Chemical kinetics and density measurements of OH in an atmospheric pressure He + O₂ + H₂O radiofrequency plasma

The RF-driven plasma source used here was described in detail in [5] and is illustrated in figure 1. Briefly, a quasi-homogenous plasma is generated between two plane-parallel electrodes with a 1 mm gap. The length of the electrodes is 30 mm and the width is 10 mm offering a surface-to-volume ratio of 2 mm⁻¹. Two MgF₂ windows close the reactor in the direction perpendicular to the plane electrodes. An RF generator (Coaxial-Power, MN 150-13.56) and an L-matching network (Coaxial-Power, MMN 150) are used to drive the discharge at a frequency of 13.56 MHz and a dissipated power of 15.0 ± 1.5 W cm⁻³. Voltage and current are measured with a PMK-14KVAC Tektronix probe (1000:1) and an Ion Physics Corp. CM-100 l probe (1 V A⁻¹) from which the dissipated power was calculated following the method of [50].

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The total gas flow is 5 slm. Helium with a purity of 99.996% was used. Most of the gas lines are made of stainless-steel to limit the impurities in the gas. Water vapour and oxygen are admixed to the gas flow using three mass flow controllers. A defined content of water vapour is added to the feed gas by guiding a fraction of the total helium flow through a glass bubbler filled with distilled water. The amount of water vapour is calculated using the vapour pressure at room temperature and the flow rate through the bubbler as in [47]. The output gas of the reactor is guided to an exhaust several meters downstream.

The hydroxyl densities are measured by absorption spectroscopy using an ultra-stable laser-driven broad-band light source (LDLS, Energetiq EQ-99) and reflective optics to limit chromatic aberrations (figure 2). The divergent light coming out of the light source is focused into the central plane of the discharge, collected and focussed onto the spectrograph's entrance slit by four parabolic mirrors (UV enhanced aluminium MPD-F01 Thorlabs). The focal beam waist is about 2 mm, so that the measured density of hydroxyl is spatially averaged over the 1 mm gap between the electrodes. The optical absorption path of 11 mm comprises the width of the electrodes and both 0.5 mm gaps between windows and electrodes. This path is perpendicular to the feed gas flow. The transmitted spectrum is recorded by a 0.5 m Czerny–Turner spectrograph (Andor SR500i) equipped with a 2400 l mm⁻¹ grating that is blazed for λ = 300 nm and a CCD camera (Andor, Newton DU940P-BU2, 2048 × 512 pixels of 13.5 × 13.5 µm² size). The spectrometer entrance slit width was set to 100 µm, resulting in an instrumental broadening of 0.03 nm (see figure 3). Additional apertures in the beam path were used to reduce unnecessary optical plasma emission reaching the entrance slit. The OH ground state density is determined from the intensity of the integrated OH (X²Π_i, ν'' = 0) → OH (A²Σ⁺, ν' = 0) rotational envelope in the wavelength range 306–310 nm.

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As previously described in several papers [9, 47, 49], the absolute density of the hydroxyl radical, ${n_{{\text{OH}}}}$, is obtained from analysis of the measured plasma absorbance spectrum A(λ) in the spectral range from 306 to 311 nm, assuming the Beer–Lambert law (equations (1) and (2)). Four signals are measured successively to correct for background and plasma emission. The first signal is recorded with both the plasma and the light source on (S_P,L), the second records the intensity of the light source only (S_L), the third records the plasma emission only (S_P) and the last signal is a background with both plasma and light source off (S_B). Each signal is integrated over a time period of 2.5 s after an equally long plasma stabilisation time.

$$\begin{align}T\left( \lambda \right) = {e^{ - A\left( \lambda \right)}} = { }\frac{{{S_{\text{T}}}\left( \lambda \right)}}{{{S_0}\left( \lambda \right)}} = \frac{{{S_{{\text{P,L}}}} - {S_{\text{P}}}}}{{{S_{\text{L}}} - {S_{\text{B}}}}}\end{align} \tag{ 1 }$$

$$\begin{align}A\left( \lambda \right) = \sigma \left( \lambda \right) \times L \times {n_{{\text{OH}}}}.\end{align} \tag{ 2 }$$

The observed absorbance is equal to the product of the absorption cross section σ(λ), the absorption path length L, and the line-of-sight averaged absorber density ${n_{{\text{OH}}}}$. A home-made spectrum simulation and fitting programme (see below) is used to find the best least-squared fit of the measured absorbance spectrum, yielding the density and rotational temperature of the lower OH (X²Π_i, ν'' = 0) state. The majority of the population of the electronic ground state lies in the lowest vibrational level, ν'' = 0; no absorption band originating from the ν'' = 1 level was observed. The accuracy of the density measurements depends on the knowledge of the absorption length, here 11 mm with 5% uncertainty. The OH(X) rotational temperature is not necessarily in equilibrium with the gas translational temperature [46].

The absorption spectrum is simulated in the following way. The ro-vibronic energy levels of the upper OH(A²Σ⁺, ν' = 0) state are calculated according to [51], with Hund's coupling case (b) since Λ = 0, and the lower OH(X²Π_i, ν'' = 0) state, with intermediate case (a) to case (b) coupling. The rotational line strength factors for the 12 electric dipole-allowed branches are calculated using the expressions in [52]. These factors are converted, firstly to relative Einstein coefficients for spontaneous emission, then to absolute ones using the value of 0.698 µs for the radiative lifetime of the lowest rotation level of the OH(A²Σ⁺, ν' = 0) state [53], and finally to spectrally integrated absorption cross sections for each individual rotational transition. The relative thermal population distribution among the rotation levels of the OH(X²Π_i, ν'' = 0) state is calculated according to the rotational Boltzmann factor normalised to the partition sum. For all rotational transitions a common spectral Gaussian line profile is assumed, which represents the resolution of the spectrograph. The intrinsic Doppler (0.93 pm at 304 K) and pressure (0.66 pm in helium at 1 atm [54]) broadening are negligible under the given experimental conditions.

Figure 3 shows an example of a measured absorbance spectrum together with the best fit simulation. All measurements were taken at the centre of the reactor, as illustrated by the position of the LDLS beam in figure 1.

In this work, the GlobalKin code is used to describe the RF quasi-homogenous discharge. It is a 0D global plasma-chemical kinetics model which is described in detail in [55]. Briefly, it consists of a reaction chemistry and transport module, a Boltzmann equation solver and an ordinary differential equation solver. In the ordinary differential equation solver, the mass continuity equations for each charged and neutral species are solved as a function of time, accounting for surface and gas phase production and consumption processes:

$$\begin{equation}\frac{{{\text{d}}{N_i}}}{{{\text{d}}t}} = \frac{S}{V}\left( { - \frac{{{D_i}{N_i}{\gamma _i}}}{{{\gamma _i}{\Lambda _{\text{D}}} + \frac{{4{D_i}}}{{{v_{{\text{th}},i}}}}}} + \sum\limits_j {{\text{ }}\frac{{{D_j}{N_j}{\gamma _j}{f_{ij}}}}{{{\gamma _j}{\Lambda _{\text{D}}} + \frac{{4{D_j}}}{{{v_{{\text{th}},j}}}}}}} } \right) + {S_i}.\end{equation} \tag{ 3 }$$

Here, $N$ represents the number density of a species i in the gas phase, $\frac{S}{V}$ is the surface area to volume ratio of the source, ${{{\Lambda }}_{\text{D}}}$ is the diffusion length, $D$ the diffusion coefficient, $\gamma$ the surface loss probability, $f$ the return fraction of species from surfaces and ${S_i}$ represents gas phase production and consumption processes. Here, the diffusion length ${{{\Lambda }}_{\text{D}}}$, is calculated as in [43] for a plasma confined in a rectangular chamber, giving a value of 0.0317 cm. Surface losses and gains are calculated using user defined surface loss probabilities as defined in [47]. In addition, the electron energy equation is also solved:

$$\begin{align}\frac{{\text{d}}}{{{\text{d}}t}}\left( {\frac{3}{2}{n_{\text{e}}}{k_{\text{B}}}{T_{\text{e}}}} \right) & = {p_{\text{d}}} - {\text{ }}\sum\limits_i {\frac{3}{2}{n_{\text{e}}}} {\nu _{{\text{m}}i}}\left( {\frac{{2{m_{\text{e}}}}}{{{M_i}}}} \right){k_{\text{B}}}\left( {{T_{\text{e}}} - {T_i}} \right) \nonumber\\ & \quad + \sum\limits_i {{n_{\text{e}}}} {k_i}{N_i}\Delta {\varepsilon _i}.\end{align} \tag{ 4 }$$

Here, ${ }{n_{\text{e}}}$ is the electron density, ${k_{\text{B}}}$ the Boltzmann constant, ${T_{\text{e}}}$ the electron temperature and ${p_{\text{d}}}$ represents the plasma power. The second and third terms on the right hand side of the equation represent electron energy changes through elastic and inelastic collisions, respectively. In these terms, ${\nu _{{\text{m}}i}}$ is the electron momentum transfer collision frequency with a species i, ${m_{\text{e}}}$ the electron mass, ${M_i}$ the mass of the collision partner, ${T_i}$ the temperature of the collision partner, $k$ the electron impact rate coefficient and ${{\Delta }}{\varepsilon _i}$ the electron energy changes due to the inelastic collision. A pseudo 1D plug flow model is used to convert the temporally varying densities and temperatures obtained from equations (3) and (4) into spatially varying quantities along the channel of the plasma source using the gas flow velocity. The electron energy distribution function, mean electron energy, electron transport coefficients, and electron impact rate coefficients, are obtained regularly during the simulation, by solving the two-term Boltzmann equation, in this case, every 0.1002 cm. At each point the Boltzmann equation is solved for a range of reduced electric fields and the results tabulated. Values of the electron impact rate coefficients and transport coefficients corresponding to the instantaneous mean electron energy calculated by equation (4) are then used to calculate the relevant source terms in the ordinary differential equations. For all cases, the gas temperature is set to 305 K. Simulations are carried out for plasma electrodes of 3 × 1 × 0.1 cm³ but a plasma volume of 3 × 1.1 × 0.1 cm³ to account for both 0.5 mm gaps between windows and electrodes where the plasma expands. The power density is 15 W cm⁻³ and the flow rate is 5 slm.

The reaction mechanism includes 46 species and 577 reactions and is described in the appendix. It is largely based on the reaction mechanism for He + H₂O mixtures presented in [47]. The reaction set was extended to account for higher O₂ concentrations by adding O-based ions and by refinements to some reaction rate coefficients based on the reaction mechanisms for He/O₂ mixtures presented in [9, 11, 12]. The species included in the model are listed in table 1.

To study the effect of water vapour and oxygen on the generation of hydroxyl, various concentrations of water vapour and oxygen are added to the feed gas. The concentration of each added species ranges from 0% to 1%. At higher contents the discharge cannot be sustained.

The density of OH measured by absorption spectroscopy is presented in figure 4 (as squares). The OH density first increases rapidly with humidity content, up to a few tenths of a percent of water vapour, after which the increase is slower. This transition has already been studied [47, 56, 57] and was attributed to the change of electron density and temperature with increasing humidity content, affecting the reaction rate coefficient for OH production. In the absence of O₂, OH is mainly produced by electron impact dissociation of water molecules. However, the water vapour dissociation frequency passes through a maximum, due to competition between an electron temperature increase (increasing the rate coefficient for electron impact dissociation) and electron density decrease (decreasing rate of electron impact dissociation) as the humidity content increases.

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Figure 4 also shows the numerical results for similar conditions (as solid lines). The experimental and simulation results agree within 50%. The trends are very similar, although the simulation results are somewhat higher by a nearly constant factor. This could be due to an overestimation of the power density (1–2 W cm⁻³, compatible with the uncertainty of the power density measurement indicated in section 2.1) and/or uncertainties in reaction sets and rate coefficients. In addition, in helium-water vapour only, the OH density increase is stronger at high humidity content compared to the experimental results. Despite these discrepancies, the comparison with experiment demonstrates very similar trends and comparable absolute densities, giving confidence in the accuracy of the numerical model.

When molecular oxygen is introduced into the feed gas, it becomes the new dominant source of O-based species, in particular atomic oxygen and ozone. The simulated densities of atomic oxygen are presented in figure 5(a). These are in good agreement with the experimental results presented in [8, 45, 47]. In [8] and [45], nitrogen was also included in the gas mixture at a 4:1 ratio. This comparison may suggest that 0.1% nitrogen has a moderate impact on the atomic oxygen density in dry conditions. However, this comparison should not be considered as a validation of the simulation but simply as a coherent comparison in relatively analogous gases. The density of atomic oxygen increases by one to two orders of magnitude in the presence of 0.1% molecular oxygen compared to He + H₂O only. This is due to the change of the main oxygen-atom production reaction already pointed out in [14]: OH + OH → O + H₂O in absence of O₂, and electron impact dissociation of O₂ in presence of significant amounts of O₂. On the other hand, as soon as few hundred ppm of water is introduced in a He + O₂ gas admixture, the oxygen-atom loss is mainly due to OH + O → O₂ + H, for any concentration of oxygen.

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The density of atomic oxygen is predominantly affected by the concentration of molecular oxygen. Nevertheless, it should be noted that a steep decrease in the O density is observed with increasing water vapour concentration up to about 0.05%, for cases where the O₂ admixture is over 0.025%. Such densities of water vapour could be in the range of impurities or diffusion of ambient humidity in the case of an open plasma source. Over 0.05% water vapour concentration, the O density stabilises progressively and is only slightly affected by further water vapour addition. This observation indicates that the introduction of known amounts of additive species to the feed gas gives better control of ROS production than relying on mixing and diffusion of the effluent into the surrounding environment in the case of open plasma sources [58].

The density of ozone is strongly related to the addition of oxygen to the feed gas (figure 5(b)). It increases by three orders of magnitude with addition of 0.1% O₂. At very low water vapour concentrations, ozone also shows a sharp decrease with water vapour addition, similar to oxygen atoms but not quite as pronounced. Over 0.05% H₂O, the ozone density is only weakly affected by increasing humidity content.

Overall, the simulation results suggest that OH, O and O₃ can be controlled quite independently by varying both water and oxygen concentrations in the feed gas over the range 0%–1% for H₂O and 0.05%–1% for O₂. The OH density depends almost entirely on the water vapour content, while the O and O₃ densities are significantly affected by presence of water vapour, but this effect is relatively independent of the concentration of water vapour for values above 0.05%. Conversely, the O and O₃ densities are more strongly affected by varying the O₂ content. More precisely, the O and O₃ densities drop by approximately an order of magnitude following the addition of up to 0.05% water vapour to a He + O₂ feed gas. If high densities of O and O₃ are required without OH species, then dry conditions appear to be better.

A detailed analysis of the hydroxyl reaction pathways in the presence of 0.025% water vapour and various concentrations of oxygen is drawn from the chemical kinetics simulation. The dominant reactions for production and consumption of OH, with and without 0.1% O₂, are presented in tables 2 and 3. The analysis is performed at 15 mm into the plasma channel, corresponding to the measurement position. Table 6 presents the corresponding net production rates of the major species as defined in the introduction.

In the absence of molecular oxygen, electron impact reactions generate almost 62% of the hydroxyl radicals. 54% of the OH molecules are produced by dissociation of water molecules by electron impact, and 74% are lost by the recombination of two OH molecules to produce hydrogen peroxide (R(10)) or water plus atomic oxygen (R(11)). The other significant reaction for production of hydroxyl involves the H₂O⁺ ion, responsible for about 30% of OH production. H₂O⁺ is almost entirely produced by two- and three-body Penning ionisation reactions of He(2³S) with water ((He) + He(2³S) + H₂O → (He) + He + H₂O⁺ + e). At low humidity content, the density of secondary species, such as HO₂, is low and the contribution of these species remains small. In particular, HO₂ contributes to only 3% of hydroxyl production and loss. These results are similar to those detailed in [47].

As soon as some oxygen is added to the mixture, the pathways for OH generation change dramatically, even though the resulting OH density remains little changed (see figure 6). The addition of 0.1% O₂ increases the density of atomic oxygen by more than one order of magnitude (figure 5(a). Consequently, the reaction rates of O-induced reactions are strongly enhanced: the reaction rate of reaction R(6): H₂O + O(¹D) → 2 OH, increases by more than two orders of magnitude, becoming the dominant production reaction of hydroxyl. Loss mechanisms are also modified. Reaction R(13), OH + O → O₂ + H, involving O is greatly enhanced and becomes responsible for 72% of hydroxyl losses. Thus, atomic oxygen is responsible for most of the production as well as the loss of hydroxyl molecules. The calculation of the net production rate of OH by processes involving atomic oxygen, i.e. production reactions R(6) and R(8) minus consumption reaction R(13) (see table 6), shows the net effect O has on the OH density. It reveals that it actually only makes a small alteration to the density of OH at 0.025% H₂O and 0.1% O₂. More precisely, R(6,8,13) have an effective rate of −2.08×10¹⁷ cm⁻³ s⁻¹ which is −8% of the total production rate of 2.53 × 10¹⁸ cm⁻³ s⁻¹. Though dramatically impacting the pathways for OH formation, the O and O(¹D) production and consumption contributions nearly cancel out, leading to a small effective (and negative) contribution to the OH density. This observation remains valid for higher added O₂ concentrations, explaining the nearly constant OH density with O₂ addition (presented in figure 4) for a given low humidity content.

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Further analysis of the reaction pathways for OH formation reveals that molecular oxygen has an indirect impact on the OH density. Apart from the O/O(¹D) reactions, the other main production reactions are through the seemingly minor reactions R(5) and R(9):

$$\begin{equation}{\text{He}}\,+\,{\text{H}}\,+\,{\text{H}}{{\text{O}}_{\text{2}}} \to {\text{He}}\,+\,{\text{OH}}\,+\,{\text{OH}}\end{equation} \tag{ 5 }$$

$$\begin{equation}{\text{H}}\,+\,{{\text{O}}_{\text{3}}}\, \to \,{\text{OH}}\,+\,{{\text{O}}_{\text{2}}}.\end{equation} \tag{ 6 }$$

The density of atomic hydrogen is very similar with and without oxygen admixture, while the densities of HO₂ and O₃ increase by one to three orders of magnitude with the addition of oxygen. The strong enhancement of HO₂ density (from about 8 × 10¹¹ to 5 × 10¹³ cm⁻³) comes from the increased rate of He + H + O₂ → He + HO₂ with addition of 0.1% molecular oxygen. This reaction is more efficient than the additional loss reaction: HO₂ + O → OH + O₂. The effective contribution of HO₂-induced reactions (R5,8,15) represents 27% of OH production at low water content (0.025% H₂O) and 0.1% O₂ (see table 6). The role of reaction (5) on the OH kinetics was also identified in a recent study of a He + O₂ + H₂O pulsed plasma jet at 1 kHz and 10 kV [59].

For relatively low concentrations of oxygen, the role played by ozone is minor. However, it grows with the increasing oxygen addition since ozone is dominantly produced by: He + O₂ + O → He + O₃. Subsequently, through reaction (6), it produces non-negligible amounts of OH radicals, representing 2% of the OH formation at 0.1% O₂ (effective rate 5.6 × 10¹⁶ cm⁻³ s⁻¹) and about 10% at 0.5% O₂ (reaction rates at 0.5% O₂ are not detailed here).

Now let us consider a similar analysis of the hydroxyl reaction pathways in the presence of 0.75% water vapour and various concentrations of oxygen. The dominant reactions for the production and consumption of OH with and without 0.1% O₂ are presented in tables 4 and 5. The corresponding net production rates of the major species can be found in table 6.

In the absence of molecular oxygen, at the highest water concentration studied (0.75% H₂O), the reaction rates of all OH related reactions increase significantly and the relative importance of the different production or loss reactions of OH change compared to low humidity conditions (0.025% H₂O). The density of most H-containing species increases dramatically, in particular for the secondary species H₂O₂ and HO₂. The H density increases to a lesser extent, while OH and O show a similar increase. Conversely, the positive ions H₂O⁺ and OH⁺ decrease by up to an order of magnitude, due to a decrease in electron density, affecting the density of their precursor species, He(2³S) which are created through electron impact reactions.

Three main points can be made. The dissociation of H₂O by electron impact remains the dominant reaction for OH production. But as mentioned above, due to the decrease in electron density as the water vapour concentration increases, OH production is reduced above a certain water vapour content. Also, with the larger increase of secondary species densities compared to other species, reactions R(5, 8, 14, 15) are promoted. In particular, reaction (5) now accounts for more than a fifth of the production of OH, against 2.7% with only 0.025% H₂O. Among the loss mechanisms of OH, reactions involving secondary species (H₂O₂, HO₂) also play a more significant role. The recombination of two OH molecules now only accounts for half of the losses, against three-quarters at 0.025% H₂O. The effective reaction rate of HO₂ has an increased role in OH production; it goes from having a negligible contribution at low water vapour content to making +14% of OH formation (see table 6). It is noteworthy that reaction R(2) has a decreased relative contribution due to the significant decrease in H₂O⁺ ion density; the ratio [H₂O⁺]/[H₂O] drops by two orders of magnitude compared to its value at 0.025% H₂O.

In the presence of 0.1% molecular oxygen and 0.75% water vapour, the plasma chemistry does not tend towards a pure water vapour chemistry, in contrast to the low humidity case (0.025%). The effect of high amounts of both water vapour and oxygen leads to the development of complex pathways for OH formation. Thus, the chemistry exhibits strong characteristics of both water vapour and oxygen-induced pathways. In this way, OH is produced approximately equally by electron-impact dissociation of water and by the O(¹D)-induced reaction R(6). The following comments compare a gas mixture with 0.1% O₂ at low humidity (He + 0.025% H₂O + 0.1% O₂) to a high humidity case (He + 0.75% H₂O + 0.1% O₂). The latter reaction R(6), largely benefits from the increase of water density: the reaction rate of R(6) increases by 193% from low to high humidity, while the reaction rate of R(13) remains almost unchanged. This means that large concentrations of water vapour are necessary for the net contribution of atomic oxygen R(6, 8, 13) to increase the production of OH. Nevertheless, O and O(¹D) contributions remain low at high humidity, about +5% of OH production rate (3.83 × 10¹⁷ out of 7.37 × 10¹⁸ cm⁻³ s⁻¹). In addition, the secondary species HO₂ which accounted for a quarter of OH formation at low humidity content has a negligible impact on OH formation (−2%) at high humidity. This is due to the strong drop of O density with higher water concentration, affecting the R(8) reaction rate.

These complex interactions show that even with only two additives, the chemistry cannot be easily tailored, and a detailed analysis of reaction rates must be undertaken for that purpose.

Focusing on O and O(¹D), their generation mechanisms are very different, and change according to the gas mixture.

If water vapour is the only admixture, both the production and loss mechanisms of ground-state atomic oxygen depend dominantly on OH and do not change significantly with increasing water content. Consequently, the density of O increases almost proportionally with OH (+490% from 0.025% to 0.75% added water vapour) and the density of O may be inferred from the density of OH if it is known in one condition. The generation mechanisms of O(¹D) are different, but again they do not change significantly with increasing water content. O(¹D) production depends on low density secondary species (mostly O₂, O₂* and O) and a little on water vapour. Therefore, the O(¹D) density does not increase significantly from low to high humidity content.

In the presence of molecular oxygen, in dry conditions, O and O(¹D) are largely produced by dissociation of O₂ by electron impact. O(¹D) mainly decays to O through: O(¹D) + O₂ → O + O₂(b¹Σ)). The densities of O and O(¹D) increase by two orders of magnitude compared to the case with water only (figures 5 and 6). O reaches 5 × 10¹⁴cm⁻³ in He + 0.02% O₂, similar to the experimental value from [8] obtained in helium with 0.1% dry air (see legend of figure 5 for other experimental conditions) and reaches 10¹⁵cm⁻³ in He + 0.1% O₂. The loss of O is mainly due to O gas-phase recombination mechanisms. Diffusion to the walls is typically the second most important loss process but it only represents a few percent of the overall losses. However, when some water vapour is introduced, OH and HO₂ become largely responsible for the loss of O. O(¹D) is very efficiently quenched by water to form 2 OH, so that the decay to O becomes less significant. O losses are significantly higher in the presence of water vapour causing the density of O to drop by an order of magnitude from dry to low humidity conditions. As the humidity content increases from 0.05% to 1%, the variations are much smaller and the O density decreases by at most a factor 3.

Finally, figure 6 illustrates the variations of the densities of the major ROS and the electron density for the most representative admixtures. It also includes the densities simulated in He + 0.025% O₂ + 0.5% O₂ in order to accentuate the role of O₂-induced reactions. It highlights that adding small (0.1%) or large (0.5%) concentrations of O₂ does not significantly affect the density of H-species (apart from HO₂). However, large concentrations of O₂ (0.5% O₂) significantly increase the production of O₃. From 0.1% to 0.5% added oxygen, the ozone density rises by a factor 4 (1–4 × 10¹³ cm⁻³).