# 3D particle simulations of positive air-methane streamers for combustion

Dennis Bouwman1, Jannis Teunissen1, Ute Ebert1,2 1 Centrum Wiskunde &
Informatica (CWI), Amsterdam, The Netherlands  2 Department of Applied
Physics, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven,
The Netherlands<EMAIL_ADDRESS>
###### Abstract

Streamer discharges can be used as a primary source of reactive species for
plasma-assisted combustion. In this research we investigate positive streamers
in a stoichiometric air-methane mixture at $1$ bar and $300$ K with a three-
dimensional particle-in-cell model for the electrons. We first discuss
suitable electron scattering cross sections and an extension of the
photoionization mechanism to air-methane mixtures. We discuss that the
addition of $9.5\%$ methane leaves electron transport and reaction
coefficients essentially unchanged, but it largely suppresses photoionization
and shortens the photon mean free path. This leads to (1) accelerated streamer
branching, (2) higher electric field enhancement at the streamer head, (3)
lower internal electric fields, and (4) higher electron densities in the
streamer channel. We also calculate the time-integrated energy density
deposited during the evolution of positive streamers in background electric
fields of 12.5 and 20 kV/cm. We find typical values of the deposited energy
density in the range of $0.5-2.5$ kJ/m3 within the ionized interior of
streamers with a length of $5$ mm; this value is rather independent of the
electric fields applied here. Finally we find that the energy deposited in the
inelastic electron scattering processes mainly produces reactive nitrogen
species: N2 triplet states and N, but also O and H radicals. The production of
H2 and O2 singlet states also occurs albeit less pronounced. Our calculation
of the primary production of reactive species can for example be used in
global chemistry models.

Keywords: Positive streamer, methane, particle-in-cell, photoionization,
plasma-assisted combustion

Accepted for publication: Plasma Sources Sci. Technol.on Apr 6, 2022

##  1 Introduction

Streamers are transient gas discharges consisting of growing plasma filaments
with field enhancement at the tip. A review covering a wide range of
investigations into streamer dynamics is presented in [1]. In industry, mainly
positive streamer discharges are found in a variety of applications such as:
plasma medicine [2], industrial surface treatment [3] and air-pollution
control [4]. In particular, positive streamers in combustible mixtures are
relevant for plasma-assisted combustion, as will be discussed in section 1.2.

###  1.1 Streamer dynamics in varying gases

The properties of streamers are determined by the electron dynamics which in
turn are governed by gas-specific photoionization, and by transport and
reaction coefficients. There are numerous investigations on streamers in
different gases which illustrate this gas-dependency, for example: CO2 [5],
N2:CH4 [6], air with artificially increased electron attachment or reduced
photoionization [7, 8], N2-O2 mixtures in various ratios [9, 10], the
atmospheres of Venus and Jupiter [11], and mixtures resembling the atmosphere
of the primordial Earth [12].

Specific to N2-O2 mixtures it was shown experimentally that streamers tend to
branch more frequently for decreasing oxygen concentrations [13, 14].
Decreasing the oxygen concentration gives different photoionization
characteristics due to a longer photon mean free path, while the number of
photoionization events stays essentially unchanged. In mixtures with low
oxygen concentrations ($<0.2\%$) stochastic fluctuations associated with the
discrete nature of photoionization then accelerate branching.

In an air-methane mixture, however, methane largely absorbs photons without
ionizing, and it also quenches excited N2 that otherwise could emit photons.
So the rate of ionizing photoabsorption decreases, and the photon mean free
path decreases as well [15]. As we will show, these two effects combined also
enhance the stochastic fluctuations in the leading edge of the streamer. In
simulations where the single-electron dynamics in the leading edge are
sufficiently resolved (e.g. in a 3D PIC-MCC model) this will ultimately
accelerate streamer branching.

###  1.2 Plasma-assisted combustion

In plasma-assisted combustion there is an interest in positive streamers
propagating through combustible mixtures, such as air-methane [16, 17, 18]. In
a streamer discharge one finds electrons with energies that exceed the gas
temperature by orders of magnitude. These energetic electrons produce excited
nitrogen states, hydrogen- and oxygen-radicals, fuel fragments and other
reactive species through collisions with neutral gas molecules. The resulting
non-equilibrium distribution of reactive species is then available for plasma-
and combustion-chemical processes taking place on slower time scales. These
processes have been studied numerically. For instance in [15] the effect of a
streamer discharge on the reduction in ignition delay was studied with an
axisymmetric fluid model. In [19, 20] a two-dimensional cartesian fluid
approach was used to investigate radical production by streamers in air-
methane (without correcting the photoionization). In [21] a similar model was
used (2D cartesian without correcting the photoionization and also without
electron attachment) to investigate the production of radicals for air-methane
streamers at $10$ bar and $600$ K. An overview of multiscale modelling for
plasma-assisted combustion is presented in [22]. Finally a comparison between
0D and axisymmetric models for the simulation of air-methane streamers
(without photoionization) is presented in [23]. The application of low-
temperature plasmas is found to have favourable effects such as: ultra-lean
combustion for emission reduction [24], increased flame propagation speed [25,
26] and flame stabilization [27]. In a single-pulse discharge the generation
of reactive species can, under the right conditions, lead to a reduction of
ignition delay time [28].

###  1.3 Content of the paper

We simulate positive streamers in a stoichiometric air-methane mixture at $1$
bar and $300$ K in background electric fields of $12.5$ kV/cm and $20$ kV/cm
in an $8$ mm gap. Such conditions are relevant for the initial stages of
plasma-assisted ignition. We will analyze the simulations from two different
viewpoints. First we will analyze how the addition of methane affects the
fundamental properties of the discharge, such as streamer branching, electric
field enhancement and electron densities and energies. Second, we analyze the
streamer discharge within the context of plasma-assisted combustion. We will
study the deposited energy density and the G-values, i.e. the efficiency with
which primary reactive species are produced. The production of these primary
species can represent an initial condition of plasma-chemical and ignition-
chemical calculations.

In section 2 we describe the particle-in-cell model, cross section sets,
simulation conditions and we correct for the suppressing influence of CH4 on
the photoionization mechanism. In section 3 we compare the dynamic properties
of a positive streamer in air and air-methane. Finally, in section 4 we
address the plasma-chemical activation of a stoichiometric air-methane mixture
by a positive streamer.

##  2 Simulation Method

We simulate a positive streamer discharge using a 3D Particle-In-Cell model
with Monte-Carlo Collisions (PIC-MCC). Our implementation is based on the
model described in [9]. Here we present a short summary and describe the
photoionization model used for discharges in air-methane mixtures.

###  2.1 Description of PIC-MCC model

Within a PIC-MCC model electrons are represented by super-particles. One
super-particle represents a variable number of physical electrons, represented
by the weight $w$ as is further elaborated in section 2.2. The motion of
electrons is then governed by an acceleration due to the electric field
combined with isotropic scattering processes due to collisions with the gas
molecules. Ions are included as a density and are assumed to be immobile on
the considered short streamer time scale. The neutral gas particles are taken
as a homogeneous background density with which the electrons can
stochastically collide. Electron collisions with ions, excited/dissociated
molecules and other electrons can be omitted since the ionization degree is
low, around $10^{-4}$ to $10^{-5}$.

An advantage of a PIC-MCC model over the conventional fluid models is that the
electron energy distribution function (EEDF) is approximated explicitly and
without the need for assumptions such as the local field approximation. For
plasma-chemical streamer applications a good approximation of electron
energies is important as this quantity determines the production rate of
reactive species. Another advantage of using PIC-MCC models is that it is
better equipped to deal with single-electron fluctuations. In air, it is known
that electron density fluctuations, in particular due to stochastic
photoionization, accelerate streamer branching [8, 29, 30]. In the case of
air-methane mixtures there is less photoionization and the photon mean free
path is shorter (see section 2.4). As a result single-electron fluctuations
occur in the active zone, i.e. the region where the electric field is above
breakdown. In order to properly resolve the influence of these single-electron
fluctuations we perform simulations with the PIC-MCC model.

The main drawback of PIC-MCC models is a high computational cost associated
with the use of a large number of particles, especially in three dimensions.
For example, typical computing times for our simulations are on the order of
days (performed on one node of Cartesius, the Dutch national supercomputer),
whereas a two-dimensional fluid simulation performed on an ‘average’ desktop
typically only requires several minutes of computation. More details about
such a comparison are found in the appendix of [30].

###  2.2 Particle weight

In a streamer discharge the number of free electrons increases rapidly to the
point where it becomes computationally infeasible to simulate every electron
individually. To overcome this limitation the weight ($w\geq 1$) of a computer
particle is dynamically updated, thereby allowing one particle to represent
one or many physical electrons. This technique, called adaptive particle
management, ensures that the computational complexity of the simulation
remains tractable at the cost of an artificial noise on the electron density.
Details on the implementation and performance of this algorithm can be found
in [31]. The central idea is that particles are merged and/or split between
time steps, thereby changing their weights in order to bring them close to a
desired weight $w_{d}$:

$w_{d}=\frac{n_{e}\Delta x^{3}}{N_{\text{ppc}}},$ (1)

where $n_{e}$ is the local electron density, $\Delta x^{3}$ is the volume of
the cell containing the particle and $N_{\text{ppc}}$ is the desired number of
particles per cell, which we have chosen at $100$.

Now we focus on the influence of artificial noise introduced by super-
particles before estimating that for our parameters this effect is small. To
that end we note that electron density fluctuations accelerate branching. This
was shown, for example, in [29] using a stochastic fluid model. A similar
conclusion was drawn in [8] and later in [30] by combining a conventional
fluid model with a stochastic version of the Zhelezniak photoionization model.
Therefore with super-particles (with $w>1$) it is inevitable that electron
density fluctuations are more prevalent compared to using only single
particles ($w=1$). As a result streamer branching would occur more often,
especially when compared with fluid models which neglect physical density
fluctuations. However, by choosing a small cell volume and a high
$N_{\text{ppc}}$ one can ensure that the artificial noise introduced by super-
particles does not dominate the fluctuations introduced by physical mechanisms
in the leading edge (such as photoionization). For example, in our simulations
the smallest cell volume, which is used in the high-field region at the
streamer head, equals $(4.0\,\mu$m$)^{3}$. Thus electron avalanches on the
finest grid with densities below $1.6\cdot 10^{18}$ m-3 are simulated using
particles with unit weight. In section 3.1 we show that this is sufficiently
accurate in order to resolve the single-electron fluctuations in the leading
edge.

###  2.3 Cross sections for electron collisions

A set of cross sections for the dominant scattering processes is required to
describe the electron kinetics. Many of the available sets have adjusted
individual cross sections in order to ensure that electron transport and
reaction coefficients are correctly reproduced in numerical swarm experiments.
A downside of this procedure is that adjusting cross sections can lead to
incorrect reaction rates. Thus the swarm-fitting procedure is limited by non-
uniqueness [32], since different modifications to inelastic cross sections can
result in the same swarm parameters but with different reaction rates. In
earlier work, we defined an unfitted cross section set and addressed the issue
of non-uniqueness for the case of CH4 [33].

Given our focus on accurately predicting the produced reactive species we only
use unfitted cross section sets. For N2 we adopt the cross section set of
Kawaguchi et al. [34], but neglect the inter-rotational (de-)excitation which
is only relevant at low reduced electric fields. For O2 we adopt the cross
sections recommended by Itikawa et al. [35]. Electron attachment by three-body
collisions with O2 are taken from [36]. For CH4 we adopt the cross section set
proposed by Bouwman et al. [33] which are based on the recommendations by Song
et al. [37] combined with cross sections for the neutral dissociation
processes. Finally, all scattering processes are assumed to be isotropic.

Figure 1: The transport and reaction coefficients for air and stoichiometric
air-methane at $1$ bar and $300$ K. The axis for the mobility $\mu$ is linear,
whereas the axis for ionization and attachment coefficients $\alpha$ and
$\eta$, respectively, is logarithmic. We conclude that a gas fraction of
$9.5\%$ CH4 introduces only minor changes to these coefficients.

For a stoichiometric air-methane mixture ($71.5\%$ N2, $19\%$ O2 and $9.5\%$
CH4) at $300$ K and $1$ bar we compared the ionization and attachment
coefficients $\alpha,\,\eta$ and the electron mobility $\mu$ with air, in
figure 1. These coefficients are calculated using BOLSIG+ (desktop version of
2019) using the temporal growth setting and assuming isotropic scattering [38,
39]. This relatively small fraction of CH4 introduces only minor quantitative
differences. Furthermore, we remind the reader that the coefficients depicted
here are only used for the purpose of illustration. Within a PIC-MCC model the
electron kinetics is directly determined by the cross sections.

###  2.4 Photoionization in air-methane mixtures

Naidis [15] approximated the influence of CH4 on photoionization in air-
methane by including an extra absorption factor in the classical Zhelezniak
photoionization model [40]. In that work, however, no corrections were made
for the changing effective quenching pressure due to the addition of methane.
An alternative to the Zhelezniak photoionization model (in air) is discussed
in [41, 42]. Here we follow the reasoning of [15] as we extend the Monte-Carlo
Zhelezniak photoionization model, such as presented in [9, 43], to air-methane
mixtures. We formulate necessary corrections to include photon-loss due to
non-ionizing photoabsorption and, notably, quenching.

####  2.4.1 Quenching:

In dry air, it is well-established that radiative transitions of excited
states of N2 emit photons in the energy range that is associated with the
photoionization of O2, namely $12.1-12.65$ eV. On the contrary, the excited
states of CH4 are all dissociative [37] and do not emit photons with energies
that can ionize O2. Thus, photons in the relevant energy range are produced in
air-methane mixtures only by the excited states of N2. However, CH4 does
suppress the total number of photons by quenching the excited states of N2.
The effective quenching pressure $p_{q}^{\text{eff}}$ can be written as:

$\frac{1}{p_{q}^{\text{eff}}}=\frac{\chi^{\text{O}_{2}}}{p_{q}^{\text{O}_{2}}}+\frac{\chi^{\text{N}_{2}}}{p_{q}^{\text{N}_{2}}}+\frac{\chi^{\text{CH}_{4}}}{p_{q}^{\text{CH}_{4}}},$
(2)

where $\chi$ denotes the respective gas number fraction. The quenching
pressures are reported in [44] to be: ${p_{q}^{\text{O}_{2}}=3.8}$ torr,
${p_{q}^{\text{N}_{2}}=91}$ torr and ${p_{q}^{\text{CH}_{4}}=1.8}$ torr.
According to the Zhelezniak model, the average number of photons $\bar{\eta}$
produced per impact ionization of an air molecule (i.e. N2 or O2) is given by:

$\bar{\eta}=\frac{p_{q}^{\text{eff}}}{p_{q}^{\text{eff}}+p}\ \xi,$ (3)

with the pressure $p$ and the factor $\xi$ relating the radiative de-
excitation rate of N2 to the ionization rate. The dependence of $\xi$ on the
local electric field is only partially tabulated in [40], so we have for
simplicity taken the constant value $\xi=0.05$. This value is within the
ranges considered in [8, 45], where it was shown that that deviations by a
factor two have little influence on most streamer properties, although lower
values can increase the probability of streamer branching.

Figure 2: The photon mean free path as a function of the CH4 fraction
$\chi^{\rm CH_{4}}$ in air-methane mixtures for various photon energies.
Stoichiometric conditions are denoted by the dashed grey line. Adding methane
shortens the photon mean free path.

####  2.4.2 Photoabsorption:

The absorption of photons is determined by the photoabsorption cross sections
$\sigma$. Kameta et al. [46] have determined the photoabsorption cross
sections for CH4. In the energy range relevant for the ionization of oxygen,
$12.1-12.65$ eV, they found that the cross section for photoionization of CH4
is negligble compared to non-ionizing photoabsorption of CH4 (and also to the
photoionization of O2). Furthermore, in this energy region the value of
$\sigma^{\text{CH}_{4}}$ is roughly constant at $3.0\cdot 10^{-17}$ cm2. On
top of that, we only have to consider the cross section for ionizing
photoabsorption of O2 in this energy range. $\sigma^{\text{O}_{2}}$ is
typically described as a power function of photon energy $\gamma$, whereas the
cross sections for N2 can be neglected [47]. This allows us to write the mean
free path of a photon with an energy $\gamma$ in interval $12.1-12.65$ eV as:

$l_{\text{mfp}}(\gamma)=\left(\sigma^{\text{CH}_{4}}n^{\text{CH}_{4}}+\sigma^{\text{O}_{2}}(\gamma)n^{\text{O}_{2}}\right)^{-1},$
(4)

with $n$ representing the number density of a gas component (the number
density $N$ of an ideal gas is $2.4\cdot 10^{25}$ m-3 at $1$ bar and $300$ K).
The mean free path plays an important role in the dynamics of electron density
fluctuations which are a result of stochastic photoionization. For that reason
we have illustrated the dependency of $l_{\rm mfp}$ on the gas fraction of CH4
in figure 2. The gas composition was determined by keeping the ratio between
N2 and O2 fixed at $79:21$ while varying the fraction of CH4.

Finally, we formulate the probability of a photon to ionize O2 (as opposed to
being lost due to absorption by CH4):

$P(\gamma)=\frac{\sigma^{\text{O}_{2}}(\gamma)n^{\text{O}_{2}}}{\sigma^{\text{CH}_{4}}n^{\text{CH}_{4}}+\sigma^{\text{O}_{2}}(\gamma)n^{\text{O}_{2}}}.$
(5)

####  2.4.3 Implementation:

The photoionization procedure in air-methane mixtures is implemented as
follows: if a super-particle with weight $w$ ionizes an N2 or O2 molecule
(note that our choice for the parameter $\xi$ in the Zhelezniak model
corresponds to air), then that produces a random number of photons which are
sampled from a Poisson distribution with mean $\bar{\eta}w$. Each of these
photons are produced individually, i.e. the use of super-photons is excluded,
and are assigned a random energy $\gamma$. Since the energy interval for
ionization of O2 is assumed to be uniformly populated, we take $\gamma$ as a
uniform random variable in the interval $12.1-12.65$ eV. Then a Bernoulli
trial with probability $P(\gamma)$ determines whether the photon is lost due
to absorption by CH4. If not, the photon is emitted isotropically with a
travel distance drawn from a Poisson distribution with mean
$l_{\text{mfp}}(\gamma)$ upon which it ionizes an O2 molecule.

####  2.4.4 Interpretation:

Figure 3: The absorption function $\Psi$ for air and an air-methane mixture
containing $9.5\%$ methane. This function illustrates the damping influence of
methane on the photoionization mechanism.

Now we illustrate the influence of CH4 on photoionization. We do this by
computing the absorption function $\Psi$, which can be expressed by the
absorption function in air, $\Psi_{\rm air}$ as:

$\displaystyle\Psi$ $\displaystyle=e^{-r\mu^{\text{CH}_{4}}}\Psi_{\rm air},$
(6) $\displaystyle=e^{-r\mu^{\text{CH}_{4}}}\cdot\frac{e^{-r\mu_{\rm
min}}-e^{-r\mu_{\rm max}}}{r\ln(\mu_{\rm max}/\mu_{\rm min})},$ (7)

with $\mu^{\text{CH}_{4}}$ the (constant) absorption coefficient due to
methane, and $\mu_{\rm min}$ and $\mu_{\rm max}$ the absorption coefficients
of dry air at $300$ K and $1$ bar according to [40, 47]. This function has
been calculated for air and an $9.5\%$ air-methane mixture (with
$\text{N}_{2}:\text{O}_{2}$ as $79:21$) and shown in figure 3. Note that for
the purpose of this illustration we have assumed that $\xi$ is constant for
both mixtures. Clearly, the addition of methane leads to a strong decay of the
absorption function on the millimeter length scale. This length scale is
relevant for the leading edge dynamics of the streamers considered in this
research. Note that $\Psi$ is not directly used in the PIC-MCC model, since
photoionization events follow from sampling of relevant distributions as
described in the previous section. Here, $\Psi$ is only used for the purpose
of illustration.

###  2.5 Computational domain and initial conditions

The simulated domain consists of a cube with a length of $10$ mm for each
Cartesian coordinate. The domain is filled with air or an air-methane mixture,
consisting of $71.5\%$ N2, $19\%$ O2 and $9.5\%$ CH4, at $300$ K and $1$ bar.
Such number fractions correspond to stoichiometric burning conditions of
methane (i.e. $\text{CH}_{4}:\text{O}_{2}$ as $1:2$ and
$\text{N}_{2}:\text{O}_{2}$ as $79:21$).

We are considering a plate-to-plate geometry with a grounded plate at the
bottom of the domain and a high-voltage plate at the top. Furthermore the
high-voltage electrode contains an axisymmetric protrusion with length of
$1.8$ mm and a radius of $200$ $\mu$m. The tip of this needle-electrode is a
hemisphere with the same radius, giving the electrode a total length of $2$
mm. To solve for the electrostatic potential, we use the multigrid solver
described in [48] which was recently generalized to include irregular
boundaries. On the top electrode (including protrusion) a constant voltage
$\phi$ of $12.5$ kV or $20$ kV is applied. Due to field enhancement near the
needle the electric field is locally above breakdown. Far away from the needle
the field relaxes to $12.5$ kV/cm and $20$ kV/cm respectively, which we will
refer to as the background field $E_{0}$. The electric field is calculated by
applying Dirichlet boundary conditions for the electric potential on the
electrodes. The boundary conditions of the electric potential on the sides of
the domain are given by homogeneous Neumann conditions, i.e. the field is
parallel to the boundary. Moreover, super-particles are removed from the
simulation if they are transported into the needle-electrode or out of the
domain. Furthermore, the numerical grid is provided by the Afivo-framework
[48] which utilizes adaptive mesh-refinement (AMR) with a minimum cell size of
$4.0$ $\mu$m. This cell size is sufficiently small in order to resolve the
dynamics in the thin space-charge layer. Moreover, in [30] it was found that
the particle model is less sensitive to the cell size than a fluid model, at
least when comparing streamer velocities.

Figure 4: The computational domain consists of a Cartesian cube with a length
of $10$ mm in each coordinate. The initial seed electrons are shown in red.

For all simulations in this work we use the same initial conditions consisting
of a neutral seed around the electron tip. The seed consists of $1000$
electron-ion pairs at coordinates that are drawn from a Gaussian distribution
centered at the tip of the electrode with a variance of $125\,\mu$m2.
Coordinates coinciding with the interior of the needle-electrode are rejected.
The electrons are represented by particles with unit weight whereas the ions
are represented as a density. Such a neutral seed is convenient when comparing
discharges under different conditions as it leads to fast inception which is
not highly stochastic (studying inception is not the focus of this work). The
electron density of the initial seed is illustrated in figure 4 in red.

##  3 Comparison of air and air-methane streamers

Here we study how the addition of methane changes streamer properties. We
focus on branching, electric field enhancement and electron energies.

###  3.1 Streamer branching

A

$L_{z}=0.1$ mm

$1.1$ mm

$2.1$ mm

$3.1$ mm

$4.1$ mm

$5.1$ mm

Air

(a) $3.00$ ns

(b) $6.25$ ns

(c) $9.25$ ns

(d) $11.75$ ns

(e) $14.00$ ns

(f) $16.25$ ns

Air-methane

(g) $5.00$ ns

(h) $8.75$ ns

(i) $11.75$ ns

(j) $14.00$ ns

(k) $16.00$ ns

(l) $18.00$ ns

Figure 5: Evolution of the electron density for positive streamers as a
function of streamer length. Streamers propagate in a background field of
$E_{0}=12.5$ kV/cm in air (top) and air-methane (bottom). The corresponding
simulated times are supplied in the sub-caption. Visualization is performed
with three partially-transparent contour surfaces. We observe that shape of
the air streamer is smooth whereas the air-methane streamer is more erratic.

We will now study the influence of methane on streamer branching. To that end
we have performed simulations in air and a stoichiometric air-methane mixture
under the conditions described in section 2.5. The background electric field
was taken as $E_{0}=12.5$ kV/cm. Furthermore the comparison will be performed
for streamers of equal (vertical) length $L_{z}$. $L_{z}$ is obtained by
calculating the maximum vertical distance between the electrode and each point
in the domain where the electron density exceeds $10^{19}$ m-3. We have chosen
this threshold value because such densities are typically obtained in (or very
near) the space-charge layer around the streamer tip. We visualize the
streamer by plotting three partially-transparent contour surfaces.

In figure 5 we show the time evolution of positive streamers in both gases. We
observe that the initial streamer formation takes about $2$ ns longer in air-
methane than in air (for $L_{z}=0.1$ m), but that the main branch of both
streamers then propagates at approximately the same instantaneous velocity
($0.47$ mm/ns for both gasses at $5.1$ mm). For a discharge in air no
branching occurs. However, in these conditions we do observe that the streamer
does not propagate in a straight line but in a meandering fashion. Its
counterpart in air-methane is more irregular. The first branching event
occurred about $3.75$ ns after inception (figure 5). Further stochastic
fluctuations occur throughout the evolution of the discharge which give the
streamer an erratic shape.

(a) Air

(b) Air-methane

Figure 6: Zoom into the electron density and the electric field on a cross
section through the streamer head, of the discharges shown in figure 5. In all
images, the contour line corresponding to an electron density of $1.6\cdot
10^{18}$ m-3 is indicated. Electrons outside this contour are represented by
individual particles (on the finest grid). Note the varying color scheme for
the electric field which are matched to the electric field of the streamer
tips. In air-methane we observe a smaller size of the electron cloud
surrounding the streamer and a higher electric field at the tip than in air.

The erratic streamer shape due to the addition of methane can be attributed to
the suppression of photoionization. In section 2.4 we have shown that a gas
fraction of $9.5\%$ methane already significantly reduced the rate of
photoionization and the photon mean free path while only having a minor
influence on the transport and reaction coefficients, as is also observed in
[15]. This has a pronounced effect on the electron density ahead of the
ionization front, as is shown in figure 6. In this figure we see slices of the
electron density and electric field along the axis of propagation for both
streamers after $3.7$ mm. Electron densities are shown on the same logarithmic
color scale, with maximum densities in the channel of $7.5\cdot 10^{19}$ m-3
for air and $25\cdot 10^{19}$ m-3 for air-methane. Electric fields are shown
on different linear color scales, which are matched to the electric field at
the streamer tip ($130$ kV/cm in air and $190$ kV/cm in air-methane). In both
gases we observe that the electron density in the leading edge decreases as we
move farther away from the ionization front. However, for a streamer in air
the electron density extends further ahead of the ionization front than its
counterpart in air-methane. This is due to the difference in the longest
photon mean free path. For sufficiently large distances $r$ we find that the
asymptotic behaviour of the function $\Psi$ satisfies:

$\Psi(r)\propto r^{-1}e^{-r/l_{\text{max}}},$ (8)

with $l_{\text{max}}$ the longest photon mean free path (cf. figure 2 at
$12.1$ eV). Concretely, the characteristic length of the leading edge electron
density, which is dominated by photoionization away from the streamer tip, is
thus determined only by the longest photoionization length. Adding methane
shortens the leading edge. Then, the electron density fluctuations ahead of
the streamer tip, which are due to the discrete nature of photons, evolve due
to impact ionization in the high electric field near the streamer tip. When
these stochastically distributed electron avalanches reach the ionization
front they accelerate branching [29].

In section 2.2 we have mentioned how to mitigate the influence of artificial
noise introduced by super-particles. We also argued that on the finest grid we
simulate leading-edge electron densities up to $1.6\cdot 10^{18}$ m-3 with
single particles only. This threshold density is denoted in figure 6 by a
black contour line. Since this contour line is close to the space-charge layer
we conclude that the leading edge dynamics are properly resolved. Thus, the
observed stochastic fluctuations and their influence on streamer branching are
physical.

###  3.2 Volume distributions of $E$ and $n_{e}$

We will analyze the quantitative difference in the electric field and electron
density between the air and air-methane streamers presented in the previous
section. From figure 5 one can already note that the electron density in the
tips of the air-methane streamers is higher. Furthermore, figure 6 shows that
electric field enhancement is higher at the main branch of the air-methane
streamer. A more complete illustration of the electric fields is shown in
figure 7. There we have shown cross sections of electric fields on a
logarithmic scale for streamers in both gases at $5.1$ mm. These cross
sections have the same perspective as in figure 5 and slice through the middle
of the electrode. Because the streamers are not perfectly axisymmetric,
sometimes streamer branches or other parts of the channel fall outside of the
plane. Nevertheless, these images convey typical behaviour of the electric
fields inside the streamer channel. In air, we observe that the electric field
just after the streamer tip is around $1.5$ kV/cm and gradually increases to
around $25$ kV/cm at the point of connection with the electrode. In air-
methane typical electric fields inside the channel are slightly lower, in the
range between $0.75-25$ kV/cm. Exceptions are the thin stagnated side-branches
in air-methane, where the smallest internal electric fields lie around $0.1$
kV/cm.

Figure 7: Cross sections of the electric fields of discharges shown in figures
5 and 5 on a logarithmic scale. Internal electric fields in the streamer are
lower in air-methane than in air. The thin stagnated side-branch has the
lowest internal electric field.

(a) Electric field

(b) Electron density

Figure 8: The volume distribution $f_{V}(E)$ of the electric field strength
and $f_{V}(n_{e})$ of the electron density. We have shown three air streamers
and three air-methane streamers at a length of $5.1$ mm under conditions
corresponding to figure 5. Air-methane streamers exhibit stronger electric
field enhancement and higher electron densities.

We will make these observations quantitative by introducing the volume-
normalized distribution of the electric field strength $f_{V}(E)$:

$f_{V}(E)=\frac{1}{V_{\mathrm{total}}}\frac{V(E)-V(E+\Delta E)}{\Delta E},$
(9)

where $V(E)$ is the volume in which the electric field strength exceeds $E$,
and $V_{\mathrm{total}}$ is the total volume. So $f_{V}(E)$ denotes the volume
of the domain where the electric field strength lies within the small interval
within $E$ and $E+\Delta E$ which is then divided by $\Delta E$ and normalized
with respect to the total volume.

$f_{V}(E)$ is shown in figure 8 for three simulations per gas mixture with
different random seeds for streamers of $5.1$ mm length. We use log-spaced bin
sizes $\Delta E$ (for computational efficiency). The region with values below
$5$ kV/cm, corresponds to internal electric fields as shown in figure 7. The
very low fields $0.1-0.5$ kV/cm are due to small side branches. Since the
streamers in air that we consider have not branched these regions are non-
existent (up to stochastic fluctuations). In the main channel of the discharge
we find values of $1.5$ kV/cm in air and $0.75$ kV/cm in air-methane (in
figure 7 it is shown that internal electric fields gradually increase to $25$
kV/cm near the electrode). This is reflected in the distribution function by
peaks at the associated values. Moving to higher fields, we find a large
volume of the domain with fields ranging from $10-15$ kV/cm. This interval is
dominated by the region away from the streamer, where the background electric
field persists. Moving to even higher fields, we find the region that
corresponds to the active zone induced by the discharge. Here we observe that
air-methane streamers exhibit higher values for the electric field in its
active zone. The maximum electric field is around $215$ kV/cm in air-methane
whereas in air we find a maximum value around $120$ kV/cm. The maximal field
in air-methane corresponds to the thin stagnated side-branch on the left (see
figure 5).

Analogously we show the volume distribution of the electron density:

$f_{V}(n_{e})=\frac{1}{V_{\mathrm{total}}}\frac{V(n_{e})-V(n_{e}+\Delta
n_{e})}{\Delta n_{e}}$ (10)

in figure 8. Here $V(n_{e})$ is the volume in which the electron density
exceeds $n_{e}$. Air-methane streamers typically exhibit higher electron
densities, which is probably due to a higher electric field at the tip.
Furthermore, the low electron density region, say below $10^{16}$ m-3,
corresponds to the electron ‘cloud’ surrounding a streamer which is a result
of photoionization. We observe that in air this electron cloud fills a much
larger volume than in air-methane. In air-methane, the size of the cloud is
reduced because CH4 shortens the photon mean free path. Thus we find that a
smaller part of the discharge is associated with this region.

To conclude, positive streamers in air-methane compared to air: (1) have
smaller electron cloud surrounding the streamer, (2) have electric fields at
the tips of their ionization fronts which are higher by a factor $1.5$, (3)
have internal electric fields that are lower by a factor two and (4) have
higher electron densities that are higher by a factor three.

###  3.3 Electron energy distribution

In figure 9 we show a comparison of the electron energy distribution function
(EEDF) of positive streamers in air and a stoichiometric air-methane mixture.
The distribution was calculated in both gases when the streamer reached a
length of $5.1$ mm. This corresponds to figures 5 and 5. The EEDF was obtained
by calculating the kinetic energy of each super-particle and making a
histogram with a bin size of $0.75$ eV.

Figure 9: The EEDF of streamers presented in section 3.1 with a length of
$5.1$ mm. A higher $E_{\text{max}}$ in the case of air-methane leads to higher
electron acceleration which enhances the tail.

For both gases we observe similar qualitative behaviour of the EEDF: most of
the electrons have energies below $5$ eV and only a few electrons have
energies sufficiently high to ionize neutral gas molecules (e.g. the lowest
ionization threshold is $12.1$ eV corresponding to O2). These electrons are
most likely to be situated in the region where ionization occurs, namely the
ionization front and the part of the channel close to the electrode. For both
gases we also find electrons with energies above $100$ eV. For air-methane we
even observe an electron energy exceeding $160$ eV, which can be considered
high for positive streamers. Electrons with such high energies are approaching
the cold-electron runaway regime [49] and challenge the assumption of
isotropic scattering, which is also made in our model.

A distinction between the two gases is that the tail of the distribution is
more enhanced in the air-methane mixture. This is to be expected due to the
higher electric field at the tip of the streamer, as was shown in section 3.2,
combined with the fact that electron energy losses in the tail hardly change
by the addition of methane [6], cf. figure 1.

##  4 Plasma-chemical activation

Here we will investigate the energy deposition and the production of reactive
species. These streamer properties are relevant for plasma-assisted
combustion. Furthermore we highlight that these properties are comparatively
insensitive to the considered electric fields.

###  4.1 Deposited energy density

Figure 10: Contour surfaces of the electron density, the deposited energy
density and the power density deposition within $0.25$ ns. We show air-methane
streamers with a length of $L_{z}=4.8$ mm in two background fields. Due to a
higher velocity in higher electric fields, the contribution of the ionization
front to $P_{\rm dep}$ appears more smeared out.

We will now investigate the energy that a positive streamer deposits to the
gas molecules in a stoichiometric air-methane mixture. To this end, we have
performed simulations under the conditions shown in figure 4 for two
background fields: $E_{0}=12.5$ and $20$ kV/cm (named ‘low field’ and ‘high
field’, respectively). From these simulations we have extracted the deposited
energy density by electron scattering $\varepsilon_{\rm dep}$. Note that this
quantity is integrated over time. The deposited energy density is calculated
by cumulatively interpolating the energy losses of all the inelastic
scattering events for each time step to the grid. Since inelastic electron
scattering is the dominant contribution to the deposited energy density, this
term represents the conversion of kinetic electron energy to chemical
activation. In a fluid approach, the deposited energy density described above
corresponds to the term $\int_{0}^{T}j\cdot E\,dt$, where $j$ is the electric
current density and $T$ is the time. Similarly we also calculate the power
density deposition $P_{\rm dep}$, which is the instantaneous energy density
deposition. This quantity is obtained by calculating the numerical time
derivative of $\varepsilon_{\rm dep}$ with a time step of $0.25$ ns. Since the
high field streamers have a higher velocity, the contribution corresponding to
the moving ionization front appears to be more smeared out compared to the low
field.

In figure 10 we have shown the contour surfaces of the electron density
$n_{e}$, the time-integrated deposited energy density $\varepsilon_{\rm dep}$
and the power density deposition $P_{\rm dep}$ for both applied electric
fields. Both streamers have an equal length of $L_{z}=4.8$ mm. In the low
field the discharge developed to this size in $18$ ns whereas in the high
field only $4.25$ ns was needed. A result of the difference in the timescales
is that the effects of electron loss in the channel due to attachment are
visible in the low field but not in the high field. For the $\varepsilon_{\rm
dep}$ we observe a peak in deposited energy in a small region close to the
electrode in both fields. In that region the energy density ranges from $10$
to $100$ kJ/m3 (not shown) and keeps growing over time. Such interactions
between the streamer channel and the needle electrode are not uncommon, see
[50, 45] for example. Away from the needle we have a typical deposited energy
density between $0.5$ and $2.5$ kJ/m3 that is comparatively insensitive to the
applied electric field. However, for thin branches (as one can see on the
sides of the discharge in the low field) the deposited energy is higher:
around $5.0$ kJ/m3. These thin side-branches have a higher deposited energy
density than the thicker main streamers due to a higher electric field
enhancement at the streamer tip. For the contour surfaces of $P_{\rm dep}$, we
have chosen the contour values such that they correspond to the deposited
energy density within the time step of $0.25$ ns. From this we can see that
energy is mainly deposited by the ionization front and the small region near
the electrode. As the ionization front propagates, it leaves behind energy in
the new region of the channel. Moreover, as the streamer grows further a
current continuously flows through the channel which also contributes to the
deposited energy density, but these illustrations indicate that this
contribution is only minor. Furthermore, the large power density deposition
near the electrode shows that the deposited energy in this region grows over
time.

In figure 11 the normalized volume distribution of the deposited energy
density $f_{V}(\varepsilon_{\rm dep})$ is shown for three simulations per
applied electric field. This quantity is defined analogously to equation (10)
but with $V_{\rm total}=V(\varepsilon_{\rm dep}>10^{-2}$ kJ/m3). In other
words we normalize with respect to the volume treated by the discharge, which
is taken as the volume where the deposited energy density exceeds $10^{-2}$
kJ/m3. This leaves the volume distribution invariant to differences in amount
of volume treated by the discharge. This is convenient when comparing
discharges that have different radii as is the case for streamers in different
electric fields. From this graph we can conclude that a higher applied
electric field does not necessarily lead to a higher deposited energy density,
as the distributions of both fields (up to $10$ kJ/m3) practically coincide.
Note that the high field discharge does in fact treat a larger volume of the
gas, but that distribution functions are invariant to such differences due to
normalization. Since the deposited energy density below $10$ kJ/m3 is
associated with the ionization front, cf. the contour surfaces of figure 10,
These results indicate that the deposited energy density dynamics of the
ionization front are quite insensitive to the applied electric fields
considered here.

Figure 11: Volume distribution of the deposited energy density for air-methane
streamers with $L_{z}=4.8$ mm. For each applied electric field we have
performed three simulations.

Furthermore, for deposited energy density values above $10$ kJ/m3 (which are
associated to the region close to the needle electrode) the distribution is
actually higher in the low electric field. One aspect that contributes to this
difference is that the width of the channel that is connected to the electrode
is much smaller for the streamer in the low field. Hence the current density
that flows through the channel of the streamer is higher near the electrode
which leads to a larger deposited energy density.

In conclusion: Away from the electrode, streamers in both fields considered
here have a similar deposited energy density. However, the high field
streamers are generally wider and therefore treat a larger volume of the gas.

###  4.2 G-values for the production of reactive species

A positive streamer produces various reactive species that are of interest to
plasma-assisted combustion. For example, electron-impact can directly
dissociate a molecule and hence produce radicals. Moreover, the electronically
excited states can also lead to further radical production by auto-
dissociation, dissociative quenching of another molecule or through chemical
reactions [51].

In this section we will calculate the efficiency with which specific primary
species are generated. With ‘primary’ we mean that we only consider the
production that follows directly from electron impact due to the streamer. The
efficiency is expressed as the number of reactive particles created per $100$
eV of input energy, named the G-value (with units 1/(100 eV)). In order to
calculate this we have cumulatively recorded the number of excitations that
have occurred for each type of collision. The spatial coordinates of an
excitation event are not stored, as that would significantly affect
computation time.

The G-values corresponding to individual species are shown in A. In this
section, the species are grouped into:

  * •

the oxygen singlet states O2$(a^{1}\Delta_{g}$, $b^{1}\Sigma_{g}^{+})$,

  * •

the nitrogen triplet states N2$(A^{3}\Sigma_{u}^{+}$, $B^{3}\Pi_{g}$,
$C^{3}\Pi_{u}$, $B^{\prime 3}\Sigma_{u}^{-}$, $E^{3}\Sigma_{g}^{+}$,
$F^{3}\Pi_{u}$, $G^{3}\Pi_{u}$, $W^{3}\Delta_{u})$, and

  * •

the radicals N, O, H and H2 that are directly created by electron impact
dissociation.

The spatial profile of the densities of all species (not shown) is similar to
that of the energy deposition depicted in section 4.1. In the region near the
needle-electrode we will find high values of the densities. Away from the
needle the reactive species are produced in comparatively fixed densities.

Figure 12: The G-values (i.e. number of particles produced per $100$ eV) of
various groups of reactive species as a function of streamer length $L_{z}$.
Dash-dotted corresponds to the low background field of $12.5$ kV/cm. Dotted
corresponds to the high background field of $20.0$ kV/cm.

In figure 12 we show the G-values produced by the discharges presented in
section 4.1, for the background fields ${E_{0}=12.5}$ and $20.0$ kV/cm. The
G-values are calculated for the whole discharge evolution until the streamer
has reached the vertical length $L_{z}$, and they are plotted as a function of
$L_{z}$. Similar to the results of the average energy deposition, we find that
the G-values for the production of reactive species are comparatively
insensitive to the applied electric field and do not change greatly as the
streamers grow. Similar observations were made regarding the calculated
G-values in flue-gases [52]. The highest G-value corresponds to nitrogen
triplet states, where $3.3$ particles are produced per $100$ eV (at a length
of $5$ mm). Direct electron impact dissociation of molecules nitrogen and
oxygen is also prevalent, with G-values of ${1.5-1.7}$ and ${0.7}$,
respectively. Furthermore, under these conditions the electron-impact
dissociation of methane produces atomic and molecular hydrogen radicals with a
typical G-value of ${0.6}$ and ${0.2}$, respectively. Finally, the low-
threshold singlet states of oxygen have a comparatively low G-value of around
${0.3}$.

These G-values show that positive streamers in stoichiometric air-methane
mixtures primarily produce large amounts of reactive nitrogen species. This
leads to further dissociation of O2, CH4 and H2 in the plasma-afterglow (which
is not accounted for here). For O2, we observe that direct dissociation by
electron impact is much more prevalent than excitation to the low-lying
singlet states. Finally, the dissociation of CH4 predominantly produces atomic
hydrogen, in amounts which are approximately equal to atomic oxygen.

In plasma-assisted combustion there is an interest in a broad range of
operating conditions, with notable emphasis on high pressures and
temperatures. In order to address that interest we sketch how to generalize
our results to a wider parameter regime. We do so by relying on scaling laws
for electric discharges, as is explained in [1]. Discharges at different
pressures and temperatures are physically similar if the reduced electric
field $E/N$ is the same. Note then that high pressure and temperature can
partially compensate each other as the relevant quantity in discharge physics
is the gas density $N$. The G-values reported here are quite insensitive to
the (reduced) electric field, suggesting that they might be representative for
a larger range of pressures and temperatures than was considered in this work.
Having said all that, it is known that the applicability of scaling laws is
limited by a number of effects. For instance, at standard temperature and
pressure the three-body attachment of electrons and the suppression of photon
emission by collisional quenching violate these scaling laws. Thus electron
loss and the production of photoelectrons do not scale with $E/N$. Another
limiting factor is that we assume that electrons collide only with ground-
state molecules. However, for very high temperatures or partially burnt
mixtures electron scattering with fragmented or excited species will no longer
be negligible.

Our results are relevant for studies using global (i.e. quasi-0D) kinetic
models. These models are used to simulated the time evolution of hundreds of
species using detailed chemical reaction mechanisms within reasonable
computation time. However, such models can not resolve the streamer discharge
phase which is strongly non-uniform, both spatially and temporally. The
G-values reported here provide insight into the initial (practically
instantaneous) production of reactive species. In this regard we mention other
modelling studies that have reported on the production of reactive species in
air-methane mixtures by a streamer discharge [19, 20, 15, 21, 23, 53, 22].
However, as far as the authors are aware the G-values for the primary
production phase have not been computed with the level of detail and as
complete as they have been reported here.

##  5 Summary and Discussion

###  5.1 Summary

We have performed three-dimensional particle-in-cell simulations of positive
streamers in air and in stoichiometric air-methane. In order to do so, we
first extended the stochastic photoionization model by Zelezniak et al. [40]
to account for quenching and non-ionizing photoabsorption due to methane. It
follows that $9.5\%$ methane already significantly suppresses photoionization
in air, decreasing both number and mean free path of photons within the
relevant energy interval significantly. Subsequently this influences streamer
properties. This model allowed us to show that:

  1. 1.

air-methane streamers branch more often than their counterparts in air. We
have attributed this to the decreased photoionization, while mobility and
effective ionization coefficients stay essentially unchanged.

  2. 2.

under the same conditions streamers in air-methane have a higher field
enhancement than in air, lower internal electric fields, a higher electron
density in the streamer channel, and higher electron energies.

On the side of plasma-assisted combustion we have shown that:

  1. 3.

the ionization front in air-methane streamers in background fields of $12.5$
and $20.0$ kV/cm typically deposits around $0.5-2.5$ kJ/m3 into plasma-
chemical activation of the gas. This quantity appeared to be quite insensitive
to the considered applied electric fields.

  2. 4.

similar to the deposited energy density, the streamers produce typical
densities of reactive species within the streamers that appear to be quite
insensitive to the electric fields applied in this study. We calculated the
G-values for the production of these species which are primarily N2 triplet
states, but also N, O and H radicals.

  3. 5.

aside from aforementioned similarities, the high field streamers have larger
radii (thus they treat a larger volume) and propagate faster.

###  5.2 Discussion

Our comparison of streamers in air and in a stoichiometric air-methane mixture
shows that it is important to correct the photoionization properties for the
presence of methane. However, the importance of a suppressed photoionization
mechanism diminishes in situations with considerable degrees of background
ionization, for instance in a pulsed discharge with a high repetition
frequency.

On the side of plasma-assisted combustion, our model represents only the
initial source of reactive species that arises due to direct electron impact
or photoionization in the streamer discharge. Important secondary production
of radicals due to chemical reactions occurs on slower time scales. The
G-values presented in this work could be used in a combustion model adapted to
resolve the subsequent chemical processes and thereby account for the non-
equilibrium excitation of the gas by a streamer. In this context we have shown
that, away from the needle-electrode, the streamer produces reactive species
in comparatively fixed densities which suggest that a volume-averaged
modelling approach with a parametrization of the streamer phase can be
considered. We refer the reader to [23] for a deeper discussion into the use
of global models.

## Acknowledgements

This work was carried out on the Dutch national e-infrastructure with the
support of SURF Cooperative. The three-dimensional renders presented in this
work have been made using VisIt [54]. DB acknowledges funding through the
Dutch TTW-project 16480 ‘Making Plasma-Assisted Combustion Efficient’.

##  Appendix A Tabulated G-values per excited species

In section 4.2 and in figure 12 the G-values of groups of excited species were
given. Here we resolve the G-values per excited species in tables 1, 2 and 3
for N2, O2 and CH4, respectively. The G-values are the number of excitation
events per $100$ eV that generate a particular excited species or group of
species. Note that to convert back to the G-values per grouped species, the
values corresponding to processes that produce multiple particles first have
to be properly weighted, e.g. the reaction:

$\text{e + \text{N}${}_{2}$ $\to$ e + N + N},$

produces two nitrogen atoms and therefore has a weight of two. For convenience
we have also supplied the activation energy $\epsilon_{a}$ for each scattering
process. Moreover, since the streamers in both electric fields are similar we
only give the G-values corresponding to streamers in $12.5$ kV/cm when they
have reached a length of $6$ mm.

| G (1/(100 eV)) |  $\epsilon_{a}$ (eV)
---|---|---
N${}_{2}(A^{3}\Sigma_{u}^{+})$ | 1.055 | 6.169
N${}_{2}(B^{\prime 3}\Sigma_{u}^{-})$ | 0.104 | 8.166
N${}_{2}(B^{3}\Pi_{g})$ | 1.007 | 7.354
N${}_{2}(C^{3}\Pi_{u})$ | 0.741 | 11.033
N${}_{2}(E^{3}\Sigma_{g}^{+})$ | 0.019 | 11.872
N${}_{2}(F^{3}\Pi_{u})$ | 0.013 | 12.986
N${}_{2}(G^{3}\Pi_{u})$ | 0.028 | 12.811
N${}_{2}(W^{3}\Delta_{u})$ | 0.365 | 7.363
N${}_{2}(a^{\prime\prime 1}\Sigma_{g}^{+})$ | 0.059 | 12.256
N${}_{2}(a^{\prime 1}\Sigma_{u}^{-})$ | 0.065 | 8.399
N${}_{2}(a^{1}\Pi_{g})$ | 0.371 | 8.550
N${}_{2}(b^{\prime 1}\Sigma_{u}^{+})$ | 0.065 | 12.855
N${}_{2}(b^{1}\Pi_{u})$ | 0.140 | 12.501
N${}_{2}(c_{3}^{1}\Pi_{u})$ | 0.055 | 12.913
N${}_{2}(c_{4}^{1}\Sigma_{u}^{+})$ | 0.038 | 12.935
N${}_{2}(o_{3}^{1}\Pi_{u})$ | 0.034 | 13.104
N${}_{2}(w^{1}\Delta_{u})$ | 0.063 | 8.896
N${}_{2}(J=2)$ | 34.406 | 0.002
N${}_{2}(v=1)$ | 13.834 | 0.288
N${}_{2}(v=2)$ | 6.337 | 0.573
N${}_{2}(v=3)$ | 3.368 | 0.855
N${}_{2}(v=4)$ | 1.731 | 1.133
N${}_{2}(v=5)$ | 0.822 | 1.408
N${}_{2}(v=6)$ | 0.365 | 1.679
N${}_{2}(v=7)$ | 0.152 | 1.947
N${}_{2}(v=8)$ | 0.059 | 2.211
N${}_{2}(v=9)$ | 0.021 | 2.471
N${}_{2}(v=10)$ | 0.007 | 2.728
N + N | 0.749 | 9.754
N${}_{2}^{+}$ | 0.260 | 15.582
N${}_{2}^{+}(A^{2}\Pi_{u})$ | 0.182 | 16.700
N${}_{2}^{+}(B^{2}\Sigma_{u}^{+})$ | 0.040 | 18.752
N + N+ | 0.023 | 24.342
N + N++ | $3.5\cdot 10^{-6}$ | 69.505
Table 1: The G-values for the generation of excited states of N2. The excitation energies of the species are given as well. | G (1/(100 eV)) |  $\epsilon_{a}$ (eV)
---|---|---
O${}_{2}($2B$)$ | 0.003 | 15.001
O${}_{2}(A^{3}\Sigma_{u}^{+},\ A^{\prime 3}\Delta_{u},\ c^{1}\Delta_{u})$ | 0.170 | 4.200
O${}_{2}(B^{3}\Sigma_{u}^{-})$ | 0.594 | 6.120
O${}_{2}($LB$)$ | 0.022 | 15.001
O${}_{2}(a^{1}\Delta_{g})$ | 0.276 | 0.977
O${}_{2}(b^{1}\Sigma_{g}^{+})$ | 0.066 | 1.627
O${}_{2}(v=1$) | 51.701 | 0.214
O${}_{2}(v=2)$ | 7.296 | 0.460
O${}_{2}(v=3)$ | 0.150 | 0.696
O + O | 0.341 | 5.100
O${}_{2}^{-}$ | 0.520 | 0.000
O + O- | 0.012 | 0.000
O${}_{2}^{+}$ | 0.113 | 12.071
O + O+ | 0.013 | 23.002
O + O++ | $1.3\cdot 10^{-6}$ | 73.006
Table 2: The G-values for the generation of excited states of O2. The excitation energies of the species are given as well. | G (1/(100 eV)) |  $\epsilon_{a}$ (eV)
---|---|---
CH4($v=1,v=3$) | 2.299 | 0.362
CH4($v=2,v=4$) | 4.621 | 0.162
CH3 \+ H | 0.526 | 7.501
CH2 \+ H2 | 0.205 | 9.101
CH + H2 \+ H | 0.003 | 15.501
C + H2 \+ H2 | $5.5\cdot 10^{-4}$ | 15.501
CH3 \+ H- | 0.004 | 0.000
H2 \+ CH${}_{2}^{-}$ | $2.5\cdot 10^{-4}$ | 0.000
CH${}_{4}^{+}$ | 0.144 | 12.631
CH3 \+ H+ | 0.002 | 21.102
H + CH${}_{3}^{+}$ | 0.089 | 12.631
CH2 \+ H${}_{2}^{+}$ | $2.4\cdot 10^{-4}$ | 22.302
H2 \+ CH${}_{2}^{+}$ | 0.007 | 16.201
H + H2 \+ CH+ | 0.002 | 22.202
H2 \+ H2 \+ C+ | 0.000 | 22.002
Table 3: The G-values for the generation of excited states of CH4. The
excitation energies of the species are given as well.

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