# A reaction mechanism for oxygen plasmas

Tiago C Dias1, Chloé Fromentin1, Luís L Alves1, Antonio Tejero-del-Caz1,2,
Tiago Silva1 and Vasco Guerra1 1Instituto de Plasmas e Fusão Nuclear,
Instituto Superior Técnico, Universidade de Lisboa, Portugal
2Departamento de Física, Facultad de Ciencias, Universidad de Córdoba, Campus
de Rabanales, Spain<EMAIL_ADDRESS>
###### Abstract

This work presents a reaction mechanism for oxygen plasmas, i.e. a set of
reactions and corresponding rate coefficients that are validated against
benchmark experiments. The kinetic scheme is validated in a DC glow discharge
for gas pressures of 0.2–10 Torr and currents of 10–40 mA, using the 0D LisbOn
KInetics (LoKI) simulation tool and available experimental data. The
comparison comprises not only the densities of the main species in the
discharge - $\mathrm{O_{2}(X^{3}\Sigma_{g}^{-})}$,
$\mathrm{O_{2}(a^{1}\Delta_{g})}$, $\mathrm{O_{2}(b^{1}\Sigma_{g}^{+})}$ and
$\mathrm{O(^{3}P)}$ \- but also the self-consistent calculation of the reduced
electric field and the gas temperature. The main processes involved in the
creation and destruction of these species are identified. Moreover, the
results show that the oxygen atoms play a dominant role in gas heating, via
recombination at the wall and quenching of
$\mathrm{O_{2}(X^{3}\Sigma_{g}^{-},v)}$ vibrations and $\mathrm{O_{2}}$
electronically-excited states. It is argued that the development and
validation of kinetic schemes for plasma chemistry should adopt a paradigm
based on the comparison against standard validation tests, as it is done in
electron swarm validation of cross sections.

††: Plasma Sources Sci. Technol.

  * April 2023

Keywords: oxygen plasma, validation, plasma modelling, glow discharge,
reaction mechanism

##  1 Introduction

Low-temperature oxygen-containing plasmas have been used in a manifold of
applications. They give an important contribution in various fields such as
semiconductor manufacturing [1, 2], thin film synthesis [3] and plasma
medicine [4, 5, 6]. The kinetics of reactive oxygen species have been studied
extensively also due to their relevance, e.g., in oxygen-iodine lasers [7, 8],
plasma-assisted combustion [9], CO2 conversion [10, 11, 12], NO production
[13] and atmospheric chemistry studies [14]. The main strength of oxygen
plasmas is the inherent ability of generating large quantities of highly
reactive species, such as $\mathrm{O(^{3}P)}$,
$\mathrm{O_{2}(a^{1}\Delta_{g})}$, $\mathrm{O_{2}(b^{1}\Sigma_{g}^{+})}$ and
O3, with a relatively small energy input. The high reactivity comes together
with great complexity and, although these plasmas were vastly studied,
important open questions remain pending.

The understanding of O2 discharges requires a combined effort of modelling and
experimental work. On the one hand, experiments allow to access and
characterize real plasma systems. However, very often the amount of physical
parameters that can be accessed by experimental diagnostics is limited and the
interpretation of the experimental measurements is not straightforward. On the
other hand, modelling provides information on both the microscopic and
macroscopic scales and access to physical quantities not directly available
from measurements.

One of the pillars of low-temperature plasma (LTP) modelling is the kinetic
scheme describing the interactions between the heavy species and their
connection with the electrons. This defines not only the densities of heavy
species but also the power required to sustain the discharge, via the electron
energy distribution function (EEDF), which is strongly correlated with the
composition of the gas mixture. Therefore, the definition of a reaction
mechanism, i.e., a set of reactions and corresponding rate coefficients that
are validated against benchmark experiments, is mandatory to make sensible
modelling predictions of the system under study. By definition, a benchmark is
a reference point against which things can be compared. In the present
context, benchmark experiments correspond to a significant ensemble of
experimental data, intended (or suited) for model validation, obtained in well
defined and reproducible conditions, using established diagnostics, and
assessing multiple quantities.

The development of reaction mechanisms is a concept introduced by the
combustion community, GRI-Mech constituting a well-known example [15].
However, this procedure is still somewhat uncommon in the LTP community, where
extensive validation of electron-impact cross sections by swarm analysis is
well established [16, 17, 18, 19] but the validation of plasma-chemistry
models against benchmark experiments remains to some extent elusive. In fact,
although many modellers follow the procedure of comparing their results
against experiment for many years, a community consensus on recommended
practices and definition of experiments as golden standards to assess the
results of the models is still missing [20]. Following [20], such a definition
requires a combined effort of experimentalists and modellers and would enhance
our ability to understand the rationale of the approximations made and bring
systematic consistency between models. Regarding $\mathrm{O_{2}}$ discharges,
there are various kinetic schemes available in the literature [21, 22, 23, 24,
25, 26, 27, 7]. Nonetheless, to the best of our knowledge, none of them can be
considered to be validated against a comprehensive set of dedicated
experiments designed as a benchmark.

In recent years, a group from the Laboratoire de Physique des Plasmas (LPP) in
Paris, in collaboration with a team from the Lomonosov Moscow State University
(MSU), performed a systematic and thorough experimental work concerning the
kinetics of $\mathrm{O(^{3}P)}$ [28], $\mathrm{O_{2}(a^{1}\Delta_{g})}$ [29]
and $\mathrm{O_{2}(b^{1}\Sigma_{g}^{+})}$ [30]. The studies were done in a DC
glow discharge in pure O2 for gas pressures of 0.2–10 Torr and currents of
10–40 mA. The experimental campaigns revealed new mechanisms for the formation
and quenching of $\mathrm{O_{2}(a^{1}\Delta_{g})}$ and
$\mathrm{O_{2}(b^{1}\Sigma_{g}^{+})}$, pointed out the elementary processes
underlying heterogeneous O-atom recombination, and made available a very
complete set of experimental data to be used for the validation of kinetic
schemes.

Our group in Lisbon developed for several years a kinetic scheme for O2
kinetics that describes rather well different sets of experiments [31, 32, 33,
34, 21]. The latest published work [21] consists on the study of inductively
coupled plasma discharges, where the influence of the vibrational kinetics of
$\mathrm{O_{2}(X^{3}\Sigma_{g}^{-},v)}$ was also included. However, a
preliminary comparison with the experimental measurements made at LPP and MSU
has shown that our model was still failing to capture important trends,
specially for the kinetics of $\mathrm{O_{2}(a^{1}\Delta_{g})}$ and
$\mathrm{O_{2}(b^{1}\Sigma_{g}^{+})}$, which motivated the present
investigation.

In this work, we leverage on recent experimental data [28, 29, 30] from LPP
and MSU to develop a reaction mechanism for oxygen plasmas. The validation is
made using a self-consistent 0-D model describing the positive column of a DC
glow discharge. This discharge is well suited for unraveling the kinetics of
oxygen species, as it generates in many conditions an axially homogeneous
plasma and is accessible to various diagnostics. The densities of the main
species - $\mathrm{O(^{3}P)}$, $\mathrm{O_{2}(X^{3}\Sigma_{g}^{-})}$,
$\mathrm{O_{2}(a^{1}\Delta_{g})}$ and $\mathrm{O_{2}(b^{1}\Sigma_{g}^{+})}$ \-
are benchmarked against experiment and the main processes of species
production/destruction are enlightened. Moreover, the validation is extended
to the prediction of the gas temperature and the reduced maintenance electric
field (the ratio of the electric field to the gas density). Bear in mind that
in some configurations DC glow discharges may exhibit striations in the axial
direction [35, 36, 37, 38], depending on the type of gas, pressure and tube
dimensions, which turns their accurate modelling deeply challenging, but in
the experimental works of [28, 29, 30] no striations are mentioned.

The calculations in this work are performed with the LisbOn KInetics (LoKI) 0D
simulation tool, constituted by two modules: a Boltzmann solver (LoKI-B) for
the two-term electron Boltzmann equation [39, 40] and a chemical solver
(LoKI-C) for the heavy-species kinetics [41, 42]. In past works, LoKI has been
used to describe with success the positive column of DC glow discharges in
different gas mixtures [41, 43, 44, 45]. However, other authors have employed
different models to describe the kinetics of glow discharges. For instance,
glow discharges have been studied with fluid models, where the spatial
dependence of the electron kinetics is accounted for by means of the local-
energy or local-field approximations [28, 30, 46, 47, 48, 49, 50]. The spatial
variation of the electron energy distribution can be considered also with the
so-called “nonlocal” approach [51, 52, 53, 54, 55, 56, 57], in conditions
where the energy relaxation length is much higher than the tube radius, which
is not the case for the conditions examined in this work [58, 59]. A more
correct, complex and time-consuming formulation describes the spatial
variation of the electron kinetics by solving the space-dependent electron
Boltzmann equation [60, 61, 62, 63, 64, 65, 66]. Here, we use the simpler 0D
approach, which was verified to provide a good description of the system under
study for most species (O3 being an exception) by comparison with 1D
calculations [67], and is further confirmed in section 3.1. Another important
point is the use of the two-term assumption for a Legendre expansion of the
electron distribution function in the velocity space [68], since it may not be
accurate when the inelastic collisions are strong or the electric field is
high [69]. Nevertheless, for the reduced electric fields typical in these
discharges, $\lesssim 100$ Td, the two-term approximation should provide good
results for the case of oxygen [70].

The paper is organized as follows. Section 2 presents (i) the discharge set-up
under study, (ii) the self-consistent 0D kinetic model, (iii) the reactions
comprising the kinetic scheme, focusing on the differences with previous
works, and (iv) the additional input data required to simulate oxygen plasmas.
Section 3 presents the validation of the model by comparison with experiment
and analyzes the main processes controlling the discharge behavior. Finally,
section 4 summarizes the main findings of this paper and sets out future work.

##  2 Methods and conditions

###  2.1 Glow discharge set-up

The experimental set-up of the glow discharge simulated in this work is
described in detail in [28, 29, 30]. The DC discharge is ignited in a Pyrex
tube 56 cm long with an internal radius of 1 cm. The electrodes are located in
side-arms separated by 52.5 cm, so as to fill the main tube only with the
positive column. The surface temperature of the tube is kept constant at 50oC
= 323.15 K using a thermostatic bath. The temperature difference between the
inner and outer surfaces of the tube is less than 2 K and therefore
negliglible [28]. The gas pressure varies from 0.2 to 10 Torr, the discharge
current from 10 to 40 mA and the flow rate is pressure dependent, being 2 sccm
at 0.2 Torr and 10 sccm at 10 Torr. The oxygen recombination probabilities at
the wall were measured in [29] and are used as input to the model, cf. section
2.3.

###  2.2 0D global model LoKI

The calculations in this work were performed with the LoKI 0D simulation tool,
constituted by two modules: a Boltzmann solver (LoKI-B) for the two-term
electron Boltzmann equation [39, 40] and a chemical solver (LoKI-C) for the
heavy-species kinetics [41, 42]. This section gives an overview of the two
modules and how they are coupled in a self-consistent way.

####  2.2.1 LoKI-B

LoKI-B [39] solves a space-independent form of the electron Boltzmann equation
under the usual two-term assumption, for a Legendre expansion of the electron
distribution function in the velocity space [68]. The simulation tool
addresses the electron kinetics in any complex gas mixture, considering
inelastic and superelastic collisions with any target state. On input, the
code requires the operating conditions (reduced electric field, gas
temperature and pressure), the gas mixture, the populations of the internal
states and the relevant sets of electron-scattering cross sections. The cross
sections can be obtained from the open-access website LXCat [71] or other
sources, and should be organized in datafiles compliant with the LXCat format.
On output, it provides (i) the EEDF, (ii) the electron power absorbed from the
electric field and transferred to the different collisional channels, (iii)
the electron swarm parameters, such as the mobility and the diffusion
coefficient and (iv) the electron-impact rate-coefficients calculated from the
integration of the corresponding cross sections over the EEDF. This
information is then used in the chemical solver as follows.

####  2.2.2 LoKI-C

LoKI-C solves a system of 0D rate-balance equations for all the heavy-species
densities ($n_{s}$) in the plasma, where the gain-loss terms due to chemical
volume reactions ($S_{s}^{\mathrm{chem}}$), transport to the wall
($S_{s}^{\mathrm{transp}}$) and gas flow ($S_{s}^{\mathrm{flow}}$) are
spatially averaged, according to

$\frac{\partial n_{s}}{\partial
t}=S_{s}^{\mathrm{chem}}+S_{s}^{\mathrm{transp}}+S_{s}^{\mathrm{flow}}\ .$ (1)

The term $S_{s}^{\mathrm{chem}}$ comprises electron-impact reactions, where
the rate coefficients from LoKI-B are used, and chemical reactions between
heavy particles (see section 2.3). The transport of neutral species to the
wall is described with the model proposed by Chantry [72], as detailed in
section 4.1 of Guerra et al. [41]. This approach takes into account both the
diffusion time of the species ($\tau_{s}^{\mathrm{diff}}$), from the reactor
volume to the wall, and the probability ($\gamma_{s}$) of destruction at the
wall. The transport of positive ions is assumed to be given by ambipolar
diffusion, as described by the high plasma-density limit of the ion transport
theories [73, 74]. Since negative ions $\mathrm{O^{-}}$ are predominant in
$\mathrm{O_{2}}$ discharges, their effect on the electron density profile is
taken into account, following[32]. The impact of different charged-particle
transport theories on the discharge kinetics is discussed and quantified
elsewhere [75]. Note that the electron mobility and diffusion coefficient
obtained in LoKI-B are used in the self-consistent calculation of the
ambipolar diffusion coefficients. Finally, $S_{s}^{\mathrm{flow}}$ entails
creation and destruction terms due to flow of particles entering and exiting
the reactor. The inflow is defined by the experimental conditions and the
outflow is determined so as to conserve the total number of atoms in the
discharge volume (see [43] for more details).

Take note that the rate-balance equation for the electrons is not solved since
the calculations are performed for a given electron density consistent with
the experimental discharge current, cf. section 2.2.3.

In addition to the equations for the species densities, LoKI-C also solves a
thermal model for the self-consistent calculation of the gas temperature $T$
[76, 77]. Under constant pressure and assuming that heat conduction is the
dominant cooling mechanism along the volume, the gas thermal balance equation
writes

$NC_{p}\frac{\partial T}{\partial
t}=Q_{\mathrm{in}}-\lambda_{\mathrm{g}}\nabla^{2}T\ ,$ (2)

where $N$ is the gas density, $C_{p}$ is the heat capacity of the gas at
constant pressure, $\lambda_{\mathrm{g}}$ is the thermal conductivity and
$Q_{\mathrm{in}}$ is the total net power per unit volume transferred to gas
heating. The latter term has two contributions: (i) the net power per unit
volume transferred from the electrons to the gas due to elastic collisions,
obtained from LoKI-B, and (ii) the sum of all the reaction rates multiplied by
the corresponding reaction enthalpies. For reactions occurring at the wall, it
is assumed that half of the energy is accommodated by the wall, while the
other half is released for gas heating. The impact of this assumption is
extensively discussed by Pintassilgo et al. [76].

In conditions of axial homogeneity and stationarity, the solution of equation
(2) corresponds to the following parabolic radial profile [76]:

$T(r)\simeq T_{0}-(T_{0}-T_{\mathrm{nw}})\frac{r^{2}}{R^{2}}\ ,$ (3)

where $R$ is the tube radius, $r\in[0,R]$, and $T_{0}$ and $T_{\mathrm{nw}}$
are the gas temperatures at the center and near the wall, respectively.
Assuming this parabolic profile, the temporal evolution of the average gas
temperature $T_{\mathrm{g}}\equiv\langle
T(r)\rangle_{r}=\frac{1}{2}(T_{0}+T_{\mathrm{nw}})$ is described by the
following equation:

$NC_{p}\frac{\partial T_{\mathrm{g}}}{\partial t}\simeq
Q_{\mathrm{in}}-\frac{8\lambda_{\mathrm{g}}(T_{\mathrm{g}}-T_{\mathrm{nw}})}{R^{2}}\
,$ (4)

where $C_{p}$, $\lambda_{\mathrm{g}}$ and $Q_{\mathrm{in}}$ are assumed
homogeneous and are calculated as a function of $T_{\mathrm{g}}$. Equation
(4), for the gas temperature, is solved coupled to equations (1) for the
species densities, with a value of $T_{\mathrm{nw}}$ to be determined. First,
we assume that the temperature drop between the inner and outer walls of the
reactor is negligible, in accordance with experiment (c.f. section 2.1), such
that a fixed inner-wall temperature $T_{\mathrm{w}}$ is taken, equal to the
outer-wall temperature of 323.15 K. Second, we consider a convective heat flux
between the gas and the inner-wall given by:

$\Gamma_{\mathrm{nw}}=h_{\mathrm{gas-wall}}(T_{\mathrm{nw}}-T_{\mathrm{w}})\
,$ (5)

where $h_{\mathrm{gas-wall}}$ is the convection coefficient between the gas
and the wall. This heat flux matches the power per unit area that flows
outward the gas by conduction, which can be estimated using the gas-
temperature profile (3):

$\Gamma_{\mathrm{nw}}=\lambda_{\mathrm{g}}|\nabla
T|\simeq\frac{4\lambda_{\mathrm{g}}}{R}(T_{\mathrm{g}}-T_{\mathrm{nw}})\ .$
(6)

Combining equations (5) and (6), we obtain:

$T_{\mathrm{nw}}=\frac{\frac{4\lambda_{\mathrm{g}}}{R}T_{\mathrm{g}}+h_{\mathrm{gas-
wall}}T_{\mathrm{w}}}{\frac{4\lambda_{\mathrm{g}}}{R}+h_{\mathrm{gas-wall}}}$
(7)

The impact of different $h_{\mathrm{gas-wall}}$ values on the determination of
$T_{\mathrm{g}}$ is studied in section 3.1. An alternative approach to obtain
the temperature jump between $T_{\mathrm{nw}}$ and $T_{\mathrm{w}}$ would be
to consider a thermal accommodation coefficient for the gas particles hitting
the tube wall, that accounts for the energy accommodation by the wall in
reactions occurring at the surface and in elastic collisions in a combined
manner, as it is done e.g. in [78, 28]. Although a direct comparison between
the coefficients used in either formulation is not straightforward, the
similarity of the calculated temperature jumps shows they are compatible.
Additionally, the parabolic profile of gas temperature considered here [76,
77], motivated by the solution of the thermal balance equation with constant
thermal properties and heat source, could be replaced by a Bessel profile,
motivated by the typical shape of the electron density profile [79, 80, 81].
It is verified a posteriori that, under the conditions of the present study,
the parabolic profile describes well the behavior evidenced in experiment (cf.
figure 8). In any case, at higher pressures ($\gtrsim$ 100 Torr), not
considered in this work, phenomena of discharge contraction may arise, where
the temperature profile is neither parabolic nor Bessel, and a space-dependent
model is required to provide an accurate description [82, 83].

####  2.2.3 Self-consistent coupling: LoKI-B+C

The workflow of LoKI is presented in various works [74, 41, 43] and is
schematized in figure 1. LoKI calculates a self-consistent solution of both
electron and heavy-species kinetics in the plasma for user-defined working
conditions: initial mixture composition ($[i]/N$), gas pressure ($p_{0}$),
discharge current ($I_{0}$), reactor radius ($R$) and length ($L$), and flow
rate ($\phi$). For initial guesses of the mixture $[i]/N$, average gas
temperature ($T_{\mathrm{g0}}$), reduced electric field ($(E/N)_{0}$) and
electron density ($n_{e0}$), the workflow starts with LoKI-B calculating the
EEDF, the electron-impact rate coefficients ($k_{\mathrm{e-i}}$), the electron
power transferred into the different collisional channels ($\Theta_{j}/N$) and
the electron transport parameters. Then, this information is used in LoKI-C to
calculate the species densities and the gas temperature until steady-state is
reached. If the steady-state pressure ($p_{f}$) is not the same as the working
pressure ($p_{0}$), the pressure at the beginning of the chemical calculation
($p_{i}=p(t=0)$) is varied until the condition is satisfied. Similarly, the
$E/N$ used in the Boltzmann solver is adjusted so as to achieve quasi-
neutrality in steady-state. Additionally, consistency in both modules is
assured by updating the steady-state gas mixture and temperature calculated by
LoKI-C into LoKI-B. Finally, the electron density is varied so as to obtain a
calculated discharge current $I=ev_{d}\pi R^{2}n_{e}$ equal to the
experimental value given as input (here, $e$ is the electron charge and
$v_{d}$ is the simulated drift velocity). Notice that, for other discharge
configurations where the current cannot be given as input, the input power
density may be used instead for the electron density calculation. In this
work, the relative tolerances for the pressure, quasi-neutrality, mixture
composition and electron density cycles are $10^{-4}$, $10^{-3}$, $10^{-4}$
and $10^{-3}$, respectively.

Figure 1: Workflow of LoKI, adapted from [43]. See text of section 2.2.3 for
details on the notation used in the figure.

###  2.3 Reaction mechanism

The reaction mechanism presented here represents an update relatively to the
work by Annušova et al. [21], and benefits from the recent experimental
campaigns by Booth and co-workers [28, 29, 30] and insights from the MSU team
[84, 7]. The current set of reactions considers the following charged and
neutral species: e, $\mathrm{O_{2}^{+}}$,$\mathrm{O^{+}}$, $\mathrm{O^{-}}$,
$\mathrm{O_{2}(X,v=0:41)}$, $\mathrm{O_{2}(a)}$, $\mathrm{O_{2}(b)}$,
$\mathrm{O_{2}(Hz)}$, $\mathrm{O(^{3}P)}$, $\mathrm{O(^{1}D)}$,
$\mathrm{O_{3}}$ and $\mathrm{O_{3}^{*}}$, where $\mathrm{O_{2}(Hz)}$ is an
effective sum of the $\mathrm{O_{2}(A^{\prime
3}\Delta_{u},A^{3}\Sigma_{u}^{+},c^{1}\Sigma_{u}^{-})}$ Herzberg states [84]
and $\mathrm{O_{3}^{*}}$ is an effective state of vibrationally-excited ozone
[34].

This section aims to summarize the reactions comprising the scheme, focusing
on the differences relatively to the previous version [21], namely: (i)
quenching of $\mathrm{O_{2}(a)}$ [7, 29] and $\mathrm{O_{2}(b)}$ [30], (ii)
inclusion of the $\mathrm{O_{2}(Hz)}$ species [84], (iii) update of the
probabilities for processes at the wall [29, 30] and (iv) revision of
coefficients with typos in [21]. The addition of $\mathrm{O_{2}(Hz)}$ is
justified by its importance in the production of $\mathrm{O_{2}(a)}$ and in
gas heating, as pointed out in [27, 84, 7] and detailed in sections 3.2.1 and
3.2.2. The updates in the kinetic scheme concern only reactions for
vibrationally-cold oxygen, i.e., involving the vibrational ground state of
oxygen $\mathrm{O_{2}(X,0)}$, and are detailed in sections 2.3.1 and 2.3.2.
The vibrational kinetics used in this work is entirely taken from [21] and is
summarized in section 2.3.3.

####  2.3.1 Electron-impact processes for vibrationally-cold oxygen

The electron-impact cross-sections of oxygen-containing plasmas used for the
solution of the electron Boltzmann equation are presented and discussed by
Alves et al. [18] and can be found in the IST-Lisbon database at LXCat [85].
The cross-section set for electron scattering with ground-state molecules was
validated by comparing calculations with experimental data from swarm
experiments. The set for $\mathrm{O_{2}}$ molecules includes the following
mechanisms: effective (elastic and inelastic) momentum-transfer; excitations
to rotational, vibrational and electronic levels; dissociation; dissociative
attachment; and ionization. The set for O atoms includes an elastic momentum-
transfer, excitations to electronic levels and ionization. For details on the
cross sections used for solving the Boltzmann equation, the reader is referred
to [18], since in the present work we focus only on the electron-impact
processes affecting the chemical kinetics.

Table 1 presents the electron-impact processes considered in the chemical
kinetics. With the exception of R17, the cross sections for electron
collisions with $\mathrm{O_{2}(X,0)}$ are originally taken from Phelps [86].
The cross sections for R3, R8 and R10 are based on the $R$-matrix calculations
by Tashiro et al. [87]. Due to lack of data, the cross sections for R9 and R11
are equal to the ones for the ground state, with the appropriate threshold
shift. The collisions with $\mathrm{O(^{3}P)}$, R4 and R15, are according to
Laher and Gilmore [88]. The cross section for electron-impact detachment of
$\mathrm{O^{-}}$ (R16) is taken from the measurements of Vejby-Christensen et
al. [89]. The cross sections for the superelastic processes of R1-R5 are
calculated through the Klein-Rosseland relation, expressing the principle of
detailed balance [90]. The sum of the rate coefficients for dissociative
recombination, R21 and R22, is according to the expression proposed by
Gudmundsson and Lieberman [91], which is adequate for electron temperatures
between 1 and 7 eV. The branching ratio 0.36:0.64 is based on the measurements
of [92], selecting the two main possible processes. Note that the value
adopted for the branching is not critical, since the typical contribution for
the production of $\mathrm{O(^{1}D)}$ is lower than $10^{-5}$. In the
conditions of this study, R21 and R22 have a small role on the destruction of
$\mathrm{O_{2}^{+}}$, lower than 2%.

Table 1: List of electron-impact processes considered in this work for vibrationally-cold oxygen. New processes compared to [21] are marked with **. Processes with rate coefficients updated in this work, with respect to [21], are marked with *. $\epsilon_{\mathrm{gh}}$ is the energy released for gas heating. Reaction |  |  Rate coefficient ($\mathrm{m^{3}s^{-1}}$) |  $\epsilon_{\mathrm{gh}}$ (eV) | Ref.
---|---|---|---|---
Electron-impact excitation |  |  |
R1 | $\mathrm{e+O_{2}(X,0)\leftrightarrow e+O_{2}(a)}$ | $f(E/N)$ | $-$ | [86, 85]
R2 | $\mathrm{e+O_{2}(X,0)\leftrightarrow e+O_{2}(b)}$ | $f(E/N)$ | $-$ | [86, 85]
R3 | $\mathrm{e+O_{2}(a)\leftrightarrow e+O_{2}(b)}$ | $f(E/N)$ | $-$ | [87, 85]
R4 | $\mathrm{e+O(^{3}P)\leftrightarrow e+O(^{1}D)}$ | $f(E/N)$ | $-$ | [88, 85]
R5∗∗ | $\mathrm{e+O_{2}(X,0)\leftrightarrow e+O_{2}(Hz)}$ | $f(E/N)$ | $-$ | [86, 85]
Electron-impact dissociation |  |  |
R6 | $\mathrm{e+O_{2}(X,0)\rightarrow e+2O(^{3}P)}$ | $f(E/N)$ | 0.84 | [86, 85]
R7 | $\mathrm{e+O_{2}(X,0)\rightarrow e+O(^{3}P)+O(^{1}D)}$ | $f(E/N)$ | 1.27 | [86, 85]
R8 | $\mathrm{e+O_{2}(a)\rightarrow e+2O(^{3}P)}$ | $f(E/N)$ | 0.84 | [87, 85]
R9 | $\mathrm{e+O_{2}(a)\rightarrow e+O(^{3}P)+O(^{1}D)}$ | $f(E/N)$ | 1.27 | [86, 85]
R10 | $\mathrm{e+O_{2}(b)\rightarrow e+2O(^{3}P)}$ | $f(E/N)$ | 0.84 | [87, 85]
R11 | $\mathrm{e+O_{2}(b)\rightarrow e+O(^{3}P)+O(^{1}D)}$ | $f(E/N)$ | 1.27 | [86, 85]
R12 | $\mathrm{e+O_{3}\rightarrow e+O(^{3}P)+O_{2}(X,0)}$ | $f(E/N)$ | 4.9 | [93, 85]
Electron-impact ionization |  |  |
R13 | $\mathrm{e+O_{2}(X,0)\rightarrow 2e+O_{2}^{+}}$ | $f(E/N)$ | $-$ | [86, 85]
R14 | $\mathrm{e+O_{2}(a)\rightarrow 2e+O_{2}^{+}}$ | $f(E/N)$ | $-$ | [93, 85]
R15 | $\mathrm{e+O(^{3}P)\rightarrow 2e+O^{+}}$ | $f(E/N)$ | $-$ | [88, 85]
R16 | $\mathrm{e+O^{-}\rightarrow 2e+O(^{3}P)}$ | $f(E/N)$ | $-$ | [89, 85]
Electron-impact dissociative ionization |  |  |
R17 | $\mathrm{e+O_{2}(X,0)\rightarrow 2e+O(^{3}P)+O^{+}}$ | $f(E/N)$ | $-$ | [93, 85]
R18 |  e + O2(a) $\rightarrow$ 2e + O(3P) + O+ | $f(E/N)$ | $-$ | [93, 85]
Electron-impact dissociative attachment |  |  |
R19 | $\mathrm{e+O_{2}(X,0)\rightarrow O^{-}+O(^{3}P)}$ | $f(E/N)$ | 1.27 | [86, 85]
R20 | $\mathrm{e+O_{2}(a)\rightarrow O^{-}+O(^{3}P)}$ | $f(E/N)$ | 1.27 | [94, 85]
Electron-impact dissociative recombination |  |  |
R21∗ | $\mathrm{e+O_{2}^{+}\rightarrow 2O(^{3}P)}$ | $0.36\times 2.2\times 10^{-14}(T_{e}[\mathrm{eV}])^{-0.5}$ | 6.98 | [91, 95, 92]
R22∗ | $\mathrm{e+O_{2}^{+}\rightarrow O(^{3}P)+O(^{1}D)}$ | $0.64\times 2.2\times 10^{-14}(T_{e}[\mathrm{eV}])^{-0.5}$ | 5.01 | [91, 95, 92]

####  2.3.2 Heavy-species processes for vibrationally-cold oxygen

Table 2 presents the heavy-species processes taken into account in the kinetic
scheme. The scheme is based on [21], with the following modifications.

  * -

The three-body quenching of $\mathrm{O_{2}(a)}$ (R24) proposed by Braginskiy
et al. [7] is now included in the kinetics, since it explains the behavior of
$\mathrm{O_{2}(a)}$ density at higher pressures, as shown in section 3.1.
Braginskiy et al. estimate a rate coefficient for this reaction in the range
$(1-3)\times 10^{-44}$ $\mathrm{m^{6}s^{-1}}$ by means of an extensive
comparison between model calculations and experimental measurements of
$\mathrm{O_{2}(a)}$ and $\mathrm{O(^{3}P)}$. In this work, we opt by the
highest value, $3\times 10^{-44}$ $\mathrm{m^{6}s^{-1}}$, since it provides a
remarkable agreement with experiment. Additionally, the probability of
$\mathrm{O_{2}(a)}$ deactivation at the wall (R70) is updated from $5\times
10^{-4}$ to $2.2\times 10^{-4}$, following the recent measurements by Booth et
al. [29].

  * -

The quenching of $\mathrm{O_{2}(b)}$ with $\mathrm{O(^{3}P)}$ (R28) is updated
according to the work by Booth et al. [30]. The coefficient is now composed by
the constant value proposed by Ionin et al. [27] together with the
$T_{\mathrm{g}}$-dependent reactive component proposed in [30]. The latter
component is based on a very complete comparison between modelling and
experiment of both average and radial profiles of $\mathrm{O_{2}(b)}$
densities, at 0.3–10 Torr gas pressure and 10–40 mA current. Furthermore,
reaction R29 is added with a coefficient consistent with its inverse reaction,
R43. As shown in section 3.2.1, this reaction starts to be important at higher
pressures. Lastly, the probability of $\mathrm{O_{2}(b)}$ deactivation at the
wall (R71) is updated from 0.02 to 0.135, according to the measurements by
Booth et al. [30].

  * -

The quenching kinetics (R30-R35) and the neutral transport (R72) of
$\mathrm{O_{2}(Hz)}$ are added in accordance with Vasiljeva et al.[84].

  * -

The three-body recombination of $\mathrm{O(^{3}P)}$ (R40) is now consistent
with Gordiets et al. [96]. However, this process is not very relevant for the
studied conditions. The $\mathrm{O(^{3}P)}$ wall recombination probability,
$\gamma_{\mathrm{O(^{3}P)}}$, strongly depends on the wall temperature, gas
pressure and discharge current. We take the values of
$\gamma_{\mathrm{O(^{3}P)}}$ measured for the corresponding conditions by
Booth et al. [29, 97], represented in figure 2. Note that the same group
measured $\gamma_{\mathrm{O(^{3}P)}}$ for the same conditions in a previous
work [28], but those values are systematically lower than in [29]. Work is in
progress to self-consistently calculate $\gamma_{\mathrm{O(^{3}P)}}$ by
coupling both the surface and volume kinetics [98]. This may help to
disentangle the reasons for the differences between the two sets of
experimental measurements.

  * -

Lastly, as referred in section 2.2.2, the loss of positive ions (R68 and R69)
is described by ambipolar diffusion, with a correction in the diffusion length
induced by the presence of the $\mathrm{O^{-}}$ ion, as described in [32]. The
loss of negative ions at the wall is not considered, as they are mostly
confined to the volume and virtually none reach the wall [99, 100, 32].

Table 2: List of heavy-species processes considered in this work for
vibrationally-cold oxygen. The decay frequencies are in $\mathrm{s^{-1}}$ and
the rate coefficients for two- and three-body reactions are in
$\mathrm{m^{3}s^{-1}}$ and $\mathrm{m^{6}s^{-1}}$, respectively. New processes
compared to [21] are marked with **. Processes with rate coefficients updated
in this work, with respect to [21], are marked with *. Processes whose rate
coefficients had a typo in [21] are marked with #. $\epsilon_{\mathrm{gh}}$ is
the energy released for gas heating and $f_{\mathrm{wall}}$ is the fraction of
energy released in wall reactions that returns to the volume.
$\mathrm{O_{2}(X)}$ includes all vibrational states of the ground-state of
molecular oxygen, $\mathrm{O_{2}(X,0:41)}$.

Reaction |  |  Rate coefficient ($\mathrm{s^{-1},m^{3}s^{-1},m^{6}s^{-1}}$) |  $\epsilon_{\mathrm{gh}}$ (eV) | Ref.
---|---|---|---|---
Neutral species collisions |  |  |
R23# | $\mathrm{O_{2}(a)+O(^{3}P)\rightarrow O_{2}(X,0)+O(^{3}P)}$ | $7\times 10^{-22}$ | 0.98 | [101, 32]
R24∗∗ | $\mathrm{O_{2}(a)+O(^{3}P)+O_{2}(X)\rightarrow O_{2}(X,0)+O(^{3}P)+O_{2}(X)}$ | $3\times 10^{-44}$ | 0.98 | [7]
R25 | $\mathrm{O_{2}(a)+O_{2}(a)\rightarrow O_{2}(b)+O_{2}(X,0)}$ | $1.81\times 10^{-24}\times(T_{\mathrm{g}}/300)^{3.8}\exp(700/T_{\mathrm{g}})$ | 0.33 | [102]
R26 | $\mathrm{O_{2}(a)+O_{2}(X)\rightarrow O_{2}(X,0)+O_{2}(X)}$ | $2.2\times 10^{-24}(T_{\mathrm{g}}/300)^{0.8}$ | 0.98 | [101]
R27 | $\mathrm{O_{2}(b)+O(^{3}P)\rightarrow O_{2}(a)+O(^{3}P)}$ | $4\times 10^{-20}$ | 0.65 | [27]
R28∗ | $\mathrm{O_{2}(b)+O(^{3}P)\rightarrow O_{2}(X,0)+O(^{3}P)}$ | $4\times 10^{-20}+10^{-16}\exp(-3700/T_{\mathrm{g}})$ | 1.63 | [30, 27]
R29∗∗ | $\mathrm{O_{2}(b)+O(^{3}P)\rightarrow O_{2}(X,0)+O(^{1}D)}$ | $3.39\times 10^{-17}\times(300/T_{\mathrm{g}})^{0.1}\exp(-4201/T_{\mathrm{g}})$ | -0.34 | [103]
R30∗∗ | $\mathrm{O_{2}(Hz)+O(^{3}P)\rightarrow O(^{3}P)+O_{2}(X,0)}$ | $4.95\times 10^{-18}$ | 4.5 | [84]
R31∗∗ | $\mathrm{O_{2}(Hz)+O(^{3}P)\rightarrow O(^{1}D)+O_{2}(a)}$ | $2.7\times 10^{-18}$ | 1.55 | [84]
R32∗∗ | $\mathrm{O_{2}(Hz)+O(^{3}P)\rightarrow O(^{1}D)+O_{2}(b)}$ | $1.35\times 10^{-18}$ | 0.9 | [84]
R33∗∗ | $\mathrm{O_{2}(Hz)+O_{2}(X)\rightarrow O_{2}(X,0)+O_{2}(X)}$ | $0.23\times 10^{-19}$ | 4.5 | [84]
R34∗∗ | $\mathrm{O_{2}(Hz)+O_{2}(X)\rightarrow O_{2}(a)+O_{2}(X)}$ | $1.86\times 10^{-19}$ | 3.52 | [84]
R35∗∗ | $\mathrm{O_{2}(Hz)+O_{2}(X)\rightarrow O_{2}(b)+O_{2}(X)}$ | $0.21\times 10^{-19}$ | 2.87 | [84]
R36 | $\mathrm{O(^{1}D)+O(^{3}P)\rightarrow 2O(^{3}P)}$ | $8\times 10^{-18}$ | 1.97 | [104]
R37 | $\mathrm{2O(^{3}P)+O_{2}(X)\rightarrow O_{2}(X,0)+O_{2}(X)}$ | $0.5\times 3.81\times 10^{-42}T_{\mathrm{g}}^{-1}\exp(-170/T_{\mathrm{g}})$ | 5.16 | [104, 105]
R38 | $\mathrm{2O(^{3}P)+O_{2}(X)\rightarrow O_{2}(a)+O_{2}(X)}$ | $0.33\times 3.81\times 10^{-42}T_{\mathrm{g}}^{-1}\exp(-170/T_{\mathrm{g}})$ | 4.18 | [104, 105]
R39 | $\mathrm{2O(^{3}P)+O_{2}(X)\rightarrow O_{2}(b)+O_{2}(X)}$ | $0.17\times 3.81\times 10^{-42}T_{\mathrm{g}}^{-1}\exp(-170/T_{\mathrm{g}})$ | 3.53 | [104, 105]
R40∗ | $\mathrm{3O(^{3}P)\rightarrow O_{2}(X,0)+O(^{3}P)}$ | $2.5\times 10^{-43}T_{\mathrm{g}}^{-0.63}$ | 5.16 | [96]
R41 | $\mathrm{O(^{1}D)+O_{2}(X)\rightarrow O(^{3}P)+O_{2}(X)}$ | $7\times 10^{-18}\exp(67/T_{\mathrm{g}})$ | 1.97 | [104]
R42∗ | $\mathrm{O(^{1}D)+O_{2}(X,0)\rightarrow O(^{3}P)+O_{2}(a)}$ | $8\times 10^{-19}\exp(67/T_{\mathrm{g}})$ | 0.99 | [106, 107]
R43 | $\mathrm{O(^{1}D)+O_{2}(X,0)\rightarrow O(^{3}P)+O_{2}(b)}$ | $2.56\times 10^{-17}\exp(67/T_{\mathrm{g}})$ | 0.34 | [104]
R44 | $\mathrm{O_{2}(b)+O_{3}\rightarrow 2O_{2}(X,0)+O(^{3}P)}$ | $1.5\times 10^{-17}$ | 0.53 | [104]
R45 | $\mathrm{O(^{1}D)+O_{3}\rightarrow 2O_{2}(X,0)}$ | $1.2\times 10^{-16}$ | 6.03 | [104]
R46 | $\mathrm{O(^{1}D)+O_{3}\rightarrow O_{2}(X,0)+2O(^{3}P)}$ | $1.2\times 10^{-16}$ | 0.87 | [104]
R47 | $\mathrm{O_{2}(a)+O_{3}\rightarrow 2O_{2}(X,0)+O(^{3}P)}$ | $5.2\times 10^{-17}\exp(-2840/T_{\mathrm{g}})$ | -0.12 | [104]
R48 | $\mathrm{O(^{3}P)+O_{3}\rightarrow 2O_{2}(X,0)}$ | $0.5\times 1.8\times 10^{-17}\exp(-2300/T_{\mathrm{g}})$ | 4.06 | [104, 105]
R49# | $\mathrm{O(^{3}P)+O_{3}\rightarrow O_{2}(a)+O_{2}(X,0)}$ | $0.33\times 1.8\times 10^{-17}\exp(-2300/T_{\mathrm{g}})$ | 3.08 | [104, 105]
R50# | $\mathrm{O(^{3}P)+O_{3}\rightarrow O_{2}(b)+O_{2}(X,0)}$ | $0.17\times 1.8\times 10^{-17}\exp(-2300/T_{\mathrm{g}})$ | 2.43 | [104, 105]
R51 | $\mathrm{O_{3}^{*}+O(^{3}P)\rightarrow O_{3}+O(^{3}P)}$ | $2\times 10^{-19}$ | 0.09 | [108]
R52 | $\mathrm{O_{3}^{*}+O_{2}(X)\rightarrow O_{3}+O_{2}(X)}$ | $3\times 10^{-21}$ | 0.09 | [108]
R53 | $\mathrm{O_{2}(a)+O_{3}^{*}\rightarrow 2O_{2}(X,0)+O(^{3}P)}$ | $5\times 5.2\times 10^{-17}\exp(-(2840-1553)/T_{\mathrm{g}})$ | -0.03 | [34]
R54 | $\mathrm{O(^{3}P)+O_{3}^{*}\rightarrow 2O_{2}(X,0)}$ | $8\times 10^{-18}\exp(-(2060-1553)/T_{\mathrm{g}})$ | 4.15 | [108]
R55 | $\mathrm{2O(^{3}P)+O_{2}(X,0)\rightarrow O_{3}+O(^{3}P)}$ | $2.1\times 10^{-46}\exp(345/T_{\mathrm{g}})$ | 1.1 | [104]
R56 | $\mathrm{O(^{3}P)+O_{2}(X,0)+O_{2}(X)\rightarrow O_{3}+O_{2}(X)}$ | $0.33\times 6.4\times 10^{-47}\exp(663/T_{\mathrm{g}})$ | 1.1 | [104, 108]
R57# | $\mathrm{O(^{3}P)+O_{2}(X,0)+O_{2}(X)\rightarrow O_{3}^{*}+O_{2}(X)}$ | $0.67\times 6.4\times 10^{-47}\exp(663/T_{\mathrm{g}})$ | 1.01 | [104, 108]
R58 | $\mathrm{O(^{3}P)+O_{2}(X,0)+O_{3}\rightarrow 2O_{3}}$ | $1.66\times 10^{-46}\exp(T_{\mathrm{g}}/300)$ | 1.1 | [109]
Ion collisions |  |  |
R59 | $\mathrm{O^{-}+O_{2}(X,0)\rightarrow O_{3}+e}$ | $10^{-18}$ | $-$ | [110]
R60 | $\mathrm{O^{-}+O_{2}(a)\rightarrow O_{3}+e}$ | $0.75\times 1.9\times 10^{-16}$ | $-$ | [104]
R61 | $\mathrm{O^{-}+O(^{3}P)\rightarrow O_{2}(X,0)+e}$ | $1.3\times 10^{-15}$ | $-$ | [110]
R62 | $\mathrm{O^{-}+O_{2}(b)\rightarrow O(^{3}P)+O_{2}(X,0)+e}$ | $6.9\times 10^{-16}$ | $-$ | [104]
R63 | $\mathrm{O^{+}+O^{-}\rightarrow 2O(^{3}P)}$ | $2.8\times 10^{-13}$ | 12.15 | [111]
R64 | $\mathrm{O^{+}+O_{3}\rightarrow O_{2}^{+}+O_{2}(X,0)}$ | $10^{-16}$ | 5.6 | [101]
R65 | $\mathrm{O^{+}+O_{2}(X,0)\rightarrow O_{2}^{+}+O(^{3}P)}$ | $2\times 10^{-17}(300/T_{\mathrm{g}})^{0.5}$ | 1.54 | [104]
R66 | $\mathrm{O^{+}+O_{2}(a)\rightarrow O_{2}^{+}+O(^{3}P)}$ | $2\times 10^{-17}(300/T_{\mathrm{g}})^{0.5}$ | 2.52 | [104, 21]
R67 | $\mathrm{O_{2}^{+}+O^{-}\rightarrow O_{2}(X,0)+O(^{3}P)}$ | $9.6\times 10^{-14}(300/T_{\mathrm{g}})^{0.5}$ | 10.61 | [104]
Ion transport |  |  |
R68 | $\mathrm{O_{2}^{+}+wall\rightarrow O_{2}(X,0)+wall}$ | Classic. ambip. diff. with neg. ions | $f_{\mathrm{wall}}\times 12.14$ | [32]
R69 | $\mathrm{O^{+}+wall\rightarrow O(^{3}P)+wall}$ | Classic. ambip. diff. with neg. ions | $f_{\mathrm{wall}}\times 13.68$ | [32]
Neutral transport |  |  |
R70∗ | $\mathrm{O_{2}(a)+wall\rightarrow O_{2}(X,0)+wall}$ | $\gamma_{\mathrm{O_{2}(a)}}=2.2\times 10^{-4}$ | $f_{\mathrm{wall}}\times 0.98$ | [29]
R71∗ | $\mathrm{O_{2}(b)+wall\rightarrow O_{2}(X,0)+wall}$ | $\gamma_{\mathrm{O_{2}(b)}}=0.135$ | $f_{\mathrm{wall}}\times 1.63$ | [30]
R72∗∗ | $\mathrm{O_{2}(Hz)+wall\rightarrow O_{2}(X,0)+wall}$ | $\gamma_{\mathrm{O_{2}(Hz)}}=1$ | $f_{\mathrm{wall}}\times 4.5$ | [84]
R73 | $\mathrm{O(^{3}P)+wall\rightarrow\frac{1}{2}O_{2}(X,0)+wall}$ | $\gamma_{\mathrm{O(^{3}P)}}=f(p,I,T_{w})$ | $f_{\mathrm{wall}}\times 2.58$ | [29]
R74 | $\mathrm{O(^{1}D)+wall\rightarrow O(^{3}P)+wall}$ | $\gamma_{\mathrm{O(^{1}D)}}=1$ | $f_{\mathrm{wall}}\times 1.97$ | [96]
R75 | $\mathrm{O_{3}^{*}+wall\rightarrow O_{3}+wall}$ | $\gamma_{\mathrm{O_{3}^{*}}}=0.1$ | $f_{\mathrm{wall}}\times 0.09$ | [108, 34]

Figure 2: Experimental values for the $\mathrm{O(^{3}P)}$ wall recombination
probability ($\gamma_{\mathrm{O(^{3}P)}}$) measured by Booth et al. [29, 97],
as a function of pressure and various discharge currents, for a wall
temperature $T\mathrm{w}=50$ oC.

####  2.3.3 Vibrational kinetics of $O_{2}(X,0:41)$

The kinetics of vibrationally-excited $\mathrm{O_{2}(X,0:41)}$ molecules is
very well detailed and validated in [21]. The vibrational kinetics contains
electron-impact (e-V), vibration-to-translation (V-T), vibration-to-vibration
(V-V) and wall de-excitation processes, as shown in table 3 and summarized in
the following paragraphs.

Table 3: List of e-V, V-T and V-V processes considered in this work to describe the vibrational kinetics of $\mathrm{O_{2}(X,v)}$. $\epsilon_{\mathrm{gh}}$ is the energy released for gas heating and $f_{\mathrm{wall}}$ is the fraction of energy released in wall reactions that returns to the volume. “$\sigma_{76}$ scaled from $\sigma_{1}$” means that the electron-scattering cross-section describing R76 ($\sigma_{76}$) is scaled from the cross-section describing R1 ($\sigma_{1}$). The same for the other cases. The scaling procedures are referred in the text of section 2.3.3. Reaction |  |  Rate coefficient ($\mathrm{s^{-1},m^{3}s^{-1}}$) |  $\epsilon_{\mathrm{gh}}$ (eV) | Ref.
---|---|---|---|---
Electron-impact excitation |  |  |
R76 | $\mathrm{e+O_{2}(X,1:6)\rightarrow e+O_{2}(a)}$ |  $f(E/N)$, $\sigma_{76}$ scaled from $\sigma_{1}$ | $-$ | [21, 85]
R77 | $\mathrm{e+O_{2}(X,1:8)\rightarrow e+O_{2}(b)}$ |  $f(E/N)$, $\sigma_{77}$ scaled from $\sigma_{2}$ | $-$ | [21, 85]
R78 | $\mathrm{e+O_{2}(X,v)\leftrightarrow e+O_{2}(X,w),\Delta v=v-w=1:30}$ | $f(E/N)$ | $-$ | [112, 85]
Electron-impact dissociation |  |  |
R79 | $\mathrm{e+O_{2}(X,1:41)\rightarrow e+O(^{3}P)+O(^{3}P)}$ |  $f(E/N)$, $\sigma_{79}$ scaled from $\sigma_{6}$ | 0.84 | [113, 21]
R80 | $\mathrm{e+O_{2}(X,1:41)\rightarrow e+O(^{3}P)+O(^{1}D)}$ |  $f(E/N)$, $\sigma_{80}=\sigma_{7}$ | 1.27 | [86, 21]
Electron-impact ionization |  |  |
R81 | $\mathrm{e+O_{2}(X,1:41)\rightarrow e+e+O_{2}^{+}}$ |  $f(E/N)$, $\sigma_{81}=\sigma_{13}$ | $-$ | [86, 21]
Electron-impact dissociative attachment |  |  |
R82 | $\mathrm{e+O_{2}(X,1:41)\rightarrow e+O^{-}+O(^{3}P)}$ |  $f(E/N)$, $\sigma_{82}$ scaled from $\sigma_{19}$ | 1.27 | [113, 21]
Vibration-to-translation relaxation |  |  |
R83 | $\mathrm{O_{2}(X,v)+O\leftrightarrow O_{2}(X,w)+O}$ | $f(\mathrm{v,w},T_{\mathrm{g}})$ | $E_{\mathrm{v}}-E_{\mathrm{w}}$ | [114]
R84 | $\mathrm{O_{2}(X,v)+O_{2}(X)\leftrightarrow O_{2}(X,v-1)+O_{2}(X)}$ | $f(\mathrm{v},T_{\mathrm{g}})$ | $E_{\mathrm{v}}-E_{\mathrm{v-1}}$ | [115, 21]
Vibration-to-vibration transfer |  |  |
R85 | $\mathrm{O_{2}(X,v)+O_{2}(X,w-1)\leftrightarrow O_{2}(X,v-1)+O_{2}(X,w)}$ | $f(\mathrm{v,w},T_{\mathrm{g}})$ | $E_{\mathrm{v}}+E_{\mathrm{w-1}}-E_{\mathrm{v-1}}-E_{\mathrm{w}}$ | [115, 21]
Wall de-excitation |  |  |
R86 | $\mathrm{O_{2}(X,v)+wall\rightarrow O_{2}(X,v-1)+wall}$ | $\gamma_{\mathrm{v}}=1.1\times 10^{-3}$ | $f_{\mathrm{wall}}\times(E_{\mathrm{v}}-E_{\mathrm{v-1}})$ | [116, 21]

The electron-impact cross-sections for the excitations of $\mathrm{O_{2}(a)}$
and $\mathrm{O_{2}(b)}$ from $\mathrm{O_{2}(X,v>0)}$, R76 and R77, were
calculated by Annušova et al. [21], using a scaling of the
$\mathrm{O_{2}(X,0)}$ cross section according to the corresponding Franck-
Condon factors, similarly to what was done by Loureiro et al. [117] in
nitrogen discharges. The electron-impact cross-sections for vibrational
excitation (R78) are taken from the work by Laporta et al. [112], where the
$R$-matrix method was used to calculate resonant interactions between the 42
bound v-levels. The cross sections for electron-impact dissociation leading to
$\mathrm{O(^{3}P)}$ (R79) and for dissociative attachment (R82) are scaled
from the vibrational ground state to higher levels [21], according to the
trends of the theoretical calculations made by Laporta et al. [113]. The cross
sections for electron-impact dissociation leading to $\mathrm{O(^{1}D)}$ (R80)
and ionization (R81) are assumed to be the same for all vibrational levels and
equal to the ones used for $\mathrm{O_{2}(X,0)}$.

The rate coefficients for V-T $\mathrm{O_{2}-O}$ processes (R83) are obtained
from the quasiclassical trajectory calculations of Esposito et al. [114].
These computations were fitted to a formula depending on several coefficients,
allowing to easily calculate the rate coefficients for multi-quanta
transitions $\mathrm{\Delta v=1:30}$ in the temperature range
$T_{\mathrm{g}}=50-10000$ K [114].

The V-T and V-V $\mathrm{O_{2}-O_{2}}$ rate coefficients were computed using
the forced harmonic oscillator (FHO) model [118], initially performed by Lino
da Silva et al. [115] and later revisited by Annušova et al. [21]. In the
latter work, the FHO calculations were redone for 42 bound vibrational levels,
instead of 46. Moreover, the rates for vibrational transitions whose energy
difference is comparable with the gas thermal energy were corrected, so as to
compensate the limitations of the FHO approach in those conditions [115]. The
final V-T and V-V rate-coefficients were fitted as a function of the
vibrational quantum numbers and $T_{\mathrm{g}}$, applicable in the
temperature range $T_{\mathrm{g}}=300-1000$ K [21].

Finally, the vibrational kinetics includes wall de-excitation of
$\mathrm{O_{2}(X,1:41)}$, assuming a deactivation probability equal to the one
measured for nitrogen [116], due to lack of information concerning this
process in oxygen.

###  2.4 Transport and thermodynamic data

In addition to a comprehensive description of the reaction mechanism, detailed
knowledge of physical parameters, e.g. mobility, diffusion coefficient,
thermal conductivity and enthalpy, is required to obtain meaningful modelling
results.

The transport of neutral species to the wall (cf. section 2.2.2) requires the
species diffusion coefficients, $D_{n}$. In this work, $D_{n}$ is given by
Wilke’s formula for multicomponent mixtures [119]:

$D_{n}=\frac{1-x_{n}}{\sum\limits_{\forall l\neq n}x_{l}/D_{n,l}}\ ,$ (8)

where $D_{n,l}$ is the binary diffusion coefficient of species $n$ in species
$l$ and $x_{n}=N_{n}/N$ is the relative concentration (molar fraction) of
species $n$. The binary diffusion coefficients $D_{n,l}$ can be calculated as
in Hirschfelder et al. [120] (section 8.2, equation 8.2-44) and Guerra et
al.[41] (section 4.1):

$D_{n,l}[\mathrm{m^{2}s^{-1}}]=\frac{1.929\times
10^{21}\sqrt{T_{\mathrm{g}}/(2\mu_{nl})}}{N\sigma_{nl}^{2}[\mathrm{\AA^{2}}]\Omega^{(1,1)*}k_{\mathrm{B}}T_{\mathrm{g}}/\epsilon_{nl}}\
,$ (9)

where $\mu_{nl}$ is the reduced mass, $T_{\mathrm{g}}$ is the gas temperature,
$\sigma_{nl}=(\sigma_{n}+\sigma_{l})/2$ and
$\epsilon_{nl}=\sqrt{\epsilon_{n}\epsilon_{l}}$ are the Lennard-Jones binary
interaction potential parameters, and the collision integral $\Omega^{(1,1)*}$
is tabulated in [120]. The Lennard-Jones parameters, $\epsilon_{n}$ and
$\sigma_{n}$, of a species $n$ are assumed to be independent of the electronic
and vibrational state, and the corresponding values for O, O2 and O3 species
are taken from [121] and presented in table 4.

The mobility and diffusion coefficients of charged species are necessary for
the calculation of ambipolar transport [32]. The ion reduced mobilities,
$\mu_{i}N$, of $\mathrm{O_{2}^{+}}$[122], $\mathrm{O^{+}}$[104] and
$\mathrm{O^{-}}$[122] are taken from experimental values reported in the
literature, for background O2 molecules, with a dependence on $E/N$, as shown
in figure 4. The reduced mobility of $\mathrm{O^{+}}$ is not available for
$E/N<60$ Td, and in this interval is taken with the value at $E/N=60$ Td. The
ion reduced diffusion coefficients, $D_{i}N$, are deduced from the Einstein
relation, $D_{i}/\mu_{i}=k_{B}T_{i}$, with the ion temperature $T_{i}$ assumed
equal to the gas temperature $T_{\mathrm{g}}$. The electron mobility and
diffusion coefficients are taken from the solution of the electron Boltzmann
equation calculated by LoKI-B.

The heat capacities and the thermal conductivities of oxygen species are
needed for the self-consistent calculation of the gas temperature. In turn,
these quantities have an important dependence with gas temperature, as
evidenced in figure 4. The heat capacity of $\mathrm{O_{2}}$ at constant
pressure $(C_{p,\mathrm{O_{2}}}$) is taken from [76, 123]. Assuming atomic
oxygen as an ideal gas, the heat capacity has a constant value
$C_{p,\mathrm{O}}=5/2k_{B}$. The heat capacity of $\mathrm{O_{3}}$ is assumed
to have a negligible impact on the results, due to the low $\mathrm{O_{3}}$
concentration in the conditions under study. Therefore, the heat capacity of
the mixture is estimated using
$C_{p}=(1-\frac{[O]}{N})C_{p,\mathrm{O_{2}}}+\frac{[O]}{N}C_{p,\mathrm{O}}$,
where $\frac{[O]}{N}$ is the O-atom density fraction. The thermal
conductivities ($\lambda$) of $\mathrm{O_{2}}$ and O are taken from [76, 124]
and [125], respectively. The thermal conductivity of the multicomponent
mixture is calculated as presented by Mason and Saxena [126]. Note that for
the steady-state calculations presented in this work, the model results are
sensitive to the thermal conductivity but not to the heat capacity, as can be
understood from equation (4).

Lastly, the enthalpies of the oxygen species are necessary to calculate the
energies released for gas heating ($\epsilon_{\mathrm{gh}}$) in the various
reactions. The values of $\epsilon_{\mathrm{gh}}$ are presented in the third
columns of tables 1, 2 and 3. The enthalpies of formation for the species
considered in this work are presented in table 5. Note that the energies of
the $\mathrm{O_{2}(X,v)}$ vibrational levels are taken from Laporta et al.
[112] and that the $\mathrm{O_{3}^{*}}$ enthalpy corresponds to the first
vibrational level of ozone [108, 34]. For reactions between heavy particles,
$\epsilon_{\mathrm{gh}}$ is given by the difference between the summed
enthalpies of formation for the reactants and the products. For reactions
occurring at the wall, the procedure is similar but the energy release is
pondered by the energy fraction transferred to the volume
($f_{\mathrm{wall}}=0.5$)[76]. Concerning electron-impact processes, only the
ones involving dissociation contribute to gas heating. In this case,
$\epsilon_{\mathrm{gh}}$ is calculated as the difference between the enthalpy
of formation of the dissociative state and the summed enthalpies of the
dissociation products. The enthalpy of formation of the dissociative state is
taken as the sum of the heavy-target enthalpy and the energy threshold of the
electron scattering cross section for the corresponding process.

Table 4: Lennard-Jones parameters ($\epsilon_{n}$ and $\sigma_{n}$) used in this work for oxygen species, taken from [121]. Species |  $\epsilon_{n}/k_{\mathrm{B}}$ (K) | $\sigma_{n}(\mathrm{\AA})$
---|---|---
O | 80.0 | 2.75
O2 | 107.4 | 3.458
O3 | 180.0 | 4.1

Figure 3: Ion reduced mobilities, $\mu_{i}N$, as a function of $E/N$. See text
for references.

Figure 4: Heat capacities at constant pressure ($C_{p}$, left) and thermal
conductivities ($\lambda$, right) of $\mathrm{O_{2}}$ and O. See text for
references.

Table 5: Enthalpies of the states considered in this work. Species | Enthalpy (eV) | Ref. | Species | Enthalpy (eV) | Ref.
---|---|---|---|---|---
$\mathrm{O_{2}(X,v)}$ | $E_{\mathrm{v}}$ | [112] | $\mathrm{O_{3}}$ | 1.48 | [104]
$\mathrm{O_{2}(a)}$ | 0.98 | [104] | $\mathrm{O_{3}^{*}}$ | 1.57 | [104, 108, 34]
$\mathrm{O_{2}(b)}$ | 1.63 | [104] | $\mathrm{O_{2}^{+}}$ | 12.14 | [104]
$\mathrm{O_{2}(Hz)}$ | 4.5 | [84] | $\mathrm{O^{+}}$ | 16.26 | [104]
$\mathrm{O(^{3}P)}$ | 2.58 | [104] | $\mathrm{O^{-}}$ | 1.05 | [104]
$\mathrm{O(^{1}D)}$ | 4.55 | [104] |  |  |

##  3 Results

###  3.1 Validation

In this section, we present the results that allow to confirm the reaction
mechanism for $\mathrm{O_{2}}$ plasmas, by comparison of the model predictions
with the experiment for gas pressures between 0.2 and 10 Torr. We start by
validating the self-consistent reduced electric field and gas temperature
(average values and radial profile), and then focus on the comparison of the
main species densities. Since the effect of gas pressure variation is rather
similar for all discharge currents, we present the results for 30 mA, where
more experimental data are available. However, we also study the impact of
discharge current on the results, for gas pressures of 0.5 and 3 Torr.

####  3.1.1 Reduced electric field and gas temperature

Figure 5 shows the comparison between modelling and experimental [28, 97]
values of the reduced electric field $E/N$. The model captures rather well the
tendency with pressure and predicts values in agreement with those found in
experiment (see figure 5). However, for pressures below 1 Torr, the model
overestimates $E/N$, with an increasing effect with decreasing pressure. This
might be related with the approximations made in assuming ambipolar diffusion,
i.e. high collisionality regime, which start to fail in this pressure range.
An analysis of the influence on the results of adopting different charged-
particle transport models is presented in [75]. Moreover, when comparing to
model results considering classical ambipolar diffusion (i.e. without the
influence of negative ions), we find that the $\mathrm{O^{-}}$ ions lead to a
small decrease in $E/N$, associated to a decrease in the loss of positive ions
at the wall. The correction induced by $\mathrm{O^{-}}$ ions is proportional
to its content relatively to the electron density, as can be verified by
inspection of figure 6. For this reason, the difference between the models is
more noticeable for pressures below 0.5 Torr and above 2 Torr. From figure 5,
we find that the model predicts a small decrease in $E/N$ with increasing
current, consistent with what is observed in experiment. A larger current /
electron density leads to higher densities of reactive species, such as
$\mathrm{O(^{3}P)}$ and $\mathrm{O_{2}(a)}$. The presence of these species
causes an increase in the high-energy tail of the EEDF, associated with weaker
inelastic collisions and stronger superelastic collisions, due to
$\mathrm{O(^{3}P)}$ and $\mathrm{O_{2}(a)}$, respectively. Therefore, for a
given $E/N$, the ionization coefficient is higher in both cases, originating a
lower maintenance electric field with increasing current.

The average values of gas temperature ($T_{\mathrm{g}}$) are compared against
experiment [28, 97] in figure 7. In figure 7, we see that a convection
coefficient between the gas and the wall ($h_{\mathrm{gas-wall}}$) of 100
$\mathrm{W/(m^{2}K)}$ provides a remarkable agreement with experiment. For
this reason, we adopt this value in all simulations. Note that if we use
$h_{\mathrm{gas-wall}}=30\ \mathrm{W/(m^{2}K)}$, the model visibly
overestimates $T_{\mathrm{g}}$, whereas if we use $h_{\mathrm{gas-
wall}}=\infty\Leftrightarrow T_{\mathrm{nw}}=T_{\mathrm{w}}$ (see also
equation 7), the model underestimates the $T_{\mathrm{g}}$ values for
pressures higher than 1 Torr, albeit the correct trend is captured in all
cases. In figure 7, $T_{\mathrm{g}}$ is plotted against the discharge power
per unit length ($E\cdot I$). This figure evidences that $T_{\mathrm{g}}$ is
nearly proportional to $E\cdot I$. At high $E\cdot I$, the deviation of model
results of $T_{\mathrm{g}}(E\cdot I)$ from the experimental trend apparent in
the figure is mainly caused by the slight overestimation of $E/N$ values
predicted by the model at high pressures, which influences directly the values
in the horizontal axis, $E\cdot I$. Nevertheless, we should underline that for
each set $(p,I)$ the agreement between the model predictions for
$T_{\mathrm{g}}$ and experiment is very good.

Note that, although the model is 0D, i.e. spatially averaged, it is possible
to reconstruct the radial profile assumed in equation (3). The calculated
profile of gas temperature is compared against experiment for a discharge
current of 30 mA and different pressures, in figure 8. Despite the significant
approximations made in the calculation of gas temperature (cf. section 2.2.2),
the accordance between the model and the measurements is quite good, and it
further confirms the choice for the value of the convection coefficient.
Notice that the radial profiles from the model are always slightly higher than
in the experiment, while the average values of gas temperature reported in
figure 7 are not. This seemingly contradictory result may stem from the
different measuring set-ups used for the determination of these temperatures.
While the radially-averaged measurements are performed across the tube, the
radially-resolved measurements are line-of-sight (for more details, see figure
1 and section 2.3 of [28]). Therefore, the experimental data reported in
figure 7 are obtained independently from the measurements shown in figure 7,
and the former are not a simple radial average of the latter (albeit not far
from that).

(a)

(b)

Figure 5: Comparison between modelling and experimental[28, 97] values of the
reduced electric field as a function of (a) the gas pressure, for a discharge
current of 30 mA, and (b) the discharge current, for gas pressures of 0.5 and
3 Torr. In figure (a), the electric field is also plotted as a function of
pressure. Figure 6: Ion densities relative to the electron density as a
function of gas pressure, for a discharge current of 30 mA.

(a)

(b)

Figure 7: Comparison between modelling and experimental[28, 97] results of the
average gas temperature ($T_{\mathrm{g}}$) as a function of (a) the gas
pressure, for a discharge current of 30 mA, and (b) the discharge power per
unit length ($E\cdot I$), for all pressures and 10-40 mA discharge currents.
Figure 8: Comparison between modelling and experimental[28] values of the
radial profile of gas temperature ($T(r)$) for different pressures and a
discharge current of 30 mA. The squares and diamonds are experimental
measurements from $\mathrm{O_{2}(b)}$ emission and TALIF, respectively[28].

####  3.1.2 Main species

The fractions of species considered in the model (defined for species $i$ as
$[i]/N$) are summarized in figure 9 as a function of gas pressure, for a
discharge current of 30 mA. In this section we focus on the validation of the
main species represented in figure 9: $\mathrm{O_{2}(X)}$,
$\mathrm{O_{2}(a)}$, $\mathrm{O_{2}(b)}$ and $\mathrm{O(^{3}P)}$.

The model calculations of the $\mathrm{O(^{3}P)}$ density and fraction are
compared with experiment in figure 10 for a discharge current of 30 mA. Take
note that the vacuum ultraviolet (VUV) [29] and actinometry [28] measurements
are radially averaged, whereas the cavity ringdown spectroscopy (CRDS) [30]
measurements are performed at the axis center. There is good accordance
between the model and the experiment. Nevertheless, we should emphasize that
the success in reproducing the experimental results is very much connected
with the values used for the recombination probability
$\gamma_{\mathrm{O(^{3}P)}}$, concerning reaction R73 of table 2, taken from
experiment [29] and represented in figure 2. For example, if we use the lower
values of $\gamma_{\mathrm{O(^{3}P)}}$ from [28] instead of [29], measured in
similar conditions, the calculated $\mathrm{[O(^{3}P)]/N}$ has a similar shape
but exhibits a maximum value at 1 Torr twice as large as shown here. This
shows that the surface conditions in these discharges have poor
reproducibility, and that further investigation is required from both
modelling and experimental points of view to accurately address this
phenomenon. The scattering between the oxygen density measurements obtained
with actinometry [28], VUV [29] and CRDS [30] is another evidence for the low
reproducibility of surface conditions and the inherent uncertainties in the
experimental determination of the atomic oxygen concentration, more noticeable
at low pressures. Furthermore, it makes clear that the validation of the model
should be established more from the prediction of the correct dependences and
orders of magnitude than by an exact agreement on the absolute values.

Figure 11 shows very good agreement between the modelling values and the VUV
experimental measurements of $\mathrm{O_{2}(X)}$ density. Similarly to
$\mathrm{O(^{3}P)}$, the model results are strongly correlated with
$\gamma_{\mathrm{O(^{3}P)}}$, since $\mathrm{O_{2}(X)}$ and
$\mathrm{O(^{3}P)}$ are the major species in the plasma (cf. figure 9).

The model densities of $\mathrm{O_{2}(a)}$ and $\mathrm{O_{2}(b)}$ are
compared with experiment in figure 12. For $\mathrm{O_{2}(a)}$, two sets of
measurements from [29] are shown: VUV and optical emission spectroscopy (OES).
The three-body quenching proposed by Braginskiy et al.[7] is fundamental to
obtain a good agreement in trend and magnitude with experimental values, as
can be seen by observing the results obtained when removing reaction R24. For
$\mathrm{O_{2}(b)}$, one set of OES measurements is shown but with two
different calibration/normalization factors [30]: absolute OES calibration and
scaling from VUV measurements. The deviation of 60% between the two
calibration factors gives an idea of the error associated with the absolute
value of the measurement. Notice that, in this case, the reactive quenching
proposed by Booth et al.[30] is crucial to predict the density saturation
after 3 Torr.

The effect of discharge current on the main species densities is summarized in
figure 13, for gas pressures of 0.5 and 3 Torr. As expected, an increase in
current / electron density leads to a larger dissociation - more
$\mathrm{[O(^{3}P)}]$ and less $\mathrm{[O_{2}(X)]}$ \- and to higher
densities of the metastables $\mathrm{O_{2}(a)}$ and $\mathrm{O_{2}(b)}$.
Besides, the model captures very well the tendencies and the absolute values
found in experiment.

(a)

(b)

Figure 9: Overview of the species fractions, $[i]/N$, in the discharge as a
function of gas pressure, for a current of 30 mA.

(a)

(b)

Figure 10: Comparison between modelling and experimental values of the
$\mathrm{O(^{3}P)}$ (a) density and (b) fraction, as a function of gas
pressure and for a current of 30 mA. The crosses, circles and squares
represent measurements using VUV[29], actinometry[28] and CRDS[30],
respectively. Figure 11: Comparison between modelling and VUV experimental
values [29] of the $\mathrm{O_{2}(X)}$ density.

(a)

(b)

Figure 12: Comparison between modelling and experimental[29, 30] values of the
(a) $\mathrm{O_{2}(a)}$ and (b) $\mathrm{O_{2}(b)}$ densities, as a function
of gas pressure, for a discharge current of 30 mA.

(a)

(b)

(c)

(d)

Figure 13: Comparison between modelling and experimental values of the main
species densities ($\mathrm{O(^{3}P)}$, $\mathrm{O_{2}(X)}$,
$\mathrm{O_{2}(a)}$ and $\mathrm{O_{2}(b)}$), as a function of discharge
current, for gas pressures of 0.5 and 3 Torr. Detailed references for the
experimental measurements [28, 29, 30] are mentioned in figures 10-12 and the
text.

###  3.2 Main processes

Benefiting from the reaction mechanism of oxygen discharges validated in
section 3.1, we can identify the main processes controlling the kinetic
behavior of the system. Here, we highlight the most important reactions for
the production/destruction of the main species and the dominant sources for
gas heating.

####  3.2.1 Species production/destruction

The main processes for the production/destruction of neutral atomic oxygen,
$\mathrm{O_{2}(X)}$, $\mathrm{O_{2}(a)}$ and $\mathrm{O_{2}(b)}$ are
represented in figure 14. The species $\mathrm{O(^{3}P)}$ and
$\mathrm{O(^{1}D)}$ are considered together in this analysis, since there are
strong excitation and quenching processes between them that do not lead to an
effective destruction or production of O atoms, merely redistributing the
population of these two states [21]. Similarly, the analysis for
$\mathrm{O_{2}(X)}$ joins the processes due to all vibrational states,
eliminating reactions that only lead to vibrational exchanges. In processes
with double arrow ($\leftrightarrow$), the contribution reflects the net value
of the direct and reverse components. On a general note, we see that the
electron-impact processes and the wall de-excitation/recombination play a key
role in the kinetics of all species.

In figure 14, we observe that the atomic oxygen is mostly created by electron
impact with $\mathrm{O_{2}(X)}$ and $\mathrm{O_{2}(a)}$ (R6-R9), and it is
mainly destroyed by $\mathrm{O(^{3}P)}$ recombination at the wall (R73). For
higher pressures, three-body recombination processes start to be important,
having an impact of $\sim 20\%$ on the $\mathrm{O}$ atom destruction at 10
Torr.

Figure 14 evidences that $\mathrm{O_{2}(X)}$ production is mostly controlled
by $\mathrm{O(^{3}P)}$ recombination at the wall (R73), quenchings of
$\mathrm{O_{2}(a)}$ (R24), $\mathrm{O_{2}(b)}$ (R28) and $\mathrm{O_{2}(Hz)}$
(R30), and de-excitation of $\mathrm{O_{2}(b)}$ at the wall (R71). The main
processes for $\mathrm{O_{2}(X)}$ destruction are electron-impact excitations
(R1, R2 and R5), electron-impact dissociations (R6 and R7) and
$\mathrm{O(^{1}D)}$ quenching (R43). Note that, for the conditions explored in
this work, the inflow and outflow processes do not play a major role, having
an influence in the production/destruction of $\mathrm{O_{2}(X)}$ of less than
8% for all conditions.

As seen in figure 14, the main production of $\mathrm{O_{2}(a)}$ is given by
electron-impact excitation from the ground state (R1) and quenching of
$\mathrm{O_{2}(b)}$ with $\mathrm{O(^{3}P)}$ (R27). The quenching of
$\mathrm{O_{2}(Hz)}$ through R31 and R34 is also relevant, with a summed
contribution of up to 15%. The $\mathrm{O_{2}(a)}$ destruction is controlled
by electron-impact processes (R3, R8 and R9), two- and three- body quenchings
(R23 and R24), and de-excitation at the wall (R70). Reaction R24 is
fundamental at high pressures, as shown also in figure 12, contributing more
than 50% for the destruction at 10 Torr.

Finally, in figure 14, we find that the $\mathrm{O_{2}(b)}$ creation is mostly
associated with electron-impact excitations (R2 and R3) and specially with the
quenching of $\mathrm{O(^{1}D)}$ (R43). For low pressures, the destruction
occurs predominantly through wall deactivation (R71), and for higher
pressures, through quenching with $\mathrm{O(^{3}P)}$ (R27 - R29).

None of the important reactions presented in figure 14 involve vibrationally-
excited states of $\mathrm{O_{2}(X)}$, although they are considered in the
calculations. This may be understood by looking at the vibrational
distribution function (VDF) of $\mathrm{O_{2}(X,v})$, represented in figure 15
for different pressures and a current of 30 mA. The VDFs are in rather good
agreement with values measured under somewhat similar conditions, see figure 9
of [41]. For all conditions, the relative populations of $\mathrm{v=1}$ and 2
are around $10^{-2}$ and $10^{-3}$, respectively. Therefore, the influence of
vibrationally-excited levels on the chemistry is very limited, at least under
the conditions studied in this work. However, the inclusion of vibrational
kinetics is fundamental for an accurate self-consistent calculation of the gas
temperature due to its relevance in the gas heating, as shown in the next
section.

(a)

(b)

(c)

(d)

Figure 14: Main processes for the creation (full circles) and destruction
(full triangles) of (a) atomic oxygen, (b) $\mathrm{O_{2}(X)}$, (c)
$\mathrm{O_{2}(a)}$ and (d) $\mathrm{O_{2}(b)}$, as a function of pressure,
for a current of 30 mA. In the legends, “wall” is abbreviated as “w”. Figure
15: VDF of $\mathrm{O_{2}(X,v)}$ for different pressures and a current of 30
mA. Figure 16: Relative contributions, $Q_{j}/Q_{\mathrm{in}}$, of the most
relevant kinetic processes for gas heating, as a function of pressure for a
current of 30 mA.

####  3.2.2 Gas heating

The relative contributions, $Q_{j}/Q_{\mathrm{in}}$ (cf. equation 4), of the
most relevant kinetic processes for gas heating, are shown in figure 16. As
discussed in section 2.2.2, each source term $Q_{j}$ is calculated by
multiplying the corresponding reaction rate by its enthalpy
($\epsilon_{\mathrm{gh}}$). Although the gas heating due to electron-neutral
elastic collisions is accounted for the calculation of $Q_{\mathrm{in}}$, its
relative contribution is always below $3\times 10^{-3}$. In the entire
pressure range, the dominant heating process is given by $\mathrm{O(^{3}P)}$
recombination on the wall. Recall that for processes occurring at the wall, we
assume that half of the energy released goes to the plasma volume
($f_{\mathrm{wall}}=0.5$), the rest being dissipated to the wall (cf. section
2.2.2). Therefore, this assumption has a significant influence on the results.
For example, for a current of 30 mA and a gas pressure of 1 Torr, the gas
temperature is 358, 378 and 396 K, for values of $f_{\mathrm{wall}}=0$, 0.5
and 1, respectively; for 30 mA and 10 Torr, $T_{\mathrm{g}}$ is 514, 540 and
565 K, for the same values of $f_{\mathrm{wall}}$.

The V-T transfers by collisions with O atoms (R83) also play a vital role in
gas heating, with a relative contribution of $\sim 17\%$. For this reason,
when the gas temperature is calculated self-consistently in an
$\mathrm{O_{2}}$ discharge model, the kinetics of vibrationally-excited states
$\mathrm{O_{2}(X,v)}$ should be properly accounted.

Finally, the quenchings of $\mathrm{O_{2}(a)}$ (R24), $\mathrm{O_{2}(b)}$
(R28) and $\mathrm{O_{2}(Hz)}$ (R30 and R34) also make a very relevant
contribution to heating, which grows with pressure, and the three-body
recombination of $\mathrm{O(^{3}P)}$ (R37) starts to have a small influence at
high pressure.

##  4 Conclusions

In this work, we present a reaction mechanism for oxygen plasmas. The reaction
mechanism was validated by comparing available experimental values [28, 29,
30] against modelling calculations of the reduced electric field, the gas
temperature and the densities of the predominant species in the glow discharge
- $\mathrm{O_{2}(X)}$, $\mathrm{O_{2}(a)}$, $\mathrm{O_{2}(b)}$ and
$\mathrm{O(^{3}P)}$ \- at gas pressures between 0.2 and 10 Torr. Moreover, we
have verified that the present oxygen reaction mechanism still describes
correctly our previous works concerning the kinetics of ground-state and
vibrationally-excited ozone [34] and vibrationally-excited
$\mathrm{O_{2}(X,v)}$ [21].

A detailed study of the main processes for species creation/destruction
quantified the relative importance of electron-impact excitation/dissociation,
wall de-excitation/recombination and two- and three- body quenching processes.
The three-body quenching (R24, $\mathrm{O_{2}(a)+O(^{3}P)+O_{2}(X)\rightarrow
O_{2}(X,0)+O(^{3}P)+O_{2}(X)}$) proposed by Braginskiy et al.[7] is
instrumental to reproduce the correct behavior of $\mathrm{O_{2}(a)}$ at high
pressure, while electron-impact processes ($\mathrm{e+O_{2}(X)\leftrightarrow
e+O_{2}(a)\leftrightarrow e+O_{2}(b)}$) are prevalent at low-pressures. The
non-reactive and reactive quenching of $\mathrm{O_{2}(b)}$ (R28,
$\mathrm{O_{2}(b)+O(^{3}P)\rightarrow O_{2}(X,0)+O(^{3}P)}$) [30] allows
obtaining the saturation of its density with increasing pressure; deactivation
at the wall is the main destruction process at low pressure; and reaction R43
involving $\mathrm{O(^{1}D)}$ constitutes always an important creation
mechanism. The kinetics of $\mathrm{O(^{3}P)}$ is strongly controlled by wall
recombination, and its density is highly dependent on the variation of the
recombination probability with the wall temperature, gas pressure and
discharge current.

The gas temperature plays a key role in the modelling, since many rate
coefficients are temperature-dependent. The calculation of the gas temperature
can be done with a self-consistent thermal model accounting for the energy
released in volume and wall reactions, the heat losses due to thermal
conduction and the gas-to-wall convection transfer. It can also adopt
alternative (simpler) approaches to introduce a temperature jump from the gas
to wall. In any case, $T_{\mathrm{g}}$ should be experimentally monitored and
model predictions for $T_{\mathrm{g}}$ should agree satisfactorily with the
measurements. In the present conditions, the $\mathrm{O(^{3}P})$ wall
recombination, the V-T transfers to O atoms and the quenching of
$\mathrm{O_{2}}$ electronic states, including the Herzberg states
$\mathrm{O_{2}(A^{\prime
3}\Delta_{u},A^{3}\Sigma_{u}^{+},c^{1}\Sigma_{u}^{-})}$, are the main
contributors for the gas heating.

We believe that the low-temperature plasma community should make an effort to
adopt a paradigm in the development of kinetic schemes to study plasma
chemistry, assuming the same rigour and systematization in validation devoted
to the development of complete and consistent electron-impact cross section
sets, and embracing the good practices from other communities. In particular,
each kinetic scheme should be compared against dedicated benchmark
experiments. In addition, an experimental effort to generate and organize such
experimental data would be welcome. In the case of oxygen discharges, the
systematic measurements from LPP and MSU [28, 29, 30] constitute a very
complete experimental data set and can be used as a validation test for other
models. However, more experimental data from various independent researchers
is necessary to evaluate possible problems of reproducibility or systematic
errors and to extend the conditions for validation.

It should be noted that the validation of a kinetic scheme by trying to put
all modelling results on top of the experimental values risks to be
meaningless. On the one hand, experimental measurements have uncertainties
associated, which in some cases are difficult to quantify. On the other hand,
model equations comprise various approximations. For example, in 0D models,
the assumption of certain density profiles to describe the transport of
neutral and charged species allows to obtain the correct trends and identify
the main elementary processes ruling the discharge, but sometimes the absolute
values are somewhat inaccurate [67]. For these reasons, it is more relevant to
ensure that the kinetic scheme is physically consistent and describes the
observed tendencies of several quantities in a wide range of conditions.

Future work will include analysis on different pressure ranges and other
discharge configurations. By extending the domain of validity of the reaction
mechanism, as distinct dominant elementary processes may be at play with rate
coefficients for different temperature ranges, the modelling results will
necessarily become more predictive and will involve the use of fewer tuning
parameters. Moreover, a surface kinetics model will soon be coupled to the
volume kinetics, in order to obtain an extended self-consistent model that
does not rely on experimental data for the recombination probabilities. This
will allow for a deeper understanding of the experimental results and which
physical parameters are controlling the surface reactions. Finally, the impact
of the two-term approximation used in LoKI-B on the plasma chemistry results
will be quantified, by comparison with the results obtained when using the
electron kinetics Monte Carlo solver LoKI-MC [70], in oxygen and other
relevant gases.

## Acknowledgments

This work was partially supported by the Portuguese FCT, under grant
PD/BD/150414/2019 (PD-F APPLAuSE) and Projects UIDB/50010/2020,
UIDP/50010/2020, EXPL/FIS-PLA/0076/2021 and MIT-EXPL/ACC/0031/2021, by the
European Union’s Horizon 2020 research and innovation programme, under grant
agreement MSCA ITN 813393, and by the European Union (NextGenerationEU) and
the Spanish Ministerio de Universidades (Plan de Recuperacion, Transformación
y Resiliencia).
We would like to thank Carlos D Pintassilgo for fruitful discussions on the
thermal model. We are also deeply thankful to Jean-Paul Booth, Olivier
Guaitella, Dmitry Voloshin, Dmitry Lopaev and Tatyana Rakhimova for invaluable
discussions on the heavy-species processes and for providing the experimental
data presented in figures 5 and 7. Finally, we acknowledge the careful
revision and suggestions by the referees, which helped us to improve the
manuscript.

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