[datatype=bibtex] [fieldset=abstract, null]

# A 3D kinetic Monte Carlo study of streamer discharges in CO2

R Marskar  SINTEF Energy Research, Sem Sælands vei 11, 7034 Trondheim, Norway.
<EMAIL_ADDRESS>
###### Abstract

We theoretically study the inception and propagation of positive and negative
streamers in CO2. Our study is done in 3D, using a newly formulated kinetic
Monte Carlo discharge model where the electrons are described as drifting and
diffusing particles that adhere to the local field approximation. Our emphasis
lies on electron attachment and photoionization. For negative streamers we
find that dissociative attachment in the streamer channels leads to appearance
of localized segments of increased electric fields, while an analogous feature
is not observed for positive-polarity discharges. Positive streamers, unlike
negative streamers, require free electrons ahead of them in order to
propagate. In CO2, just as in air, these electrons are supplied through
photoionization. However, ionizing radiation in CO2 is absorbed quite rapidly
and is also weaker than in air, which has important ramifications for the
emerging positive streamer morphology (radius, velocity, and fields). We
perform a computational analysis which shows that positive streamers can
propagate due to photoionization in CO2. Conversely, photoionization has no
affect on negative streamer fronts, but plays a major role in the coupling
between negative streamers and the cathode. Photoionization in CO2 is
therefore important for the propagation of both positive and negative
streamers. Our results are relevant in several applications, e.g., CO2
conversion and high-voltage technology (where CO2 is used in pure form or
admixed with other gases).

††: Plasma Sources Sci. Technol.

##  1 Introduction

As with other gases, electric discharges in CO2 begin with one or more initial
electrons that accelerate in an electric field. If the electron velocity
becomes sufficiently high, collisions with CO2 molecules lead to net
ionization when the ionization probability exceeds the attachment probability.
As the process cascades through further ionization by acceleration of
secondary electrons, build-up of space charge from the electrons and residual
ions modifies the electric field in which the electrons originally
accelerated. This modification marks the onset of a streamer discharge [1],
which is a filamentary type of low-temperature plasma. Streamers have the
peculiar property that they continuously modify the electric field in which
they propagate, and thus exhibit a substantial degree of self-propagation.

Streamer discharges are categorized as positive or negative, depending on
their direction of propagation relative to the electric field. Negative
streamers propagate in the direction of the electrons (hence opposite to the
electric field), and are characterized by a negative space charge layer
surrounding their channels. Streamers that propagate opposite to the electron
drift direction are called positive streamers, and unlike negative streamers
they require a source of free electrons ahead of them. In air and other N2-O2
mixtures, this source is photoionization. CO2 is another molecule that is
relevant in multiple fields of research involving electrical discharges. In
high-voltage (HV) technology, for example, manufacturers of HV equipment are
currently transitioning from the usage of SF6 to environmentally friendlier
alternatives, such as pure CO2 or mixtures of CO2 and C4F7N (also relevant are
mixtures of air and C5F10O). However, photoionization in CO2 is known to be
much weaker than in air (for an overview, see [2]). It is now also accepted
that photoionization sensitively affects the morphology of positive streamers
in air [3, 4, 5] since it produces electron-ion pairs in regions where the
plasma density is low, which exacerbates noise at the streamer front. Positive
streamer branching thus occurs much more frequently in gases with lower
amounts of photoionization [6]. Since photoionization in CO2 is lower than in
air, one may expect that positive streamer discharges propagate quite
irregularly.

Few experimental studies have addressed streamer propagation in CO2.
Experiments by [7] showed that the DC breakdown voltage of CO2 is different
for positive and negative polarities. The authors investigated DC discharges
in non-uniform fields for both polarities, and showed that the breakdown
voltage for positive polarity is lower than for negative polarity at pressures
$p\leq$1\text{\,}\mathrm{bar}\text{/}$$. At higher pressures
$p>$1\text{\,}\mathrm{bar}\text{/}$$ this trend was reversed, and breakdown at
negative polarity consistently occured at a lower applied voltage than
breakdowns at positive polarity. This behavior is quite unlike that of air,
where positive streamers propagate more easily than negative streamers over a
wide range of pressures. Large statistical time lags were also observed for
positive streamers, but not for negative streamers. Inception did not always
occur for positive streamers, despite waiting times up to several minutes,
indicating that initiatory electrons are quite rare in CO2. No similar effect
was reported for negative streamers, which suggests that the source of the
initiatory electron could be different for the two polarities. A more thorough
investigation of inception times in CO2 was recently presented by [8], who
investigated pulsed discharges with $10\text{\,}\mathrm{Hz}\text{/}$
repetition rates.

Theoretically, [9] studied streamer propagation in CO2 gas using a Particle-
In-Cell (PIC) model with Monte Carlo Collisions (MCC), ignoring
photoionization and elucidating the intricate details of the electron velocity
distribution. [10] studied positive streamers in CO2 and air using a fluid
model. For CO2, the authors claim that photoionization is an irrelevant
mechanism, and in the computer simulations they replace it by a uniform
background ionization. The above theoretical studies were done in Cartesian 2D
[9] and axisymmetric 2D [10], and 3D simulations have not yet been reported.

In this paper we study the formation of positive and negative streamer
discharges in pure CO2, in full 3D. Our focus lies on the emerging morphology
of the streamers, and in particular on the roles of electron attachment and
photoionization. We show that currently reported photoionization levels for
CO2 [2] can facilitate positive streamer propagation. As we artificially
decrease the level of photoionization, we find that higher voltages are
required in order to initiate positive streamer discharges. Negative streamers
are also examined, and we show that the comparatively low levels of
photoionization in CO2 has virtually no effect on the dynamics of negative
streamer heads. However, photoionization is shown to play a role in the
coupling of the negative streamer to the cathode.

This paper is organized as follows: Our computational model is presented in
section 2, where we include the physical model and a brief overview of the
numerical discretization that we use. Results are presented in section 3.1 and
section 3.2 for negative streamers, and in section 3.3 and section 3.4 for
positive streamers. The paper is then concluded in section 4.

##  2 Theoretical model

###  2.1 Physical model

We use a physical model where the electrons are described as microscopic
particles that drift and diffuse according to the local field approximation
(LFA), i.e., we use a microscopic drift-diffusion model rather than a fluid
drift-diffusion model [11]. The transport equation for the electrons occurs in
the form of Îto diffusion:

$\textrm{d}\bm{X}=\bm{V}\textrm{d}t+\sqrt{2D\textrm{d}t}\bm{N},$ (1)

where $\bm{X}(t)$ is the electron position, and $\bm{V}$ and $D$ are the
electron drift velocity and diffusion coefficient. $\bm{N}$ indicates a normal
distribution with a mean value of zero and a standard deviation of one, and we
close the velocity relation in the LFA as

$\displaystyle\bm{V}$
$\displaystyle=\bm{v}_{\rme}\mathopen{}\mathclose{{}\left(\bm{X}}\right),$ (2)
$\displaystyle D$
$\displaystyle=D_{\rme}\mathopen{}\mathclose{{}\left(\bm{X}}\right),$ (3)

where $\bm{v}_{\rme}=-\mu_{\rme}\bm{E}$ is the fluid drift velocity where
$\mu_{\rme}$ is the electron mobility and $\bm{E}$ is the electric field, and
$D_{\rme}$ is the fluid diffusion coefficient. Our model is quite similar to a
conventional macroscopic drift-diffusion model, except that we replace the
electron transport kernel by a microscopic drift-diffusion process (i.e., an
Îto process) and the reactions by a kinetic Monte Carlo (KMC) algorithm.
Further details regarding the Îto-KMC algorithm and its association with fluid
drift-diffusion models is given in [11].

We use a fluid drift-diffusion model for ions, whose densities are indicated
by $n_{i}$ where $i$ is some species index. The equation of motion for the
ions is

$\frac{\partial n_{i}}{\partial
t}=-\nabla\cdot\mathopen{}\mathclose{{}\left(\bm{v}_{i}n_{i}-D_{i}\nabla
n_{i}}\right)+S_{i}.$ (4)

where $\bm{v}_{i}$, $D_{i}$, and $S_{i}$ are the drift velocities, diffusion
coefficients, and source terms for ions of type of $i$. The electric field
$\bm{E}=-\nabla\Phi$ is obtained by solving the Poisson equation for the
potential $\Phi$:

$\nabla^{2}\Phi=-\frac{\rho}{\epsilon_{0}},$ (5)

where $\rho$ is the space charge density and $\epsilon_{0}$ is the vacuum
permittivity.

###  2.2 Chemistry

We consider a comparatively simple kinetic scheme for CO2 consisting only of
ionizing, attaching, and photon-producing reactions, see table 1. Excited
states of CO2 are not tracked in our model as we are presently only interested
in the main ion products. We also remark that while the KMC algorithm uses
chemical propensities rather than the more conventional reaction rate
coefficient, all reactions in this paper are first-order reactions and in this
case the reaction rates in the KMC and fluid formulations are numerically
equivalent. The connection between the rates that occur in the chemical
propensities and conventional reaction rates can otherwise be quite subtle for
higher-order reactions, see e.g. [11, 12, 13] for further details.

Table 1:  CO2 plasma chemistry used in this paper.

  * Reaction | Rate | Ref.
---|---|---
$\rme+\ch{CO2}\rightarrow\rme+\rme+\ch{CO2^{+}}$ | $k_{\alpha}\mathopen{}\mathclose{{}\left(E/N}\right)$ | [14]
$\rme+\ch{CO2}\rightarrow\ch{CO}+\ch{O^{-}}$ | $k_{\eta}\mathopen{}\mathclose{{}\left(E/N}\right)$ | [14]
$\rme+\ch{CO2}\rightarrow\rme+\ch{CO2}+\gamma$ | $k_{\gamma}\mathopen{}\mathclose{{}\left(E/N}\right)$ | [14]

Transport coefficients and reaction rates for the electrons are computed using
BOLSIG+ [14] and the Phelps database (retrieved Oct. 16, 2023) [15]. The ion
mobility is set to
$2\text{\times}{10}^{-4}\text{\,}{\mathrm{m}}^{2}\text{/}\text{(}\mathrm{V}\text{\,}\mathrm{s}\text{)}$.
All rates are calculated at standard atmosphere, i.e.
$N\approx=$2.446\text{\times}{10}^{25}\text{\,}{\mathrm{m}}^{-3}$$.

###  2.3 Electron attachment in CO2

In the transport data we notice a peculiar feature that is relevant on longer
timescales (tens of nanoseconds). Figure 1 shows the ionization and attachment
coefficients ($k_{\alpha}$ and $k_{\eta}$), and the effective ionization rate
$\mathopen{}\mathclose{{}\left(k_{\alpha}-k_{\eta}}\right)$ for the attachment
region $E/N\leq$90\text{\,}\mathrm{Td}\text{/}$$. For fields
$E/N\approx$70\text{\,}\mathrm{Td}\text{/}$$ there is a global minimum in the
effective ionization coefficient where dissociative attachment is particularly
effective. At atmospheric pressure, which is what we study, the attachment
lifetime at $E/N\approx$70\text{\,}\mathrm{Td}\text{/}$$ is
$1/\mathopen{}\mathclose{{}\left|k_{\alpha}-k_{\eta}}\right|\approx$20\text{\,}\mathrm{ns}\text{/}$$.
If such fields appear in the streamer channel, dissociative attachment can
potentially reduce the electron density by a factor of $1/\text{e}$ every
$20\text{\,}\mathrm{ns}\text{/}$. We mention this feature because an analogous
phenomenon exists for streamer discharges in air, where it is known as the
attachment instability [16].

Figure 1:  Ionization and attachment rate coefficients as functions of $E/N$
at atmospheric pressure.

The physical explanation of the attachment instability is based on the
tendency of streamer channels to become quasi-stationary due to the short
relaxation time of the channel, in which case the current through the channel
is constant [17]. We then obtain
$\nabla\cdot\mathopen{}\mathclose{{}\left(\sigma\bm{E}}\right)=0$ for the
current density $\bm{J}=\sigma\bm{E}$, where $\sigma$ is the electric
conductivity. When electron attachment reduces the conductivity the channel
responds by increasing the electric field such that the current through the
channel remains constant. Because the effective attachment rate is field
dependent with a local maximum around
$E/N\approx$70\text{\,}\mathrm{Td}\text{/}$$, this process is self-
reinforcing. Suppose for a moment that some region in the streamer channel
initially has an internal field $E/N=$50\text{\,}\mathrm{Td}\text{/}$$. When
dissociative attachment sets in, the field in the channel will start to
increase as the conductivity is reduced. However, as we move rightwards in
figure 1 from $E/N=$50\text{\,}\mathrm{Td}\text{/}$$ the effective attachment
rate increases further, which simply accelerates the rate of dissociative
attachment and thus increases the field in the channel further. In recent
calculations we showed that this mechanism is responsible for column glows and
beads in so-called sprite discharges in the Earth atmosphere [18]. [19] also
propose that the attachment instability is the reason why pilot systems [20]
and space leaders [21] appear in metre-scale discharges, as they lead to
optical emission and heating of localized segments of the streamer channel.
However, it is not yet clear under which conditions the attachment instability
begins to manifests since it requires a comparatively high initial electric
field, well above the fields commonly observed in unperturbed streamer
channels (at least for positive streamers in air). Our transport data
nonetheless suggests that an attachment instability is also present in CO2,
which is of particular relevance to laboratory discharges as well as sprites
in the Venusian atmosphere (which is mostly composed of CO2).

###  2.4 Photoionization in CO2

In contrast to the case of air where photoionization data is abundant and the
primary states involved in the emission process have been identified,
photoionization data for CO2 is scarce. Photoionization in air primarily
occurs due to a Penning effect where N2 is first excited to the Carrol-Yoshino
and Birge-Hopfield II bands, which have excitation energies higher than the
ionization potential of O2. The de-excitation pathways from excited N2 are
collisional relaxation (i.e., collisional quenching), and spontaneous
emission. In the context of air, spontaneous emission rates are found in [22]
(predissociation is also a relevant relaxation mechanism for N2). When excited
N2 emits radiation through spontaneous emission it can ionize O2, and this
supplies an efficient photoionization mechanism that produces free electrons.
However, this mechanism relies on the availability of two molecular components
with different ionization potentials, so there can be no pure effect like this
in single-component gases like pure CO2. Photoionization in pure molecular
gases must accordingly proceed first by formation of excited dissociation
products, which is then followed by spontaneous emission of ionizing
radiation. In CO2, this may occur due to emission from O_I, O_II, C_II, CO,
and CO^+ [23]. Emission from these fragments, which form due to dissociative
excitation of CO2, can thus ionize CO2 which has an ionization potential
corresponding to $90.5\text{\,}\mathrm{nm}\text{/}$ radiation.

The emission cross sections for the dissociative fragments (O_I, O_II, C_II,
CO, CO^+) that produce extreme ultraviolet (EUV) ionizing radiation below
$90.5\text{\,}\mathrm{nm}\text{/}$ are incomplete, which prevents us from
using cross sections when deriving a photoionization model. [23] provide
emission cross sections for $200\text{\,}\mathrm{eV}\text{/}$ electrons and
identify spectral peaks corresponding to emissions from O_I, O_II, C_II, CO,
and CO^+. For the $83.4\text{\,}\mathrm{nm}\text{/}$ peak which corresponds to
emission from O_II, the authors also present energy-resolved cross sections.
The data in [23] is not available in tabulated form, but for an electron
energy of $76.5\text{\,}\mathrm{eV}\text{/}$ we may extract an approximate
emission cross section of
$1.78\text{\times}{10}^{-20}\text{\,}{\mathrm{cm}}^{2}\text{/}$ (from figure 2
in [23]). The corresponding ionization cross section that we use at
$76.5\text{\,}\mathrm{eV}\text{/}$ energy is approximately
$3.5\text{\times}{10}^{-16}\text{\,}{\mathrm{cm}}^{2}\text{/}$, so the
production of O_II emissions is considerably lower than the rate of electron
impact ionization. Unfortunately, the precision in the figures by [23] makes
it difficult to extract cross sections at lower electron energies and,
furthermore, energy resolved cross sections are not available for the other
EUV emissions.

The only available experiments that provide data for photoionization in CO2
are due to [24] who performed experiments at pressures of
$13\text{\,}\mathrm{Torr}\text{/}$. Collisional quenching is most likely
negligible at these pressures, and we have not been able to obtain data that
describes the quenching rates of the involved EUV-emitting fragments. Even in
air, quenching rates for the Carrol-Yoshino and Birge-Hopfield II bands of N2
are not known individually (collisional de-excitation may occur at different
rates for the two bands), but one may describe quenching by an approximate
quenching pressure $p_{q}\sim$40\text{\,}\mathrm{mbar}\text{/}$$. This leads
to a correction in the photoionization level by $p_{q}/(p+p_{q})\sim 0.04$ at
$1\text{\,}\mathrm{bar}\text{/}$ gas pressure, and this approach describes
experiments with an acceptable level of accuracy [5].

Figure 2:  Coefficient $\xi(E/N)$, describing the number of ionizing photons
produced per electron impact ionization event.

The present situation for CO2 is not ideal: Appropriate energy-resolved
emission cross sections at relevant electron energies are not available, and
experimental data is only available for low-pressure CO2. The atomic fragments
that emit the EUV radiation might be quenched differently, implying that
collisional quenching does not only reduce the number of ionizing photons, but
potentially also their spatial distribution. As we do not know of any data
that provides an equivalent quenching pressure in CO2, we introduce a free
parameter $\nu_{q}$ that adjusts the amount of photoionization in our
simulations, which is to be interpreted as follows: The quenching behavior of
the EUV-emitting fragments (e.g., O_I) following impact dissociation of CO2
obeys

$\partial_{t}\ch{O_{I}}=-\frac{\ch{O_{I}}}{\tau}-(k_{q}N)\ch{O_{I}},$ (6)

where $\tau$ is the radiative lifetime and $k_{q}N$ is the quenching rate.
Quenching occurs due to collisions between O_I and neutral CO2 molecules, so
the quenching rate grows linearly with neutral density $N$. The number of
photoemission events per de-excitation of O_I is then
$\frac{\tau^{-1}}{\tau^{-1}+k_{q}N}$, and as $N$ is proportional to pressure
($p=Nk_{\text{B}}T$), collisional quenching can reduce the amount of
photoionization at higher pressures. Similar relations could be formulated for
the other fragments, but as none of the corresponding rate constants
($\tau^{-1}$ and $k_{q}$) are known, we lump this factor into a single term
$\nu_{q}$.

The photon production rate $k_{\gamma}$ in our calculations is then calculated
as

$k_{\gamma}(\nu_{q})=\nu_{q}\xi\mathopen{}\mathclose{{}\left(E/N}\right)k_{\alpha},$
(7)

where $\nu_{q}\leq 1$ phenomenologically describes a reduction in the
production of ionizing photons due to collisional quenching, and $\xi(E/N)$ is
a field-dependent proportionality factor that describes the number of
photoionization events per electron impact ionization event as originally
measured by [24]. For air, $\xi\mathopen{}\mathclose{{}\left(E/N}\right)$ is
approximately $0.06$, while for CO2 the reported value is at least one order
of magnitude smaller. We have presented this data in figure 2 versus $E/N$.
The experimental data is limited to
$E/N\in\mathopen{}\mathclose{{}\left[$220\text{\,}\mathrm{Td}\text{/}$,$1\,000\text{\,}\mathrm{Td}\text{/}$}\right]$,
so we linearly extrapolate the data as indicated in the figure. This
extrapolation is done because we observe that very high fields develop in
computer simulations with low values of $\nu_{q}$, while emission cross
sections generally peak at around $200\text{\,}\mathrm{eV}\text{/}$ [23].

Figure 3: a) Ionization, attachment, and photon production rates for various
quenching efficiencies $\nu_{q}$. b) Inverse rates (i.e., lifetimes).

Figure 3a) shows the rates $k_{\alpha}$, $k_{\eta}$, and $k_{\gamma}(\nu_{q})$
for $\nu_{q}=${10}^{-4}{10}^{-2}1$$ as functions of the reduced electric field
$E/N$. We also include the inverse rates (i.e., lifetimes) of these reactions
in figure 3b). The reaction lifetimes describe the average time before an
electron triggers the reaction, and we can see that each electron generates
one impact ionization collision every
$k_{\alpha}^{-1}\sim$10\text{\,}\mathrm{ps}\text{/}$$ at
$E/N\sim$600\text{\,}\mathrm{Td}\text{/}$$. However, the lifetime
$k_{\gamma}\mathopen{}\mathclose{{}\left(\nu_{q}=1}\right)$ is approximately
$10\text{\,}\mathrm{ns}\text{/}$ at the same field strength, and
photoionization events are thus rare compared to ionization events.

CO2 absorbs quite strongly in the $8389\text{\,}\mathrm{nm}\text{/}$ spectral
range, where the pressure-reduced mean absorption coefficient is between
$\kappa_{\text{min}}/p=$0.34\text{\,}\text{/}\text{(}\mathrm{cm}\text{\,}\mathrm{Torr}\text{)}$$
and
$\kappa_{\text{max}}/p=$2.2\text{\,}\text{/}\text{(}\mathrm{cm}\text{\,}\mathrm{Torr}\text{)}$$
[2]. At atmospheric pressure this corresponds to mean photon absorption
lengths between $6\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ and
$38\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$. This is shorter than in
air, where mean absorption lengths are between
$30\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ and
$2\text{\,}\mathrm{mm}\text{/}$ at atmospheric pressure.

When computational photons are generated in our simulations, their mean
absorption coefficient is computed as

$\kappa=\kappa_{\text{min}}\mathopen{}\mathclose{{}\left(\frac{\kappa_{\text{max}}}{\kappa_{\text{min}}}}\right)^{U},$
(8)

where $U$ is a random number sampled from a uniform distribution on the
interval $[0,1]$, and $\kappa_{\text{min}}$ and $\kappa_{\text{max}}$ are as
given above. Only a few photons are generated per time step and cell. A rough
estimate may be obtained from figure 3a) with
$E/N=$600\text{\,}\mathrm{Td}\text{/}$$, where
$k_{\gamma}\sim$2.75\text{\times}{10}^{5}\text{\,}{\mathrm{s}}^{-1}$$, while
typical plasma densities $n_{\rme}$ at streamer tips are
${10}^{18}{10}^{20}\text{\,}{\mathrm{m}}^{-3}$. Time steps are typically
$\Delta t\sim k_{\alpha}^{-1}\approx$10\text{\,}\mathrm{ps}\text{/}$$ and grid
cell volumes in the streamer head are $\Delta
V\sim$8\text{\times}{10}^{-18}\text{\,}{\mathrm{m}}^{3}\text{/}$$. The mean
number of photons generated per cell and time step is roughly
$k_{\gamma}n_{\rme}\Delta t\Delta V$ which evaluates to between
$2\text{\times}{10}^{-5}$ and $2\text{\times}{10}^{-3}$ photons on average.
Note that this estimate is per grid cell; the total number of ionizing photons
emitted from a streamer head will be substantially higher. We point out that
the computational photons in our calculations correspond to physical photons,
so there is no artificial elevation of discrete particle noise due to
photoionization.

###  2.5 Simulation conditions

Our simulations are performed in the protrusion-plane geometry shown in figure
4, which has dimensions of
$$12\text{\,}\mathrm{cm}\text{/}$\times$3\text{\,}\mathrm{cm}\text{/}$\times$12\text{\,}\mathrm{cm}\text{/}$$.
The discharges initiate at the tip of a $5\text{\,}\mathrm{mm}\text{/}$ long
electrode that protrudes downwards along the $z$-axis. The gap distance
between the electrode and the ground plane is
$25\text{\,}\mathrm{mm}\text{/}$. The protrusion radius is
$1\text{\,}\mathrm{mm}\text{/}$, and narrows along a conical section with a
full opening angle of $30$ degrees and a tip radius of
$200\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$. All calculations are
performed for a standard atmosphere (i.e., pressure
$p=$101\,325\text{\,}\mathrm{Pa}\text{/}$$ and temperature
$T=$300\text{\,}\mathrm{K}\text{/}$$).

For the electric potential $\Phi$ we use homogeneous Neumann boundary
conditions ($\partial_{n}\Phi=0$) on the side faces and Dirichlet boundary
conditions on the top and bottom faces. The lower face is always grounded
($\Phi=0$) while the upper face and the protrusion are always at live voltage
$\Phi=V$ where $V$ is a constant applied voltage which is varied in our
computer simulations. We also define the average electric field between the
electrode and the ground plane as

$E_{\text{avg}}=\frac{V}{L},$ (9)

where $L=$25\text{\,}\mathrm{mm}\text{/}$$. The baseline quenching efficiency
that we use in our simulations is $\nu_{q}=$1$$, but we vary this in section
3.2 and section 3.4.

Figure 4:  Sketch (not to scale) of the computational domain with
electrostatic boundary conditions. The rounding radius at the needle tip is
$200\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ and the full opening angle
of the conical section on the electrode is $30$ degrees.

All simulations begin by sampling $100$ physical electrons with random
positions inside a $200\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ radius
sphere centered at the electrode tip. Initial electrons whose positions end up
inside the electrode are discarded before the simulation begins. The
initializing particles are unique to each simulation.

###  2.6 Numerical discretization

We use the chombo-discharge code [12] for performing our computer simulations.
As the full discretization and implementation of the model are quite
elaborate, we only discuss the basic features here.

In time, we use a Godunov operator splitting between the plasma transport and
reaction steps, where the transport step is semi-implicitly coupled to the
electric field (see [25] for another type semi-implicit coupling for fluid
discretizations). After the transport step we resolve the reactions in each
grid cell using a KMC algorithm. Unlike the deterministic reaction rate
equation, the KMC algorithm is fully stochastic and operates with the number
of particles in each grid cell rather than the particle densities. Complete
details are given in [11]. Constant time steps $\Delta
t=$10\text{\,}\mathrm{ps}\text{/}$$ are used in our simulations.

In space, we use an adaptive Cartesian grid with an embedded boundary (EB)
formalism for solid boundaries. EB discretization, also known as cut-cell
discretization, is a special form of boundary discretization and brings
substantial complexity into the discretization schemes (e.g., see [26]). In
return it permits use of almost regular Cartesian data structures, and allows
us to apply adaptive mesh refinement (AMR) in the presence of complex
geometries with comparatively low numerical overhead. Special handling of
discretization routines is introduced at cut-cell and refinement boundaries.
For example, we always enforce flux matching for the Poisson equation [27],
and particles are deposited using custom deposition methods near refinement
boundaries [11].

We discretize the 3D domain using $$256$\times$64$\times$256$$ cells and add 9
levels of grid refinement which are dynamically adapted as simulations
proceed. The refinement factor between adjacent grid levels is always 2, so
the finest representable grid cell in our simulation is $\Delta
x\approx$0.91\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$$. Grid cells are
refined every 5 time steps (i.e., every $50\text{\,}\mathrm{ps}\text{/}$) if

$\overline{\alpha}\Delta x\geq 1,$ (10)

where $\overline{\alpha}$ is the effective Townsend ionization coefficient.
Likewise, grid cells are coarsened if

$\overline{\alpha}\Delta x\leq 0.1$ (11)

and $\Delta x$ was no larger than $8\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{/}$.

Figure 5: Negative streamer evolution using various applied voltages. Top row:
$V=$-25\text{\,}\mathrm{kV}\text{/}$$
($E_{\text{avg}}=$1\text{\,}\mathrm{kV}\text{/}\mathrm{mm}$$). Middle row:
$V=$-30\text{\,}\mathrm{kV}\text{/}$$
($E_{\text{avg}}=$1.2\text{\,}\mathrm{kV}\text{/}\mathrm{mm}$$). Bottom row:
$V=$-35\text{\,}\mathrm{kV}\text{/}$$
($E_{\text{avg}}=$1.4\text{\,}\mathrm{kV}\text{/}\mathrm{mm}$$), where the
final frame shows a snapshot when the streamer connects to the ground plane
after $t=$43\text{\,}\mathrm{ns}\text{/}$$. The dashed lines shows the average
vertical velocities (between the indicated markers in each figure).

The Îto-KMC model we use is a particle-based model for the electrons, and
particle re-balancing is required since the number of physical electrons at
the streamer tips grows exponentially in time. Particle merging and splitting
is done following our previous approach discussed in [11] where bounding
volume hierarchies are used for group partitioning of particles within a grid
cell. The algorithm is run at every time step, and ensures that computational
particle weights within a grid cell differ by at most one physical particle.
In all simulations we limited the maximum number of computational particles to
$32$.

The calculations in this paper were performed on $880$ nodes on the Betzy
supercomputer, where each node consists of dual AMD EPYC 7742 CPUs. Each node
has $$2$\times$64$$ CPU cores, corresponding to a total of $1\,02410\,240$ CPU
cores for the various simulations. Meshes ranged up to
$2.5\text{\times}{10}^{9}$ grid cells and ${10}^{10}$ computational particles,
with various simulations completing in $0.55$ days.

##  3 Results

###  3.1 Negative streamers versus voltage

In this section we present results for the evolution of negative streamers for
voltages
$V\in\mathopen{}\mathclose{{}\left[-$25\text{\,}\mathrm{kV}\text{/}$,-$30\text{\,}\mathrm{kV}\text{/}$,-$35\text{\,}\mathrm{kV}\text{/}$}\right]$.
These voltages corresponds to average electric fields of
$1\text{\,}\mathrm{kV}\text{/}\mathrm{mm}$,
$1.2\text{\,}\mathrm{kV}\text{/}\mathrm{mm}$, and
$1.4\text{\,}\mathrm{kV}\text{/}\mathrm{mm}$. We performed a single 3D
simulation for each voltage, and present the results in figure 5.

####  3.1.1 Streamer propagation field

The top row in figure 5 shows that a negative streamer started at a voltage of
$V=$-25\text{\,}\mathrm{kV}\text{/}$$, but the discharge did not propagate
very far during the $50\text{\,}\mathrm{ns}\text{/}$ simulation time. The
streamer was not electrically connected (with plasma) to the cathode either,
and propagated as a diffuse cloud of electrons that gradually broadened and
weakened with propagation distance. While we did not run the simulation
further, we expect that the discharge would eventually fade out and decay. The
middle row in figure 5 shows negative streamers at a voltage of
$V=$-30\text{\,}\mathrm{kV}\text{/}$$, corresponding to an average electric
field of $E_{\text{avg}}=$1.2\text{\,}\mathrm{kV}\text{/}\mathrm{mm}$$. The
streamer is characterized by a main branch with numerous side branches, many
of which stagnate early and do not propagate further. This streamer did not
cross the discharge gap in the course of the simulation, although we expect
that it would have if the simulation was run further. Finally, the bottom row
shows the streamer development with an applied voltage of
$V=$-35\text{\,}\mathrm{kV}\text{/}\mathrm{mm}$$, corresponding to
$E_{\text{avg}}=$1.4\text{\,}\mathrm{kV}\text{/}\mathrm{mm}$$. The discharge
consists of multiple branches that form a broad discharge tree approximately
$15\text{\,}\mathrm{mm}\text{/}$ wide, and crossed the discharge gap in
$43\text{\,}\mathrm{ns}\text{/}$.

From the results we conclude that negative streamers in our simulations
propagate if the average electric field is
$E_{\text{avg}}\in$11.2\text{\,}\mathrm{kV}\text{/}\mathrm{mm}$$.
Incidentally, [7] report that the negative streamer stability field derived
from their experiments is $11\pm
2\text{\,}\mathrm{V}\text{/}\text{(}\mathrm{m}\text{\,}\mathrm{Pa}\text{)}$,
which translates to $11\pm 2\text{\,}\mathrm{kV}\text{/}\mathrm{mm}$ at
$1\text{\,}\mathrm{bar}\text{/}$ gas pressure. We thus find quantitative
agreement between experiments [7] and our observed propagation fields.

####  3.1.2 Velocity

In the computer simulations we observe that the front velocity of the
discharge is voltage dependent, and that it varies during the discharge
evolution. We have indicated the velocities in figure 5, which are calculated
by estimating how far the vertical front position of the discharge has moved
between the frames. At the lowest voltage
($V=$-25\text{\,}\mathrm{kV}\text{/}$$) the discharge propagated with an
average velocity of $0.1\text{\,}\mathrm{mm}\text{/}\mathrm{ns}$, but as we
mentioned above the discharge does not represent a propagating streamer. For
$V=$-30\text{\,}\mathrm{kV}\text{/}$$ the observed velocity remained fairly
constant throughout the propagation phase, with an approximate value of
$v=$0.340.36\text{\,}\mathrm{mm}\text{/}\mathrm{ns}$$. The bottom row in
figure 5 shows that at the highest simulated voltage
$V=$-35\text{\,}\mathrm{kV}\text{/}$$ the front velocity varied a great deal
throughout the streamer development. For $t<$20\text{\,}\mathrm{ns}\text{/}$$
the streamer velocity was approximately was
$0.52\text{\,}\mathrm{mm}\text{/}\mathrm{ns}$, which increased to
approximately $1\text{\,}\mathrm{mm}\text{/}\mathrm{ns}$ as the streamer
approach the ground plane.

[7] have measured approximate streamer velocities in CO2, using a single PMT
for estimating the propagation time of the streamer and an ICCD camera for
measuring the streamer length. The results for negative streamers were
obtained for a field distribution slightly different from ours, and the
authors report negative streamer velocities in the range of
$0.20.6\text{\,}\mathrm{mm}\text{/}\mathrm{ns}$ at
$1\text{\,}\mathrm{bar}\text{/}$ gas pressure. We also point out that this
velocity interval represents average streamer velocities rather than
instantaneous velocities. Our simulation results nonetheless agree quite well
with the experimental values, despite the fact that the experimentally
estimated streamer velocities contain uncertainties due to the measurement
method.

####  3.1.3 Radius

Figure 5 shows that the negative streamers branch frequently, and many of the
branches also stagnate, which makes it difficult to extract a single streamer
radius in our calculations. Since negative streamers can broaden quite
efficiently, a range of streamer radii can probably be observed also in
experiments. In experiments, only the optical radius of the streamers are
available, and [7] report experimentally obtained negative streamer radii as
$25\pm 3\text{\,}\mathrm{m}\text{\,}\mathrm{Pa}\text{/}$, which translates to
$250\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ at atmospheric pressure.
This radius was obtained for streamer filaments that did not branch, and thus
correspond to the minimal streamer radius. Our simulations do not model
optical emission in the CO2 plasma, so only the electrodynamic radius is
available. These measures can differ substantially. For positive streamers in
air it is estimated that the electrodynamic radius is twice that of the
optical radius [28], but no corresponding relation has been reported for
negative streamers.

Figure 6:  Determination of the minimum negative streamer radius for an
applied voltage $V=$-35\text{\,}\mathrm{kV}\text{/}$$ after
$t=$40\text{\,}\mathrm{ns}\text{/}$$, using the plasma density $n_{\rme}$ as a
proxy for the electrodynamic radius. a) Full view, showing plasma densities
$n_{\rme}\geq${10}^{18}\text{\,}{\mathrm{m}}^{-3}$$. b) Inset of the indicated
region in a).

Figure 6 shows the plasma density for the simulation with
$V=$-35\text{\,}\mathrm{kV}\text{/}$$ after
$t=$40\text{\,}\mathrm{ns}\text{/}$$. In figure 6b) we have also included
various length indicators, as well as the diameter of a specific branch which
initially propagated but later stagnated. Examining the various branches in
the figure we find that the smallest electrodynamic diameter of the filaments
is at least $420\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$, in good
agreement with experimentally reported values [7].

Figure 7:  Isosurface $n_{\rme}=${10}^{18}\text{\,}{\mathrm{m}}^{-3}$$ for
$V=-$35\text{\,}\mathrm{kV}\text{/}$$ after
$t=$30\text{\,}\mathrm{ns}\text{/}$$. The surface is colored by the reduced
electric field $E/N$.

####  3.1.4 Field distribution

In order to obtain an estimate for the range of electric fields that occur on
negative streamer tips, figure 7 shows an isosurface
$n_{\rme}=${10}^{18}\text{\,}{\mathrm{m}}^{-3}$$ of the plasma after
$t=$30\text{\,}\mathrm{ns}\text{/}$$ for the simulation with
$V=$-35\text{\,}\mathrm{kV}\text{/}$$. Depending on the radius and position of
the negative streamer tips, we find that the electric field at the negative
streamer tips is $250500\text{\,}\mathrm{Td}\text{/}$. For comparison,
reported electric fields for negative streamers in air are approximately
$300\text{\,}\mathrm{Td}\text{/}$ [29].

Figure 8:  Field lines in the plasma colored by electric field (in units of
$\mathrm{Td}\text{/}$, for $V=$-35\text{\,}\mathrm{kV}\text{/}$$ applied
voltage. The color range is truncated to
$E/N\in$080\text{\,}\mathrm{Td}\text{/}$$ with alpha channels that reduce the
opacity of the field lines with lower $E/N$.

Next, we examine the evolution of the electric field in the streamer channels.
Figure 8 shows snapshots of electric field lines at various time instants for
the simulation with $V=$-35\text{\,}\mathrm{kV}\text{/}$$. Field lines are
pruned from the plot if the electron density is
$n_{\rme}<${10}^{18}\text{\,}{\mathrm{m}}^{-3}$$ so that all field lines in
figure 8 pass through the plasma. The field lines are colored by $E/N$, and
transparency channels are added such that field lines with
$E/N\geq$80\text{\,}\mathrm{Td}\text{/}$$ are opaque and field lines with
$E/N=$0\text{\,}\mathrm{Td}\text{/}$$ are completely transparent. Figure 8
shows that localized regions in the streamer channel with initially low
electric fields later develop comparatively high fields
$E/N>$70\text{\,}\mathrm{Td}\text{/}$$. We have indicated one of these regions
by a dashed circle in figure 8, but other regions can also be identified. The
field enhancement in the channels is caused by dissociative attachment which
reduces the conductivity of the channel, as discussed in section 2. The
conductivity reduction is then compensated by an increased electric field,
similar to how the attachment instability operates in air [16].

Figure 9:  Cathode sheath region details at
$t=$2.5\text{\,}\mathrm{ns}\text{/}$$. The data in each figure shows a slice
through the $z$-plane in the simulation with $\nu_{q}=1$,
$V=$-35\text{\,}\mathrm{kV}\text{/}$$. a) Electric field magnitude. b)
Electron density. c) Space charge density.

####  3.1.5 Cathode sheath

Negative streamers propagate away from the cathode and leave behind positive
space charge composed of positive ions, which can lead to a sheath immediately
outside of the cathode surface. The sheath is electron-depleted because the
electrons in it propagate away from the cathode (and thus out of the sheath).
Analogous sheaths also exist for positive streamers propagating over
dielectric surfaces [30, 31, 12]. Unfortunately, we can not study the details
in the sheath with desired accuracy because of the inherent limitations of the
LFA. Physically, secondary electrons that appear in the sheath are due to
photoionization, cathode emission, or electron impact ionization. The
secondary electrons arising from these processes are low-energy electrons that
do not generate further impact ionization until they have been sufficiently
accelerated to above-ionization energies. But in LFA-based models these
electrons are always born with artificially high energies, parametrically
given as a function of $E/N$. In our model, photoelectrons that appear in the
sheath can thus immediately ionize the gas, which is non-physical since their
true energy is
$\mathcal{O}\mathopen{}\mathclose{{}\left($1\text{\,}\mathrm{eV}\text{/}$}\right)$.
Our model therefore predicts an artificially high level of impact ionization
in the sheath region, and we can thus only make a qualitative assessment of
the sheath features (such as its thickness).

Figure 9 shows some details of the cathode sheath region for the computer
simulation with $\nu_{q}=1$, $V=$-35\text{\,}\mathrm{kV}\text{/}$$, where we
include slice plots of the electric field magnitude, the electron density, and
the space charge density. From the figure we find a sheath thickness of
approximately $50\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$, and fields
that range up to $2\,000\text{\,}\mathrm{Td}\text{/}$. Figure 9c) shows the
reason why this high field region appears, which is due to CO2^+ ions that
have accumulated just outside the cathode surface. Since the cathode surface
is charged negatively and the space charge is positive, there is a
corresponding high field region between these two features. This field will
not persist indefinitely because as the ions move slowly towards the cathode,
the space charge layer is gradually absorbed by the cathode and the field in
the sheath will correspondingly decrease. We have not shown this process in
detail but point out that it occurs on a comparatively long time scale (tens
to hundreds of nanoseconds) due to the comparatively low ion mobility. For
sheaths along dielectric surfaces, this can lead to charge saturation, as
demonstrated by [32].

Figure 10: Evolution of a negative streamer with
$V=$-35\text{\,}\mathrm{kV}\text{/}$$ with photoionization ($\nu_{q}=1$) and
without photoionization ($\nu_{q}=0$).

###  3.2 Negative streamers without photoionization

Few publications have addressed the role of photoionization in negative
streamers. Most of the available results are for air and with continuum
approximations for the photons [33, 34]. [34] provide a qualitative
explanation on the role of photoionization for negative streamers: Seed
electrons that appear ahead of negative streamers turn into avalanches that
propagate outwards from the streamer tip, which facilitates further expansion
of the streamer head. When the negative streamer head expands, field
enhancement and thus impact ionization at the streamer tip decreases, which
leads to a slower streamer. Similar conclusions were reached by [29], who also
point out that this broadening can also lead to negative streamer decay
(similar to figure 5 for $V=$-25\text{\,}\mathrm{kV}\text{/}$$).

It is important to note that the above cited results all use continuum
approximations for photoionization, which is not a valid approximation for our
conditions. The role of discrete photoionization for positive streamers in air
has been reported [3, 4], and the studies show that positive streamer
morphologies depend sensitively on the photoionization parameters. Analogous
studies for negative streamers in air have not yet been reported. However,
since photoionization can provide seed electrons ahead of negative streamers
in precisely the same way as for positive streamers, photoionization might
also play a role in the branching of negative streamers.

Figure 10 shows the plasma density for a case where photoionization is fully
turned off ($\nu_{q}=0$). The applied voltage is
$V=$-35\text{\,}\mathrm{kV}\text{/}$$, i.e. corresponding to the bottom row in
figure 5, which is included for the sake of comparison. Without
photoionization, the negative streamer propagates much slower than its
photoionization-enabled counterpart, and it eventually also decays.

The computer simulations also show that the cathode sheath dynamics are
affected by photoionization. Figure 10 shows that the cathode is partially
covered by plasma when photoionization is enabled ($\nu_{q}=1$). This plasma
is a positive streamer that propagates upwards along the cathode, and it
leaves behind a positive space charge layer outside the cathode (as seen in
figure 9). The sheath is thus affected by the appearance of seed electrons in
the cathode region, in particular seed electrons that appear in the cathode
fall since these electrons initiate new avalanches that leave behind
additional space charge. As this process cascades, it leads to inception of a
positive streamer that propagates towards and finally along the cathode
surface. Without photoionization, the necessary seed electrons that are
required in order to faciliate the positive streamer no longer appear. The
upwards propagating positive streamer does not manifest without this source of
electrons, and this reduces the intensity of the space charge layer close to
the cathode and also the field in the cathode sheath.

Figure 11: Evolution of a negative streamer with
$V=$-35\text{\,}\mathrm{kV}\text{/}$$, using $\nu_{q}=1$ for
$z>$24.5\text{\,}\mathrm{mm}\text{/}$$ and $\nu_{q}=0$ elsewhere. Figure 12:
Positive streamer evolution with various applied voltages. Top row:
$V=$25\text{\,}\mathrm{kV}\text{/}$$
($E_{\text{avg}}=$1\text{\,}\mathrm{kV}\text{/}\mathrm{mm}$$). Middle row:
$V=$30\text{\,}\mathrm{kV}\text{/}$$
($E_{\text{avg}}=$1.2\text{\,}\mathrm{kV}\text{/}\mathrm{mm}$$). Bottom row:
$V=$35\text{\,}\mathrm{kV}\text{/}$$
($E_{\text{avg}}=$1.4\text{\,}\mathrm{kV}\text{/}\mathrm{mm}$$). The final
frame shows the plasma at $t=$66.5\text{\,}\mathrm{ns}\text{/}$$, immediately
before the streamer connected to the ground plane. The dashed lines shows the
average vertical velocities (between the indicated markers in each figure).

While figure 10 shows that photoionization is important for the negative
streamer evolution, it does not answer whether or not this is due to
conditions at the negative streamer tip, or due to absence of the upwards
positive streamer. The positive streamer feeds a current into the system, and
consequently it affects the potential distribution and field enhancement of
the negative streamer head.

In order to determine whether or not the decay of the negative streamer seen
in figure 10 is due to lack of photoionization at the negative streamer tip or
absence of a positive cathode-directed streamer, we run another computer
simulation where generation of ionizing photons is turned off in all regions
where $z<$24.5\text{\,}\mathrm{mm}\text{/}$$. This is equivalent to turning
off photoionization for the propagating negative streamer, but maintaining
photoionization for the upwards positive streamer that propagates along the
cathode surface. Figure 11 shows the results for this simulation, and should
be contrasted with the left and right columns in figure 10. The evolution of
the negative streamer for this case is qualitatively similar to the left
column of figure 10 where photoionization was enabled everywhere.
Consequently, photoionization at negative streamer tips in CO2 does not appear
to play a major role in the streamer evolution. However, appropriate coupling
to the cathode still requires inclusion of photoionization; the upwards
positive streamer feeds a current into the tail of the negative streamer which
increases field enhancement at the negative streamer tip, and thus facilitates
its propagation.

###  3.3 Positive streamers versus voltage

In this section we consider propagation of positive streamers for voltages
$V\in\mathopen{}\mathclose{{}\left[$20\text{\,}\mathrm{kV}\text{/}$,$25\text{\,}\mathrm{kV}\text{/}$,$30\text{\,}\mathrm{kV}\text{/}$,$35\text{\,}\mathrm{kV}\text{/}$}\right]$.
We perform the study in the same way as we did for negative streamers: A
single computer simulation is performed for each voltage application, and we
extract velocities, radii, and field distributions. The evolution of the
corresponding discharges is shown in figure 12.

####  3.3.1 Streamer propagation field

The top row in figure 12 shows the evolution of positive streamers in CO2 at
$V=$25\text{\,}\mathrm{kV}\text{/}$$, which corresponds to an average electric
field of $E_{\text{avg}}=$1\text{\,}\mathrm{kV}\text{/}\mathrm{mm}$$. We did
not include the simulation with $V=$20\text{\,}\mathrm{kV}\text{/}$$
($E_{\text{avg}}=$0.8\text{\,}\mathrm{kV}\text{/}\mathrm{mm}$$) in figure 12
because a positive streamer failed to develop at this voltage. The middle row
in figure 12 shows the evolution for $V=$30\text{\,}\mathrm{kV}\text{/}$$
($E_{\text{avg}}=$1.2\text{\,}\mathrm{kV}\text{/}\mathrm{mm}$$) and the bottom
row shows the evolution with $V=$35\text{\,}\mathrm{kV}\text{/}$$
($E_{\text{avg}}=$1.4\text{\,}\mathrm{kV}\text{/}\mathrm{mm}$$).
Qualitatively, we observe that with increasing voltage the streamers evolve
into broader and faster discharge trees. The discharges also grow much more
irregularly than for negative polarity (see figure 5).

Our baseline simulations show that positive streamers develop at
$V=$25\text{\,}\mathrm{kV}\text{/}$$ but not
$V=$20\text{\,}\mathrm{kV}\text{/}$$, indicating that the streamer propagation
field in our calculations is $0.81\text{\,}\mathrm{kV}\text{/}\mathrm{mm}$.
This is, in fact, substantially lower than what is observed in experiments
where the reported streamer propagation field at
$1\text{\,}\mathrm{bar}\text{/}$ is approximately
$1.3\text{\,}\mathrm{kV}\text{/}\mathrm{mm}$ [7]. For positive streamers, our
calculations of the streamer propagation field therefore contain an error of
about $3050\text{\,}\mathrm{\char 37\relax}\text{/}$.

####  3.3.2 Velocity

Like we found with negative streamers, figure 12 shows that the front velocity
of the discharge depends on the applied voltage and also that it varies during
the discharge evolution. With $V=$25\text{\,}\mathrm{kV}\text{/}$$ the
discharge propagated with an average velocity of approximately
$0.2\text{\,}\mathrm{mm}\text{/}\mathrm{ns}$, which is lower than the slowest
negative streamer we observed ($V=$-30\text{\,}\mathrm{kV}\text{/}$$). For
$V=$30\text{\,}\mathrm{kV}\text{/}$$ the observed velocity increased to
approximately $0.230.27\text{\,}\mathrm{mm}\text{/}\mathrm{ns}$, while for
$V=$35\text{\,}\mathrm{kV}\text{/}$$ the velocity was
$0.240.44\text{\,}\mathrm{mm}\text{/}\mathrm{ns}$. Our simulations show that
positive streamers propagate slower than negative streamers, in agreement with
experiments [7]. For comparison, the corresponding average velocities deduced
from experiments are $0.10.5\text{\,}\mathrm{mm}\text{/}\mathrm{ns}$, so our
velocity calculations are in comparatively good agreement.

####  3.3.3 Radius

Figure 12 shows that positive streamers in CO2 can develop into tree
structures that have a distribution of radii, i.e. there is no unique streamer
radius. As we did for negative streamers, we only extract the radius for
streamer filaments that do not branch, using regions where
$n_{\rme}\geq${10}^{18}\text{\,}{\mathrm{m}}^{-3}$$ as a proxy for the
electrodynamic radius. Figure 13 shows the simulation data for
$V=$25\text{\,}\mathrm{kV}\text{/}$$ after
$t=$50\text{\,}\mathrm{ns}\text{/}$$. Various length indicators are included,
as well as the diameter of a specific branch that did not branch, but whose
path fluctuated. From this branch we extract an approximate radius of
$140\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$. This agrees quite well
with the experiments by [7] who report that the optical radius of positive
streamers is at least $13.4\pm
3\text{\,}\mathrm{m}\text{\,}\mathrm{Pa}\text{/}$, which translates to
$130\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$ at atmospheric pressure.

Figure 13:  Determination of the minimum positive streamer radius for an
applied voltage $V=$25\text{\,}\mathrm{kV}\text{/}$$ after
$t=$50\text{\,}\mathrm{ns}\text{/}$$, using the plasma density $n_{\rme}$ as a
proxy for the electrodynamic radius.

####  3.3.4 Field distribution

As we discussed in section 3.1 we found that on longer timescales the field in
negative streamer channels gradually increase due to an attachment instability
that reduces the channel conductivity and hence increases the field in the
channel. We have not found any corresponding field increase in positive
streamer channels, which suggests that the field is too low for the attachment
instability to manifest at the positive streamer evolution time scale. Figure
14 shows the field distribution along field lines in the streamer channels,
i.e. regions where $n_{\rme}\geq${10}^{18}\text{\,}{\mathrm{m}}^{-3}$$. In
positive streamer channels we find that the internal electric field is
$1020\text{\,}\mathrm{Td}\text{/}$. For comparison, this is the same value as
positive streamers in atmospheric air, which is usually reported as being
around $20\text{\,}\mathrm{Td}\text{/}$ [29].

Figure 14:  Field lines in the plasma colored by electric field (in units of
$\mathrm{Td}\text{/}$, for $V=$35\text{\,}\mathrm{kV}\text{/}$$ applied
voltage after $t=$39\text{\,}\mathrm{ns}\text{/}$$. The color range is
truncated to $E/N\in$080\text{\,}\mathrm{Td}\text{/}$$ with alpha channels
that reduce the opacity of the field lines with lower $E/N$.

Figure 15 shows an isosurface $n_{\rme}=${10}^{18}\text{\,}{\mathrm{m}}^{-3}$$
for the simulation with $V=$30\text{\,}\mathrm{kV}\text{/}$$ after
$t=$56\text{\,}\mathrm{ns}\text{/}$$. The isosurface is colored by the reduced
electric field $E/N$, and shows the reduced electric field at positive
streamer tips. Typical fields are $600900\text{\,}\mathrm{Td}\text{/}$, but we
point out that the positive streamer fronts are quite irregular with local
enhancements of the field at their tips, as can be seen in figure 15b) which
shows a closeup near one of the positive streamer tips. At this tip the field
is locally enhanced to $E/N\geq$1\,200\text{\,}\mathrm{Td}\text{/}$$, and the
plasma irregularity at the tip can be identified. We believe that this
irregularity is caused by the low amount of photoionization, in which the
incoming avalanches that grow toward the streamer tips lead to a fine-grained
local field enhancement.

Figure 15:  Isosurface $n_{\rme}=${10}^{18}\text{\,}{\mathrm{m}}^{-3}$$ for
$V=$30\text{\,}\mathrm{kV}\text{/}$$ after
$t=$56\text{\,}\mathrm{ns}\text{/}$$. The surface is colored by the reduced
electric field $E/N$. a) Full view. b) Closeup of the indicated region from
a).  Figure 16:  Comparison of positive streamers using an applied voltage
$V=$30\text{\,}\mathrm{kV}\text{/}$$ with $\nu_{q}=$1$$ (top row) and
$\nu_{q}=$0.1$$ (bottom row).

###  3.4 Positive streamers with varying photoionization

We now consider the evolution of positive streamers when we vary the amount of
photoionization to $\nu_{q}=$0.1$$ and $\nu_{q}=$0.01$$. We first ran
simulations with $V=$25\text{\,}\mathrm{kV}\text{/}$$ using these parameters,
but at this voltage the streamers failed to develop. Recalling that the
streamer propagation field was $0.81\text{\,}\mathrm{kV}\text{/}\mathrm{mm}$
with $\nu_{q}=$1$$, we find that the streamer propagation field increases to
at least $11.2\text{\,}\mathrm{kV}\text{/}\mathrm{mm}$ with $\nu_{q}=0.1$.
This is closer to the experimentally reported propagation field at
$1\text{\,}\mathrm{bar}\text{/}$ pressure which is
$1.21.3\text{\,}\mathrm{kV}\text{/}\mathrm{mm}$ [7].

Next, computer simulations using a slightly higher applied voltage of
$V=$30\text{\,}\mathrm{kV}\text{/}$$ showed that streamers failed to develop
with $\nu_{q}=$0.01$$, but fully developed using $\nu_{q}=$0.1$$. Figure 16
shows the evolution of this discharge, where the baseline simulation
($\nu_{q}=$1$$) is included for the sake of comparison. For the simulation
with $\nu_{q}=$0.1$$ the positive streamer propagates at about half the
velocity of the baseline simulation, corresponding to a velocity of
approximately $0.10.2\text{\,}\mathrm{mm}\text{/}\mathrm{ns}$. The discharge
is also highly irregular with a higher plasma density in the filaments, which
lies in the range $n_{\rme}=${10}^{21}{10}^{22}\text{\,}{\mathrm{m}}^{-3}$$.
In the baseline simulation the plasma density in the channels was typically
${10}^{20}{10}^{21}\text{\,}{\mathrm{m}}^{-3}$. The corresponding field at the
positive streamer tips were also higher, ranging up to
$2\,000\text{\,}\mathrm{Td}\text{/}$ for the filaments with the smallest
diameters.

Another difference between the streamer evolution with $\nu_{q}=$0.1$$ and
$\nu_{q}=$1$$ is that as the amount of photoionization is lowered the streamer
grows even more irregularly. A qualitative demonstration of this is given in
figure 17 which shows a closeup of the plasma density in the vicinity of the
anode. The baseline data ($\nu_{q}=$1$$) in this figure corresponds to the
data at $t=$40\text{\,}\mathrm{ns}\text{/}$$ in figure 16. Electron density
fluctuations can be seen for the simulation with $\nu_{q}=$1$$, which
manifests as a fuzziness in the plasma density in the channels. This
irregularity is due to individually incoming avalanches, caused by stochastic
fluctuations due to photoionization at the front of the discharge. For the
simulation with $\nu_{q}=$0.1$$ this feature is much more pronounced, and
individual avalanches can be identified.

Figure 17:  Closeup of the irregularity of positive streamers using an applied
voltage $V=$30\text{\,}\mathrm{kV}\text{/}$$ after
$t=$40\text{\,}\mathrm{ns}\text{/}$$, using $\nu_{q}=$1$$ (top) and
$\nu_{q}=$0.1$$ (bottom).

##### The role of photoionization in CO2

Despite our initial underestimation of the positive streamer propagation
field, we find that positive streamers in CO2 require substantially higher
background fields than positive streamers in air. The breakdown field in CO2
is around $86\text{\,}\mathrm{Td}\text{/}$ while in air the corresponding
value is approximately $120\text{\,}\mathrm{Td}\text{/}$, so the breakdown
field of CO2 is only $71\text{\,}\mathrm{\char 37\relax}\text{/}$ that of air.
However, positive streamers in atmospheric air can propagate if the average
background field is greater than $0.5\text{\,}\mathrm{kV}\text{/}\mathrm{mm}$,
but positive streamers in CO2 require background fields greater than
$1\text{\,}\mathrm{kV}\text{/}\mathrm{mm}$. As we showed, the internal field
in positive streamers channels is about the same in CO2 and air, which implies
that positive streamers of similar lengths in air and CO2 carry the same
electric potential at their tips. The comparatively large difference in
propagation fields in these two gases must therefore be due to conditions at
the streamer head rather than conditions inside the streamer channels. Our
calculations showed that an increasingly higher voltage is required for
propagating positive streamers when the amount of photoionization is lowered,
which suggests that the higher propagation field for positive CO2 streamers
could be due to photoionization mechanisms at the streamer tip. Production of
ionizing photons in CO2 is, relatively speaking, lower than in air.
Furthermore, in CO2 most of the ionizing photons are absorbed very close to
the streamer head, with a mean absorption distance on the order of
$30\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$. Ionizing photons in air
propagate longer before they are absorbed, up to
$2\text{\,}\mathrm{mm}\text{/}$. This difference in absorption length implies
that free electrons that appear due to ionizing radiation in air multiply
exponentially over a longer distance than corresponding free electrons in CO2.
Since positive streamers grow due to incoming electron avalanches, and the
size of these avalanches depend on where the free electrons initially
appeared, a shorter absorption length effectively leads to a weaker
photoionization coupling.

Experimentally, [7] found that the pressure-reduced streamer propagation field
for negative CO2 streamers is constant. In other words, if the pressure is
doubled then the minimum applied voltage that is necessary in order to
initiate and propagate negative streamers is also doubled. For positive
streamers, however, the authors observed the same behavior that is commonly
observed in air: The pressure-reduced streamer propagation field grows with
pressure. This implies that if the pressure is doubled, the applied voltage
must be more than doubled in order to initiate and propagate a positive
streamer discharge. Based on the observations made above, we conjecture that
this is caused by a reduction in the photoionization level at higher
pressures, potentially due to collisional quenching of the EUV-emitting
fragments involved in the photoionization process.

##  4 Conclusion

###  4.1 Summary

We have presented a 3D computational study of positive and negative streamer
discharges in pure CO2, using a microscopic drift-diffusion particle model
based on kinetic Monte Carlo. From the transport data we showed that the
existence of a local maximum in the effective attachment rate affects the
conductivity, and thus electric field, in the streamer channels. The reduction
in the conductivity leads to a corresponding increase in the electric field.
This occurs on the timescale of $20\text{\,}\mathrm{ns}\text{/}$ for
$E/N\approx$70\text{\,}\mathrm{Td}\text{/}$$ at atmospheric pressure, and thus
took place on the time scale of our computer simulations. We suggest that this
mechanism is analogous to the attachment instability in air [16], and that it
may play an important role in the further evolution of the discharge. In air,
the attachment instability is associated with increased optical emission and
presumably also increased localized heating in the channel, which to a coarse
approximation is given by $\bm{E}\cdot\bm{J}$ where only $\bm{J}$ is constant
through a channel. While we have not modeled optical emission nor heating, it
is known that the attachment instability is responsible for the long-term
optical emission of sprite discharges in the Earth atmosphere (e.g., column
glows and beads) [17, 18]. Analogous emissions may thus exist for sprites in
atmospheres mainly composed of CO2, such as the Venusian atmosphere.

In the computer simulations we observed very high electric fields
$E/N\sim$2\,000\text{\,}\mathrm{Td}\text{/}$$ in the cathode sheath and on
some positive streamer tips (in particular for lower photoionization levels).
We also used a very fine spatial resolution $\Delta
x\lesssim$1\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$$. At these
conditions, a standard Courant condition for the maximum permissible time step
in fluid-based methods is $\Delta t\leq\Delta
x/\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left|v_{x}}\right|+\mathopen{}\mathclose{{}\left|v_{y}}\right|+\mathopen{}\mathclose{{}\left|v_{z}}\right|}\right)$,
which would imply using time steps below $0.5\text{\,}\mathrm{ps}\text{/}$.
This is $20$ times shorter than the actual time step we used in our
calculations, and would imply taking over $100\,000$ time steps in our
calculations, which would lead to prohibitively expensive calculations.
However, the particle-based LFA model does not have a Courant condition, and
alleviated the need for such a small time step. A partial reason for the
success of this study was due to this feature, as it allowed us to obtain
self-consistent solutions for comparatively large 3D streamer discharges
without incurring unacceptable computational costs.

###  4.2 Negative streamers

For negative streamers we obtain a satisfactory agreement with the experiments
by [7]. Our calculations indicate that the negative streamer propagation field
at $1\text{\,}\mathrm{bar}\text{/}$ pressure is
$1.11.2\text{\,}\mathrm{kV}\text{/}\mathrm{mm}$. The streamer velocity is
voltage dependent and ranges between
$0.30.6\text{\,}\mathrm{mm}\text{/}\mathrm{ns}$ in our simulations. The
minimum streamer diameter was at least $420\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{/}$. Velocities, propagation fields, and radii were in good agreement
with experiments. The channel field in negative CO2 streamer could become
quite high, exceeding $70\text{\,}\mathrm{Td}\text{/}$ in the channel, which
we suggested was due to the attachment instability. Photoionization was shown
to be negligible at the negative streamer head, but nonetheless had a major
impact on the streamer evolution since it affects the connection of the
negative streamer to the cathode.

###  4.3 Positive streamers

For positive streamers we obtained partial agreement with experiments [7]. The
reported streamer propagation field for positive streamers at
$1\text{\,}\mathrm{bar}\text{/}$ pressure is approximately
$1.21.3\text{\,}\mathrm{kV}\text{/}\mathrm{mm}$, whereas our baseline
calculations gave a propagation field of
$0.81\text{\,}\mathrm{kV}\text{/}\mathrm{mm}$. This discrepancy indicates that
we probably overestimated the amount of photoionization in our calculations,
which could either be due to disregard of collisional quenching, lack of
reliable photoionization data, or extrapolation of the available
photoionization data outside of the experimentally obtained range. Although we
were unable to answer which of these factors were incorrect, we found that
higher voltages were required in order to sustain positive streamer
propagation as we reduced the amount of photoionization.

The reported photoionization values by [24] are sufficient for sustaining
positive discharges in CO2 at $1\text{\,}\mathrm{bar}\text{/}$, despite the
fact that photoionization is weaker than in air and that the photons are
absorbed very close to the streamer head. The smallest observed positive
streamer radius in the computer simulations was approximately
$140\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}$, and streamers with this
radius did not branch, so they might correspond to the minimal-diameter
streamers observed by [7]. The positive streamers were slower than the
negative streamers, with typical velocities in the range
$0.20.4\text{\,}\mathrm{mm}\text{/}\mathrm{ns}$. This is contrary to behavior
in air where positive streamers propagate faster than negative streamers [35].
In CO2, simulations and experiments [7] both show that negative streamers are
faster than positive streamers.

###  4.4 Outlook

We discussed the lack of reliable photoionization data for CO2, which is in
contrast to air where the photoionization process is well identified, and even
simplified models provide sufficient accuracy [5]. Part of the reason for this
situation is the lack of energy-resolved emission cross sections for the
fragmented products that appear when CO2 molecules are dissociated through
electron impact. Experimentally, data is only available at low pressures and
it is not known if the reported data by [24] represents total photoionization,
or photoionization per steradian [2]. In the latter case, the photoionization
efficiency $\xi\mathopen{}\mathclose{{}\left(E/N}\right)$ that we used in this
paper needs to be multiplied by a factor of $4\pi$. The usage of our parameter
$\nu_{q}$ is then reinterpreted to include the factor of $4\pi$ and the role
of quenching. Lack of data on the role of collisional quenching of the EUV-
emitting fragments was artificially compensated for by reducing
photoionization levels by a factor of $10$ and $100$, respectively. We
observed that positive streamers could develop even at such low levels of
photoionization, but that their initiation required a higher applied voltage.
We speculate that this is at least partially the reason why experiments show
that positive polarity is the dominant breakdown mechanism in CO2 below
atmospheric pressure, while negative polarity dominates at higher pressure
[7].

## Acknowledgements

This study was partially supported by funding from the Research Council of
Norway through grant 319930. The computations were performed on resources
provided by UNINETT Sigma2 - the National Infrastructure for High Performance
Computing and Data Storage in Norway.

## Data availability statement

The calculations presented in this paper were performed using the chombo-
discharge computer code [12] (git hash 540f105). Although the results included
in this paper are stochastic, the input scripts containing simulation
parameters and chemistry specifications that were used in this paper are
available in both the chombo-discharge results repository at
https://github.com/chombo-discharge/discharge-papers/tree/main/ItoKMC-CO2, and
at the following URL/DOI: https://doi.org/10.5281/zenodo.10219863.

## References

  * [1] Sander Nijdam, Jannis Teunissen and Ute Ebert  “The physics of streamer discharge phenomena”  In _Plasma Sources Science and Technology_ , 2020  DOI: 10.1088/1361-6595/abaa05
  * [2] Sergey Pancheshnyi  “Photoionization produced by low-current discharges in O2, air, N2 and CO2”  In _Plasma Sources Science and Technology_ 24, 2015  DOI: 10.1088/0963-0252/24/1/015023
  * [3] B. Bagheri and J. Teunissen  “The effect of the stochasticity of photoionization on 3D streamer simulations”  In _Plasma Sources Science and Technology_ 28 {IOP} Publishing, 2019, pp. 45013  DOI: 10.1088/1361-6595/ab1331
  * [4] Robert Marskar  “3D fluid modeling of positive streamer discharges in air with stochastic photoionization”  In _Plasma Sources Science and Technology_ , 2020  DOI: 10.1088/1361-6595/ab87b6
  * [5] Zhen Wang et al.  “Quantitative modeling of streamer discharge branching in air”  In _Plasma Sources Science and Technology_ 32, 2023, pp. 085007  DOI: 10.1088/1361-6595/ace9fa
  * [6] T..P. Briels et al.  “Positive and negative streamers in ambient air: Measuring diameter, velocity and dissipated energy”  In _Journal of Physics D: Applied Physics_ 41, 2008  DOI: 10.1088/0022-3727/41/23/234004
  * [7] M. Seeger, J. Avaheden, S. Pancheshnyi and T. Votteler  “Streamer parameters and breakdown in CO2”  In _Journal of Physics D: Applied Physics_ 50, 2017  DOI: 10.1088/1361-6463/50/1/015207
  * [8] S. Mirpour and S. Nijdam  “Investigating CO2streamer inception in repetitive pulsed discharges”  In _Plasma Sources Science and Technology_ 31, 2022  DOI: 10.1088/1361-6595/ac6a0e
  * [9] Dmitry Levko, Michael Pachuilo and Laxminarayan L. Raja  “Particle-in-cell modeling of streamer branching in CO2 gas”  In _Journal of Physics D: Applied Physics_ 50, 2017  DOI: 10.1088/1361-6463/aa7e6c
  * [10] Behnaz Bagheri, Jannis Teunissen and Ute Ebert  “Simulation of positive streamers in CO2 and in air: The role of photoionization or other electron sources”  In _Plasma Sources Science and Technology_ 29, 2020  DOI: 10.1088/1361-6595/abc93e
  * [11] Robert Marskar  “Stochastic and self-consistent 3D modeling of streamer discharge trees with Kinetic Monte Carlo”, 2023  DOI: 10.48550/arXiv.2307.01797
  * [12] Robert Marskar  “chombo-discharge: An AMR code for gas discharge simulations in complex geometries”  In _Journal of Open Source Software_ 8, 2023, pp. 5335  DOI: 10.21105/joss.05335
  * [13] Daniel T. Gillespie  “Stochastic simulation of chemical kinetics”  In _Annual Review of Physical Chemistry_ 58, 2007  DOI: 10.1146/annurev.physchem.58.032806.104637
  * [14] G..M. Hagelaar and L.. Pitchford  “Solving the Boltzmann equation to obtain electron transport coefficients and rate coefficients for fluid models”  In _Plasma Sources Science and Technology_ 14, 2005, pp. 722–733  DOI: 10.1088/0963-0252/14/4/011
  * [15] “Phelps Database”, 2023  URL: www.lxcat.net/Phelps
  * [16] D.. Douglas-Hamilton and Siva A. Mani  “An electron attachment plasma instability”  In _Applied Physics Letters_ 23, 1973  DOI: 10.1063/1.1654978
  * [17] A. Luque, H.. Stenbaek-Nielsen, M.. McHarg and R.. Haaland  “Sprite beads and glows arising from the attachment instability in streamer channels”  In _Journal of Geophysical Research: Space Physics_ 121, 2016, pp. 2431–2449  DOI: 10.1002/2015JA022234
  * [18] Robert Marskar  “Genesis of column sprites: Formation mechanisms and optical structures”, 2023  DOI: 10.48550/arXiv.2310.08254
  * [19] A. Malagón-Romero and A. Luque  “Spontaneous Emergence of Space Stems Ahead of Negative Leaders in Lightning and Long Sparks”  In _Geophysical Research Letters_ 46, 2019  DOI: 10.1029/2019GL082063
  * [20] Pavlo Kochkin, Nikolai Lehtinen, Alexander P.J. Deursen and Nikolai Østgaard  “Pilot system development in metre-scale laboratory discharge”  In _Journal of Physics D: Applied Physics_ 49, 2016  DOI: 10.1088/0022-3727/49/42/425203
  * [21] P.. Kochkin, A..J. Deursen and U. Ebert  “Experimental study of the spatio-temporal development of metre-scale negative discharge in air”  In _Journal of Physics D: Applied Physics_ 47, 2014  DOI: 10.1088/0022-3727/47/14/145203
  * [22] J. Stephens, M. Abide, A. Fierro and A. Neuber  “Practical considerations for modeling streamer discharges in air with radiation transport”  In _Plasma Sources Science and Technology_ , 2018  DOI: 10.1088/1361-6595/aacc91
  * [23] I. Kanik, J.. Ajello and G.. James  “Extreme ultraviolet emission spectrum of CO2 induced by electron impact at 200 eV”  In _Chemical Physics Letters_ 211, 1993  DOI: 10.1016/0009-2614(93)80137-E
  * [24] A Przybylski  “Untersuchung über die gasionisierende Strahlung einer Entladung II”  In _Zeitschrift für Physik_ 168, 1962, pp. 504–515  DOI: 10.1007/BF01378146
  * [25] Peter L.. Ventzek, Robert J. Hoekstra and Mark Kushner  “Two-dimensional modeling of high plasma density inductively coupled sources for materials processing”  In _Journal of Vacuum Science and Technology B: Microelectronics and Nanometer Structures_ 12, 1994, pp. 461  DOI: 10.1116/1.587101
  * [26] Robert Marskar  “An adaptive Cartesian embedded boundary approach for fluid simulations of two- and three-dimensional low temperature plasma filaments in complex geometries”  In _Journal of Computational Physics_ 388, 2019, pp. 624–654  DOI: 10.1016/j.jcp.2019.03.036
  * [27] Robert Marskar  “Adaptive multiscale methods for 3D streamer discharges in air”  In _Plasma Research Express_ 1, 2019, pp. 015011  DOI: 10.1088/2516-1067/aafc7b
  * [28] S. Pancheshnyi, M. Nudnova and A. Starikovskii  “Development of a cathode-directed streamer discharge in air at different pressures: Experiment and comparison with direct numerical simulation”  In _Physical Review E - Statistical, Nonlinear, and Soft Matter Physics_ 71 American Physical Society, 2005, pp. 016407  DOI: 10.1103/PhysRevE.71.016407
  * [29] Alejandro Luque, Valeria Ratushnaya and Ute Ebert  “Positive and negative streamers in ambient air: Modelling evolution and velocities”  In _Journal of Physics D: Applied Physics_ 41, 2008  DOI: 10.1088/0022-3727/41/23/234005
  * [30] H..H. Meyer, R. Marskar, H. Gjemdal and F. Mauseth  “Streamer propagation along a profiled dielectric surface”  In _Plasma Sources Science and Technology_ , 2020  DOI: 10.1088/1361-6595/abbae2
  * [31] H K H Meyer, R Marskar and F Mauseth  “Evolution of positive streamers in air over non-planar dielectrics: experiments and simulations”  In _Plasma Sources Science and Technology_ 31, 2022, pp. 114006  DOI: 10.1088/1361-6595/aca0be
  * [32] Hans Kristian Meyer et al.  “Streamer and surface charge dynamics in non-uniform air gaps with a dielectric barrier”  In _IEEE Transactions on Dielectrics and Electrical Insulation_ 26, 2019, pp. 1163–1171  DOI: 10.1109/TDEI.2019.007929
  * [33] Alejandro Luque, Ute Ebert, Carolynne Montijn and Willem Hundsdorfer  “Photoionization in negative streamers: Fast computations and two propagation modes”  In _Applied Physics Letters_ 90, 2007  DOI: 10.1063/1.2435934
  * [34] A. Starikovskiy and N.. Aleksandrov  “How pulse polarity and photoionization control streamer discharge development in long air gaps”  In _Plasma Sources Science and Technology_ , 2020  DOI: 10.1088/1361-6595/ab9484
  * [35] T..P. Briels, E.. Veldhuizen and U. Ebert  “Positive streamers in air and nitrogen of varying density: Experiments on similarity laws”  In _Journal of Physics D: Applied Physics_ 41, 2008  DOI: 10.1088/0022-3727/41/23/234008