><td>C, O</td><td>1.80 × 10<sup>21</sup></td></tr><tr><td>C<sub>2</sub>, O<sub>2</sub>, CO, CO<sub>2</sub></td><td>1.20 × 10<sup>21</sup></td></tr><tr><td colspan="6">Neutral exchange reactions</td></tr><tr><td>CO<sub>2</sub> + O ↔ O<sub>2</sub> + CO</td><td></td><td><i>T</i><sub>ir</sub></td><td>2.71 × 10<sup>14</sup></td><td>0</td><td>33 800</td></tr><tr><td colspan="6">Electron-impact ionization</td></tr><tr><td>CO + e<sup>-</sup> ↔ CO<sup>+</sup> + e<sup>-</sup> + e<sup>-</sup></td><td></td><td><i>T</i><sub>ve</sub></td><td>4.50 × 10<sup>14</sup></td><td>0.275</td><td>163 000</td></tr><tr><td>O<sub>2</sub> + e<sup>-</sup> ↔ O<sub>2</sub><sup>+</sup> + e<sup>-</sup> + e<sup>-</sup></td><td></td><td><i>T</i><sub>ve</sub></td><td>2.19 × 10<sup>10</sup></td><td>1.16</td><td>130 000</td></tr></tbody></table>

<table><caption>**Table 10.** Cruden's model, for velocities up to 6.6 km s⁻¹. *A* is in cm³ mole⁻¹ s⁻¹ and *E*<sub>a</sub> is in *K*. Modifications from the Johnston model are in green.</caption><thead><tr><th>Reactions</th><th><i>M</i></th><th><i>A</i></th><th><i>n</i></th><th><i>E</i><sub>a</sub></th></tr></thead><tbody><tr><td colspan="5">Dissociation reactions</td></tr><tr><td>C<sub>2</sub> + M ↔ C + C + M</td><td>All</td><td>1.82 × 10<sup>15</sup></td><td>0</td><td>64 000</td></tr><tr><td>O<sub>2</sub> + M ↔ O + O + M</td><td>All</td><td>1.20 × 10<sup>14</sup></td><td>0</td><td>54 246</td></tr><tr><td>CO + M ↔ C + O + M</td><td>All</td><td>7.99 × 10<sup>38</sup></td><td>-5.5</td><td>129 060</td></tr><tr><td>CO<sub>2</sub> + M ↔ CO + O + M</td><td>All</td><td>7.47 × 10<sup>12</sup></td><td>0.5</td><td>52 321</td></tr><tr><td colspan="5">Neutral exchange reactions</td></tr><tr><td>C<sub>2</sub> + O ↔ CO + C</td><td></td><td>3.61 × 10<sup>14</sup></td><td>0</td><td>0</td></tr></tbody></table>

<table><caption>**Table 11.** Reactions equilibrated after 1 millisecond after imposing a temperature drop from 9000 K to 6500 K and using the Johnston model to determine the chemical evolution of the mixture (initially at equilibrium). The reactions are sorted by the value of the relative difference between the forward and reverse reaction rates (4th column). A value close to 1 indicates a one-sided reaction (fully forward or fully backward) whereas a value close to 0 indicates that the reaction is equilibrated. The threshold for considering the reaction to be equilibrated was fixed to 0.05.</caption><thead><tr><th>Reactions</th><th><i>R</i><sub>f</sub> (mole cm<sup>-3</sup> s<sup>-1</sup>)</th><th><i>R</i><sub>r</sub> (mole cm<sup>-3</sup> s<sup>-1</sup>)</th><th><i>|R<sub>f</sub> - R<sub>r</sub>|</i><br><i>R<sub>f</sub> + R<sub>r</sub></i></th><th>Rank</th></tr></thead><tbody><tr><td>Ar<sup>+</sup> + O ↔ Ar + O<sup>+</sup></td><td>3.963 × 10<sup>-5</sup></td><td>3.958 × 10<sup>-5</sup></td><td>6.5 × 10<sup>-4</sup></td><td>1</td></tr><tr><td>C + O ↔ CO<sup>+</sup> + e<sup>-</sup></td><td>5.170 × 10<sup>-4</sup></td><td>5.161 × 10<sup>-4</sup></td><td>9.1 × 10<sup>-4</sup></td><td>2</td></tr><tr><td>Ar<sup>+</sup> + CO ↔ Ar + CO<sup>+</sup></td><td>2.129 × 10<sup>-4</sup></td><td>2.138 × 10<sup>-4</sup></td><td>2.1 × 10<sup>-3</sup></td><td>3</td></tr><tr><td>Ar<sup>+</sup> + C ↔ Ar + C<sup>+</sup></td><td>8.999 × 10<sup>-5</sup></td><td>8.918 × 10<sup>-5</sup></td><td>4.5 × 10<sup>-3</sup></td><td>4</td></tr><tr><td>O + O ↔ O<sub>2</sub><sup>+</sup> + e<sup>-</sup></td><td>2.775 × 10<sup>-6</sup></td><td>2.810 × 10<sup>-6</sup></td><td>6.3 × 10<sup>-3</sup></td><td>5</td></tr><tr><td>C<sup>+</sup> + CO ↔ CO<sup>+</sup> + C</td><td>4.907 × 10<sup>-6</sup></td><td>4.972 × 10<sup>-6</sup></td><td>6.6 × 10<sup>-3</sup></td><td>6</td></tr><tr><td>CO + C ↔ C<sub>2</sub> + O</td><td>2.763 × 10<sup>-6</sup></td><td>2.814 × 10<sup>-5</sup></td><td>9.1 × 10<sup>-3</sup></td><td>7</td></tr><tr><td>CO<sub>2</sub> + M ↔ CO + O + M</td><td>1.971 × 10<sup>-6</sup></td><td>2.019 × 10<sup>-6</sup></td><td>1.2 × 10<sup>-2</sup></td><td>8</td></tr><tr><td>CO<sub>2</sub> + O ↔ O<sub>2</sub> + CO</td><td>1.029 × 10<sup>-6</sup></td><td>9.816 × 10<sup>-7</sup></td><td>2.3 × 10<sup>-2</sup></td><td>9</td></tr><tr><td>O<sub>2</sub> + M ↔ O + O + M</td><td>3.471 × 10<sup>-5</sup></td><td>3.726 × 10<sup>-5</sup></td><td>3.5 × 10<sup>-2</sup></td><td>10</td></tr><tr><td>O<sub>2</sub> + e<sup>-</sup> ↔ O<sub>2</sub><sup>+</sup> + e<sup>-</sup> + e<sup>-</sup></td><td>2.76 × 10<sup>-14</sup></td><td>3.00 × 10<sup>-14</sup></td><td>4.2 × 10<sup>-2</sup></td><td>11</td></tr></tbody></table>
<table><caption>**Table 12.** Reactions not equilibrated after 1 millisecond after imposing a temperature drop from 9000 K to 6500 K and using the Johnston model to determine the chemical evolution of the mixture (initially at equilibrium).</caption><thead><tr><th>Reactions</th><th>R<sub>f</sub> (mole cm<sup>-3</sup> s<sup>-1</sup>)</th><th>R<sub>r</sub> (mole cm<sup>-3</sup> s<sup>-1</sup>)</th><th>|R<sub>f</sub> - R<sub>r</sub>| / (R<sub>f</sub> + R<sub>r</sub>)</th></tr></thead><tbody><tr><td>C<sup>+</sup> + O<sub>2</sub> ↔ O<sub>2</sub><sup>+</sup> + C</td><td>4.820 × 10<sup>-8</sup></td><td>2.866 × 10<sup>-8</sup></td><td>0.254</td></tr><tr><td>O<sub>2</sub><sup>+</sup> + O ↔ O<sup>+</sup> + O<sub>2</sub></td><td>1.399 × 10<sup>-9</sup></td><td>2.371 × 10<sup>-9</sup></td><td>0.258</td></tr><tr><td>Ar<sup>+</sup> + O<sub>2</sub> ↔ Ar + O<sub>2</sub><sup>+</sup></td><td>3.622 × 10<sup>-8</sup></td><td>2.134 × 10<sup>-8</sup></td><td>0.258</td></tr><tr><td>CO + O ↔ C + O<sub>2</sub></td><td>3.509 × 10<sup>-6</sup></td><td>6.066 × 10<sup>-6</sup></td><td>0.267</td></tr><tr><td>C<sub>2</sub> + M ↔ C + C + M</td><td>6.180 × 10<sup>-7</sup></td><td>1.126 × 10<sup>-6</sup></td><td>0.291</td></tr><tr><td>C + e<sup>-</sup> ↔ C<sup>+</sup> + e<sup>-</sup> + e<sup>-</sup></td><td>1.551 × 10<sup>-6</sup></td><td>2.834 × 10<sup>-6</sup></td><td>0.293</td></tr><tr><td>O + e<sup>-</sup> ↔ O<sup>+</sup> + e<sup>-</sup> + e<sup>-</sup></td><td>7.044 × 10<sup>-8</sup></td><td>1.297 × 10<sup>-7</sup></td><td>0.297</td></tr><tr><td>Ar<sup>+</sup> + e<sup>-</sup> + e<sup>-</sup> ↔ Ar + e<sup>-</sup></td><td>1.356 × 10<sup>-8</sup></td><td>7.350 × 10<sup>-9</sup></td><td>0.297</td></tr><tr><td>CO + e<sup>-</sup> ↔ CO<sup>+</sup> + e<sup>-</sup> + e<sup>-</sup></td><td>4.49 × 10<sup>-12</sup></td><td>8.32 × 10<sup>-12</sup></td><td>0.299</td></tr><tr><td>CO + M ↔ C + O + M</td><td>2.466 × 10<sup>-5</sup></td><td>4.575 × 10<sup>-5</sup></td><td>0.300</td></tr></tbody></table>

ORCID iDs

Corentin H C Grimaldi ID 0009-0000-1080-6719

Sean McGuire ID 0000-0002-5663-0357

Christophe O Laux ID 0000-0002-9129-1186

References

[1] Gu S, Morgan R G, McIntyre T J and Brandis A M 2021 An experimental study of CO₂ thermochemical nonequilibrium AIAA J. **60** 1283–92

[2] Brandis A M, Saunders D A, Johnston C O, Cruden B A and White T R 2020 Radiative heating on the after-body of Martian entry vehicles J. Thermophys. Heat Transf. **34** 66–77

[3] Maillard J et al Time-resolved optical emission spectroscopy measurements of electron density and temperature in CO₂ nanosecond repetitively pulsed discharges AIAA SCITECH 2022 Forum

[4] Cruden B A, Prabhu D and Martinez R 2012 Absolute radiation measurement in Venus and Mars entry conditions J. Spacecr. Rockets **49** 1069–79

[5] Gildfind D E, Morgan R G and Jacobs P A 2016 Expansion tubes in Australia Experimental Methods of Shock Wave Research ed O Igra and F Seiler (Springer) pp 399–431

[6] McGilvray M et al T6: the Oxford University stalker tunnel 20th AIAA Int. Space Planes and Hypersonic Systems and Technologies Conf. (American Institute of Aeronautics and Astronautics)

[7] Huczko A and Szymański A 1984 Thermal decomposition of carbon dioxide in an argon plasma jet Plasma Chem. Plasma Process. **4** 59–72

[8] Yang T, Shen J, Ran T, Li J, Chen P and Yin Y 2018 Understanding CO₂ decomposition by thermal plasma with supersonic expansion quench Plasma Sci. Technol. **20** 065502

[9] Li J, Zhang X, Shen J, Ran T, Chen P and Yin Y 2017 Dissociation of CO₂ by thermal plasma with contracting nozzle quenching J. CO₂ Util. **21** 72–76

[10] Brabbs T A and Belles F E 1967 Recombination of carbon monoxide and atomic oxygen at high temperatures Symp. (International) on Combustion vol 11 pp 125–35

[11] Hardy J E, Gardiner W C Jr and Burcat A 1978 Recombination of carbon monoxide and oxygen atoms Int. J. Chem. Kinet. **10** 503–17

[12] Kondratiev V N and Intezarova E I 1969 Interaction between CO and O Int. J. Chem. Kinet. **1** 105–11

[13] Slanger T G, Wood B J and Black G 1972 Kinetics of O(3P) + CO + M recombination J. Chem. Phys. **57** 233–8

[14] Tibère-Inglesse A C, McGuire S D, Mariotto P and Laux C O 2018 Validation cases for recombining nitrogen and air plasmas Plasma Sources Sci. Technol. **27** 115010

[15] Grimaldi C, McGuire S and Laux C Temperature and density measurements in a recombining CO₂ mixture Plasma Sources Sci. Technol. under review

[16] Grimaldi C 2023 Experimental study and modeling of a recombining CO₂ plasma École doctorale Sciences Mécaniques et Energétiques Matériaux et Géosciences (SMEMAG) (Université Paris-Saclay)

[17] Turner M M 2015 Uncertainty and error in complex plasma chemistry models Plasma Sources Sci. Technol. **24** 035027

[18] Berthelot A and Bogaerts A 2017 Modeling of CO₂ plasma: effect of uncertainties in the plasma chemistry Plasma Sources Sci. Technol. **26** 115002

[19] Park C, Howe J T, Jaffe R L and Candler G V 1994 Review of chemical-kinetic problems of future NASA missions. II—Mars entries J. Thermophys. Heat Transf. **8** 9–23

[20] Annaloro J 2013 Modèles collisionnels-radiatifs appliqués aux situations d'entrée atmosphérique martienne et terrestre (Université de Rouen)

[21] Kozák T and Annemie B 2014 Splitting of CO₂ by vibrational excitation in non-equilibrium plasmas: a reaction kinetics model Plasma Sources Sci. Technol. **23** 045004

[22] Johnston C O and Brandis A M 2014 Modeling of nonequilibrium CO fourth-positive and CN violet emission in CO₂–N₂ gases J. Quant. Spectrosc. Radiat. Transf. **149** 303–17

[23] Annaloro J, Bultel A and Omaly P 2014 Detailed kinetic of CO₂ dissociation and C ionization: application to atmospheric Martian entries J. Phys.: Conf. Ser. **511** 012053

[24] Bogaerts A, Kozák T, van Laer K and Snoeckx R 2015 Plasma-based conversion of CO₂: current status and future challenges Faraday Discuss. **183** 217–32

[25] Capitelli M, Colonna G, D'Ammando G and Pietanza L D 2017 Self-consistent time dependent vibrational and free electron kinetics for CO 2 dissociation and ionization in cold plasmas Plasma Sources Sci. Technol. **26** 055009

[26] Cruden B A, Brandis A M and MacDonald M E 2018 Characterization of CO thermochemistry in incident shockwaves 2018 Joint Thermophysics and Heat Transfer Conf. (American Institute of Aeronautics and Astronautics)

[27] Heijkers S, Martini L M, Dilecce G, Tosi P and Bogaerts A 2019 Nanosecond pulsed discharge for CO₂ conversion:
kinetic modeling to elucidate the chemistry and improve the performance *J. Phys. Chem. C* **123** 12104–16

[28] Pannier E 2019 CO₂ Dissociation with Nanosecond Discharges (Université Paris-Saclay)

[29] Fromentin C et al 2023 Study of vibrational kinetics of CO₂ and CO in CO₂–O₂ plasmas under non-equilibrium conditions *Plasma Sources Sci. Technol.* **32** 024001

[30] Pietanza L D, Colonna G and Capitelli M 2024 Self-consistent state-to-state kinetic modeling of CO₂ cold plasmas: insights on the role of electronically excited states *Plasma Chem. Plasma Process.* **44** 1431–68

[31] Miller R A, Tang C Y, Santos J A B, White T R and Cruden B A 2023 Aftbody heat flux measurements during Mars 2020 entry *J. Spacecr. Rockets* **61** 369–82

[32] Tibere-Inglesse A C, West T K, Jelloian C C, Minesi N Q, Spearrin R M, Clarke J, Di Mare L, McGilvray M and Cruden B A 2024 Reconstruction of Mars 2020 backshell radiative heating via infrared emission spectroscopy of CO₂/N₂/Ar shockwaves *J. Thermophys. Heat Transf.* **38** 568–85

[33] Goodwin D G, Moffat H K, Schoegl I, Speth R L and Weber B W 2024 Cantera: an object-oriented software toolkit for chemical kinetics, thermodynamics, and transport processes *Zenodo* (https://doi.org/10.5281/zenodo.45206)

[34] McBride B J, Zehe M J and Gordon S NASA Glenn coefficients for calculating thermodynamic properties of individual species

[35] Owano T, Kruger C and Beddini R Electron-ion three body recombination coefficient of argon 22nd *Fluid Dynamics, Plasma Dynamics and Lasers Conf.*

[36] Beuthe T G B T G and Jen-Shih Chang J-S C 1997 Chemical kinetic modelling of non-equilibrium Ar-CO₂ thermal plasmas *Jpn. J. Appl. Phys.* **36** 4997

[37] Dresvin S V, Donskoi A V, God'dfarb V M and Klubnikin V S 1971 *Physics and Technology of Low-Temperature Plasma*

[38] Candler G et al 1997 *Numerical simulation of a nonequilibrium nitrogen plasma experiment 28th Plasmadynamics and Lasers Conf.* (American Institute of Aeronautics and Astronautics)

[39] Laux C 1993 Optical diagnostics and radiative emission of air plasmas *Dept. Of Mechanical Engineering* (Stanford University)

[40] Mariotto P 2023 *Study of Nonequilibrium Recombining N₂/Ar Plasmas: Application to after Body Radiative Heating for*

*Reentry into Earth's Atmosphere* (Université Paris-Saclay)

[41] Van Poyer H M S, Tsonev I, Maerivoet S J R, Albrechts M C K and Bogaerts A 2025 Influence of radial transport and turbulent effects on CO₂ conversion in a microwave plasma reactor: insights from a multi-dimensional thermo-chemical flow model *Chem. Eng. J.* **507** 160688

[42] Ibragimova L B, Smekhov G D, Shatalov O P, Eremin A V and Shumova V V 2000 Dissociation of CO₂ molecules in a wide temperature range *High Temp.* **38** 33–36

[43] Balakrishnan N and Chen W S 1999 *Handbook of Tables for Order Statistics from Lognormal Distributions with Applications* (Springer)

[44] Crow E L and Shimizu K 1988 *Lognormal Distributions: Theory and Applications* 1st edn (CRC press)

[45] Kenney J F and Keeping E S 1964 *Mathematics of Statistics* 3rd edn (Van Nostrand)

[46] Niederreiter H 1978 *Quasi-Monte Carlo methods and pseudo-random numbers* *Bull. Am. Math. Soc.* **84** 957–1041

[47] Sobol I M 1967 On the distribution of points in a cube and the approximate evaluation of integrals *USSR Comput. Math. Math. Phys.* **7** 86–112

[48] Shui V H, Appleton J P and Keck J C 1970 Three-body recombination and dissociation of nitrogen: a comparison between theory and experiment *J. Chem. Phys.* **53** 2547–58

[49] Bédié S and Tchen H 1976 Étude théorique de la recombinaison à trois corps: cas particuliers des réactions du type $H + H + M \rightarrow H_2 + M, O + O + M \rightarrow O_2 + M$ et $N + N + M \rightarrow N_2 + M$ *J. Chim. Phys.* **73** 61–74

[50] Colonna G and Capitelli M 2001 Self-consistent model of chemical, vibrational, electron kinetics in nozzle expansion *J. Thermophys. Heat Transf.* **15** 308–16

[51] Park C 2008 Rate parameters for electronic excitation of diatomic molecules II. Heavy particle-impact processes *46th AIAA Aerospace Sciences Meeting and Exhibit (American Institute of Aeronautics and Astronautics)*

[52] Dubuet U 2023 *Modeling of a nonequilibrium recombining air/argon plasma* (Université Paris-Saclay)

[53] Grimaldi C, McGuire S and Laux C 2025 Study of CO₂ recombination kinetics *Recherche Data Gouv* (https://doi.org/10.57745/V00CSP)

[54] Cenian A, Chernukho A and Borodin V 1995 Modeling of plasma-chemical reactions in gas mixture of CO₂ lasers. II. Theoretical model and its verification *Contrib. Plasma Phys.* **35** 273–96