Electrically isolated propagating streamer heads formed by strong electron attachment

We used a drift–diffusion-reaction type fluid model to simulate positive streamers in artificial air, composed of 80% nitrogen and 20% oxygen, at standard temperature and pressure. Twelve reactions, listed in table 1, were considered: electron impact ionization (k₁, k₂), electron attachment (k₃, k₄), electron detachment (k₅, k₆), ion conversion (k₇–k₁₁), and electron–ion recombination (k₁₂). The temporal evolution of the electron density (n_e) is given by the continuity equation

$$\begin{equation}\frac{\partial {n}_{\mathrm{e}}}{\partial t}=\nabla \cdot \left({n}_{\mathrm{e}}{\mu }_{\mathrm{e}}\mathbf{E}+{D}_{\mathrm{e}}\nabla {n}_{\mathrm{e}}\right)+{S}_{\mathrm{i}}-{S}_{\eta }+{S}_{\mathrm{p}\mathrm{h}}+{S}_{\text{ion}},\end{equation} \tag{ 1 }$$

where E is the electric field, μ_e is the electron mobility, D_e is the electron diffusion coefficient, S_i is the electron impact ionization source term, S_η is the electron attachment source term, S_ph is the non-local photoionization source term, and S_ion contains all electron detachment reactions from the ions minus the electron–ion recombination reaction.

Table 1. Reactions included in the model, with reaction rate coefficients and references. M is any molecule, either O₂ or N₂. T_e in reaction number 12 depends on E/N through the mean electron energy $\left\langle {{\epsilon}}_{\mathrm{e}}\right\rangle$ as ${T}_{\mathrm{e}}=2\left\langle {{\epsilon}}_{\mathrm{e}}\right\rangle /3{k}_{\mathrm{B}}$ [24], and $\left\langle {{\epsilon}}_{\mathrm{e}}\right\rangle$ is calculated with Bolsig+.

Reaction No.	Reaction	Reaction rate coefficient	Reference
1	$\mathrm{e}+{\mathrm{N}}_{2} \rightarrow 2\mathrm{e}+{\mathrm{N}}_{2}^{+}$	${k}_{1}\left(E/N\right)$	[21]
2	$\mathrm{e}+{\mathrm{O}}_{2} \rightarrow 2\mathrm{e}+{\mathrm{O}}_{2}^{+}$	${k}_{2}\left(E/N\right)$	[21]
3	$\mathrm{e}+{\mathrm{O}}_{2}+{\mathrm{O}}_{2} \rightarrow {\mathrm{O}}_{2}^{-}+{\mathrm{O}}_{2}$	${k}_{3}\left(E/N\right)$	[21]
4	e + O2 → O + O−	${k}_{4}\left(E/N\right)$	[21]
5	$\mathrm{M}+{\mathrm{O}}_{2}^{-} \rightarrow \mathrm{e}+{\mathrm{O}}_{2}+\mathrm{M}$	${k}_{5}\left(E/N\right)=1.24{\times}1{0}^{-17}$ exp $\left(-{\left(\frac{179}{8.8+E/N}\right)}^{2}\right)\enspace {\mathrm{m}}^{3}\enspace {\mathrm{s}}^{-1}$	[22]
6	N2 + O−→e + N2O	${k}_{6}\left(E/N\right)=1.16{\times}1{0}^{-18}$ exp $\left(-{\left(\frac{48.9}{11+E/N}\right)}^{2}\right)\enspace {\mathrm{m}}^{3}\enspace {\mathrm{s}}^{-1}$	[22]
7	${\mathrm{O}}_{2}+{\mathrm{O}}^{-} \rightarrow {\mathrm{O}}_{2}^{-}+\mathrm{O}$	${k}_{7}\left(E/N\right)=6.96{\times}1{0}^{-17}$ exp $\left(-{\left(\frac{198}{5.6+E/N}\right)}^{2}\right)\enspace {\mathrm{m}}^{3}\enspace {\mathrm{s}}^{-1}$	[22]
8	${\mathrm{O}}_{2}+{\mathrm{O}}^{-}+\mathrm{M} \rightarrow {\mathrm{O}}_{3}^{-}+\mathrm{M}$	${k}_{8}\left(E/N\right)=1.10{\times}1{0}^{-42}$ exp $\left(-{\left(\frac{E/N}{65}\right)}^{2}\right)\enspace {\mathrm{m}}^{6}\enspace {\mathrm{s}}^{-1}$	[22]
9	${\mathrm{N}}_{2}^{+}+{\mathrm{N}}_{2}+\mathrm{M} \rightarrow {\mathrm{N}}_{4}^{+}+\mathrm{M}$	k9 = 5.0 × 10−41 m6 s−1	[23]
10	${\mathrm{N}}_{4}^{+}+{\mathrm{O}}_{2} \rightarrow 2{\mathrm{N}}_{2}+{\mathrm{O}}_{2}^{+}$	k10 = 2.50 × 10−16 m3 s−1	[23]
11	${\mathrm{O}}_{2}^{+}+{\mathrm{O}}_{2}+\mathrm{M} \rightarrow {\mathrm{O}}_{4}^{+}+\mathrm{M}$	k11 = 2.40 × 10−42 m6 s−1	[23]
12	$\mathrm{e}+{\mathrm{O}}_{4}^{+} \rightarrow 2{\mathrm{O}}_{2}$	${k}_{12}\left(E/N\right)=1.40{\times}1{0}^{-12}{\left(300\enspace \mathrm{K}/{T}_{\mathrm{e}}\right)}^{1/2}\enspace {\mathrm{m}}^{3}\enspace {\mathrm{s}}^{-1}$	[24]

The impact ionization source term S_i, the electron attachment source term S_η , and the photoionization source term S_ph contain reactions of electrons with neutral molecules. These are the relevant plasma-chemical reactions in the ionization zone ahead of the streamer. They are calculated using the rate coefficients given in table 1:

$$\begin{equation}{S}_{\mathrm{i}}={k}_{1}{n}_{\mathrm{e}}\left[{\mathrm{N}}_{2}\right]+{k}_{2}{n}_{\mathrm{e}}\left[{\mathrm{O}}_{2}\right],\end{equation} \tag{ 2 }$$

$$\begin{equation}{S}_{\eta }={k}_{3}{n}_{\mathrm{e}}{\left[{\mathrm{O}}_{2}\right]}^{2}+{k}_{4}{n}_{\mathrm{e}}\left[{\mathrm{O}}_{2}\right].\end{equation} \tag{ 3 }$$

Note that we do not include the three-body attachment reaction, $\mathrm{e}+{\mathrm{O}}_{2}+{\mathrm{N}}_{2}\to {\mathrm{O}}_{2}^{-}+{\mathrm{N}}_{2}$, in our model because its reactions rates are about three orders of magnitude smaller than k₃ [24].

S_i and S_η are linear in the electron density n_e. As the degree of ionization at standard pressure and temperature and below is small within a streamer, the densities of neutral molecules [O₂] and [N₂] can be assumed to be constant. Therefore the impact ionization and attachment reactions can be written as the product of the electron flux μ_e En_e and the coefficients α and η respectively [3]. The effective ionization source term can then be defined as

$$\begin{equation}{S}_{\mathrm{i}}-{S}_{\eta }=\left(\alpha -\eta \right)\enspace {\mu }_{\mathrm{e}}E\enspace {n}_{\mathrm{e}},\hspace{15.0pt}{\alpha }_{\text{eff}}=\alpha -\eta .\end{equation} \tag{ 4 }$$

The sign of α_eff determines whether the local electron density increases or decreases in the streamer ionization front.

The photoionization source term S_ph contributes only a small correction to the local reaction rates. However, it is very important due to its nonlocal nature, which allows for the liberation of electrons in the non-ionized region. It is given by

$$\begin{equation}{S}_{\mathrm{p}\mathrm{h}}\left(\mathbf{r}\right)=\int {\mathrm{d}}^{3}{r}^{\prime }\enspace \frac{I\left({\mathbf{r}}^{\prime }\right)f \left(\vert \mathbf{r}-{\mathbf{r}}^{\prime }\vert \right)}{4\pi \vert \mathbf{r}-{\mathbf{r}}^{\prime }{\vert }^{2}},\end{equation} \tag{ 5 }$$

where $I\left(\mathbf{r}\right)$ is the source of ionizing photons, f(r) is the absorption function, and 4π|r − r'|² is a geometric factor. We employed the commonly used Zheleznyak's model [25], where $I\left(\mathbf{r}\right)$ is proportional to the electron impact ionization source term (S_i)

$$\begin{equation}I\left(\mathbf{r}\right)=\frac{{p}_{\mathrm{q}}}{p+{p}_{\mathrm{q}}}\xi {S}_{\mathrm{i}}\left(\mathbf{r}\right).\end{equation} \tag{ 6 }$$

Here p is the actual gas pressure, p_q = 40 mbar is a gas-specific quenching pressure, and ξ is a proportionality factor that was set to ξ = 0.075. We approximated the integral in equation (5) using a set of Helmholtz differential equations [26, 27] with Bourdon's three-term parameters [27]. In addition to the original papers [26, 27], the reader is also referred to the appendix of [28] for more details.

In the ionized region inside the streamer, reactions involving ions are significant as well. The relevant terms affecting the electron density are

$$\begin{equation}{S}_{\text{ion}}={k}_{5}\left[\mathrm{M}\right]\left[{\mathrm{O}}_{2}^{-}\right]+{k}_{6}\left[{\mathrm{N}}_{2}\right]\left[{\mathrm{O}}^{-}\right]-{k}_{12}{n}_{\mathrm{e}}\left[{\mathrm{O}}_{4}^{+}\right],\end{equation} \tag{ 7 }$$

where [M] is the density of all neutral molecules: [M] = [N₂] + [O₂].

Other species Z_i in our model are ionized or non-ionized molecules or atoms. According to table 1, these are ${\mathrm{N}}_{2}^{+}$, ${\mathrm{N}}_{4}^{+}$, ${\mathrm{O}}_{2}^{+}$, ${\mathrm{O}}_{4}^{+}$, O⁻, ${\mathrm{O}}_{2}^{-}$, ${\mathrm{O}}_{3}^{-}$, and the neutrals O and N₂O. Their densities are also calculated from a continuity equation

$$\begin{equation}\frac{\partial \left[{Z}_{\mathrm{i}}\right]}{\partial t}=-{s}_{\mathrm{i}}\nabla \cdot \left(\left[{Z}_{\mathrm{i}}\right]{\mu }_{\mathrm{i}}\mathbf{E}\right)+{S}_{{Z}_{\mathrm{i}}},\end{equation} \tag{ 8 }$$

where s_i = ±1 is the sign of the electric charge of species i and μ_i is the ion mobility. The neutral species are treated as immobile. The drift of ions is neglected in most sections of the paper as their mobility is much smaller than that of electrons, but in section 3.5, we include the ion drift and discuss its effect on the previous results. All ion mobilities are assumed to be 2.2 ×  10⁻⁴ m² V⁻¹ s⁻¹ [29] in that set of simulations. As ${\mathrm{O}}_{3}^{-}$, O, and N₂O do not drive further reactions in our model, their densities do not need to be calculated in practice, except to account for the contribution of the ${\mathrm{O}}_{3}^{-}$ ion to the space charge density.

The electric potential and the electric field are calculated as

$$\begin{equation}{\nabla }^{2}\phi =-\frac{\rho }{{{\epsilon}}_{0}},\hspace{25.0pt}\mathbf{E}=-\nabla \phi ,\end{equation} \tag{ 9 }$$

where ₀ is the vacuum permittivity, $\rho =e\left({n}_{\mathrm{i}}-{n}_{\mathrm{e}}\right)$ is the space charge density, e is the elementary charge, and n_i is the density of all positive ions minus the density of all negative ions.

Most reaction rate coefficients in table 1 are a function of the reduced electric field E/N. Electron-neutral scattering cross sections for nitrogen and oxygen are taken from the Phelps database [21]. Bolsig+ [30] (version 03/2016 for Linux), an electron Boltzmann equation solver, was used to calculate the mean electron energy, transport coefficients μ_e and D_e, and the reaction rates for electron impact ionization and electron attachment (i.e. k₁, k₂, k₃, and k₄). Bolsig+ solved the Boltzmann equation under the assumption that the electron density grows exponentially in space without time dependence.

To identify the effect of strong attachment in regions below the breakdown field, the effective ionization coefficient was modified in such a way that the attachment rate in high electric field regions is unchanged and it is only enhanced in regions below the electric breakdown value. The photoionization rate is made unaffected by the changes in the coefficients.

This is accomplished in two steps. First, the effective ionization coefficient was multiplied by a factor m below the electric breakdown field E_c, which is about 28 kV cm⁻¹ in our case.

$$\begin{equation}{\alpha }_{m}\left(E\right)=\begin{cases}_{\text{eff}}\left(\vert \mathbf{E}\vert \right)\quad \hfill & \text{when}\;\vert \mathbf{E}\vert {\geqslant}{E}_{\mathrm{c}},\hfill \\ m\cdot {\alpha }_{\text{eff}}\left(\vert \mathbf{E}\vert \right)\quad \hfill & \text{when}\;\vert \mathbf{E}\vert {< }{E}_{\mathrm{c}}.\hfill \end{cases}\end{equation} \tag{ 10 }$$

This translates to multiplying reaction rate coefficients k₁, k₂, k₃, and k₄ with the factor m when used in regions below the breakdown field. Above the breakdown field, the effective ionization coefficient is unchanged. Figure 1 shows the reduced effective ionization coefficient α_m /N as a function of the reduced electric field strength E/N for different values of m. These so-called reduced quantities are scaled with the gas density N. The reduced effective ionization coefficient for air at 26 bar is included for comparison, and we see that its values match the m = 26 case for reduced electric fields below 30 Td, where three body attachment is dominating. As we describe later in section 3, the reduced electric fields inside streamer channels are found to be below 30 Td. Furthermore, the attachment rate in the m = 37 case, which has the highest attachment rate in this study, is still about one order of magnitude smaller than the respective rate in SF₆ at standard temperature and pressure [21, 31].

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Second, the photoionization source term (6) was set to continue to use the unmodified ionization coefficient α.

The model is implemented in afivo-streamer [32, 33], which employs geometric multigrid techniques to solve Poisson's equation and OpenMP parallelism. We simulate cylindrically symmetric positive streamers in a volume spanned by radius r and axis z. The domain has a length of 50 mm and a radius of 50 mm. Adaptive mesh refinement is employed with the grid set to have a minimum size of 2 μm. The refinement and derefinement criteria are based on the local electric field value as in [32] with an additional criterion based on the charge density: refine if α(1.2 × E)Δx > 0.5 and derefine if $\frac{\alpha \left(1.2{\times}E\right)}{1.2}{\Delta}x{< }6.25{\times}1{0}^{-2}$ and $\vert \frac{{{\epsilon}}_{0}\cdot 4.5{\times}1{0}^{10}}{\rho }\vert { >}0.5$, where α(E) is the field-dependent ionization coefficient, E is the electric field strength, and Δx is the grid spacing. We apply a homogeneous background electric field by fixing the electric potential at z = 0 mm and z = 50 mm. Neumann zero boundary conditions are applied to the electric potential at r = 0 mm and r = 50 mm, and to the electron density at all boundaries. The electric field points in $-\hat{z}$ direction with a magnitude of 15 kV cm⁻¹, which is about half of the breakdown field. No background ionization is incorporated in the domain.

To initiate a positive streamer, we placed two neutrally charged cylindrical seeds on the axis of symmetry. The first seed serves as the initiation point of the streamer. It is 1 mm long with 2.25 × 10²⁰ m⁻³ electrons and positive ions and has a width of 0.25 mm. It extends from z = 50 mm to 49 mm. The second seed supplies additional electrons during initiation until photoionization provides sufficient electrons for continued streamer propagation. This seed is 2 mm long with 1.0 × 10¹⁷ m⁻³ electrons and positive ions and has a width of 0.2 mm. It extends from z = 49 mm to 47 mm. The seeds decay with a Gaussian profile along the r-axis throughout their length and radially decay with a Gaussian profile in all directions at the caps.

For air, which is the case of m = 1 in equation (10), a streamer initiates in such setup. However, for strongly enhanced attachment conditions with large values of m, a streamer does not successfully initiate. Therefore, we first ran our simulation with the normal attachment rate m = 1 for 20 ns to get a streamer and then take this stage, that is illustrated in figure 2, as the initial condition for all runs with values of m from 1 up to 37. At this stage, a streamer has clearly formed and grown to a length of about 3.53 mm and a radius of 153 μm.

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Figures 3 and 4 show the evolution of the streamer with m = 1 on the left hand side and the streamer with m = 26 on the right hand side. For m = 26, electron attachment is strongly enhanced when the electric field is below the breakdown value, while it is the same as in air when the field is above breakdown. The background field is 15 kV cm⁻¹, which is about half of the breakdown field. The evolution of the m = 1 streamer is shown in time steps of 15 ns, and the evolution of the m = 26 streamer is shown in time steps of 30 ns. The m = 1 streamer propagates over a length of 30.4 mm within 45 ns, and the m = 26 streamer propagates over a length of 22.8 mm within 90 ns.

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The panels in figures 3 and 4 show the electron number density n_e, the number density of all negative ions $\left[{\mathrm{O}}^{-}\right]+\left[{\mathrm{O}}_{2}^{-}\right]+\left[{\mathrm{O}}_{3}^{-}\right]$, the space charge density ρ/e, and the electric field strength E = |E| with white equipotential lines ϕ. Within each row, the same color coding is used for the respective densities and the field strength. The figures zoom into the region r ⩽ 3 mm and 15 mm ⩽ z ⩽ 50 mm. The overall simulation volume extends to 50 mm in the r direction and 50 mm in the z direction.

3.1.1.  m = 1

3.1.2.  m = 26

In figure 5, the maximal electric field E_max, streamer velocity v, and streamer radius R for different values of m are plotted as a function of streamer length L. The plot finishes at a length of 40 mm, while the electrode is at z = 50 mm. Hence, effects of electrode proximity are not shown. The streamer length is defined as 50 mm − z_max, where z_max is the location of the maximum electric field. As earlier in [28], the radius is measured as the location where the radial component E_r of the electric field is maximal.

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Three different cases of streamer growth can be distinguished in figure 5:

• Accelerating streamers. For the case of artificial air (m = 1) as shown in figures 3 and 4, the streamer radius R increases with streamer length L from 0.146 to 1.00 mm and the streamer velocity v from 2.20 × 10⁵ m s⁻¹ to 1.12 × 10⁶ m s⁻¹, while the maximal electric field E_max at the streamer head decreases slightly from 158 kV cm⁻¹ to 116 kV cm⁻¹. The same tendency of increasing velocity and radius and decreasing maximal field as a function of streamer length is seen for all values of the attachment parameter m up to 26, though the rates of increase or decrease diminish with growing m.
• Uniformly translating streamers. The case of m = 26, that is also displayed in figures 3 and 4, is a limiting case: radius, velocity, and maximal field stay approximately constant while the streamer propagates. It is remarkable that this streamer head is dynamically stable, and that the motion is really a uniform translation where the background field is restored after the streamer head has passed and the nonconducting state is re-established.
• Decelerating and stagnating streamers. For m > 26, radius and velocity decrease with growing streamer length while the maximal electric field at the streamer tip grows very rapidly. In fact, the simulation stops, because the increasing electric field requires a too small numerical time step. Due to the CFL condition [33], the numerical time step has gone down to less than 10⁻¹⁴ s from 10⁻¹² s with the maximal electric field suddenly reaching to values greater than 300 kV cm⁻¹ in a limited region ahead of the streamer. From an investigation on negative streamers in [34], the local field approximation was found to no longer be valid at electric field values above 200 kV cm⁻¹ at 1 bar, and this limit is greatly exceeded by our stagnating streamers before the simulations stop.

The way how the positive streamers for m > 26 decelerate, and eventually stop, is reminiscent of earlier observations of stagnating and 'dying' positive streamers [35, 36].

Now, streamers of 30 mm length for different values of m are analyzed in more detail; more precisely, streamers that have propagated from z = 46.5 mm up to z = 20 mm. This occurs after a time of 44.75 ns for m = 1, after 87.25 ns for m = 17, and after 125 ns for m = 26. The cases of m = 30 and 37 are excluded, as these streamers do not reach this length.

Figure 6 zooms into the region 18 mm ⩽ z ⩽ 24 mm, which is around the streamer head. It shows the electric field and electron density on the streamer axis for different values of m.

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The electric field in the streamer front shows the familiar structure: the thin and weakly curved space charge layer at z = 20 mm (see figure 4) causes the field to jump from a highly enhanced value to a quite low one. Behind the space charge layer in the streamer interior (i.e. for larger z), the electric field is largely screened. The decay length of the electric field ahead of the space charge layer (i.e. for smaller z) is determined by the radius of curvature of the space charge layer. The maximum of the field is determined dynamically. It occurs to be higher for the slower and thinner streamers with higher m values.

The electron density is shown in the lower panel of figure 6 on a linear scale in the streamer head region. Where the electric field is maximal, the electron density is growing rapidly, almost discontinuously, in particular, for large m. The second panel of figure 7 shows the electron density on a logarithmic scale over a wider spatial range for m = 1, 17, and 26. Ahead of the front in the range of z < 17 mm, the electron density grows exponentially in space towards the front region, like exp [z/ℓ], with the same length scale ℓ = 1.1 mm, according to a fit in the range of z = 4 to 15 mm, for all three values of m. An analytical estimate shows that this length scale ℓ should be set by the largest photon absorption length in the photoionization model, and this length is indeed 1.2 mm [27, 28]). In the high field region at z ≈ 20 mm, the electron density increases steeply as in the linear plot of figure 6.

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Behind the ionization region, the electron density is nearly constant for m close to 1, but it decreases for larger m with growing distance behind the front. For m = 17 and m = 26, it saturates to about 10¹⁷ m⁻³ at z > 33 mm and z > 25 mm respectively, which is the result of attachment and detachment reactions compensating each other. By inspecting the density of the ionic species in the streamer channel for the three cases shown in the third, fourth, and fifth panels of figure 7, we observe that the densities of ${\mathrm{O}}_{2}^{+}$, ${\mathrm{N}}_{2}^{+}$, and ${\mathrm{N}}_{4}^{+}$ differ significantly in m = 1, 17, and 26, whereas the densities of the other ionic species are quite similar. For m = 17 and m = 26, the density of ionic species also approach an identical constant value for both cases.

Surprisingly, the maximum value of the electron density generated in the ionization front is highest for the largest m, i.e. when the electron attachment rate in the region below breakdown is highest. Common estimates relate the maximal electron density behind the front to the maximal electric field at the front [34, 37, 38] though this relation somewhat deviates from numerical observations for positive streamers with photoionization [39], as discussed in section 3.4 of [3]. The present observation confirms that a higher maximal field creates a higher electron density behind the ionization front.

Figure 7 shows the electric field, electron density, and ion species on the streamer axis throughout the whole ionized region. Here, the streamer heads have arrived at z = 20 mm as before.

The upper plot shows the electric field on axis, and three constant values are inserted for reference: the breakdown field of 28 kV cm⁻¹ (where α_eff = 0), the applied background field of 15 kV cm⁻¹, and the field of 5 kV cm⁻¹ that is frequently observed in the interior of single positive streamers in air and interpreted as the so-called stability field. Far ahead of the streamers at z → 0, the electric field essentially has the background value, and then it increases strongly towards the streamer head, as described above. Behind the ionization front in the streamer interior, the electric field approaches 5 kV cm⁻¹ for m = 1, while it approaches the background field of 15 kV cm⁻¹ in the cases of m = 17 and 26. The axis range of 45 mm < z < 50 mm shows remainders of the initiation process and is not discussed further.

The different interior fields of 5 or 15 kV cm⁻¹ are related to the different interior conductivities of the streamers. While for m = 1, there is a sufficient electron density to screen the electric field from the streamer channel, this is not the case for the middle and back regions of the streamers with m = 17 or 26 as we will discuss now.

The simulation results for electron attachment and ion conversion are shown in the lower three panels of figure 7. The reactions listed in table 1 involve electrons, neutrals, the positive ions ${\mathrm{N}}_{2}^{+}$, ${\mathrm{O}}_{2}^{+}$, ${\mathrm{N}}_{4}^{+}$, ${\mathrm{O}}_{4}^{+}$, and the negative ions O⁻, ${\mathrm{O}}_{2}^{-}$, ${\mathrm{O}}_{3}^{-}$. As described in [23], all positive ions in air rapidly convert to ${\mathrm{O}}_{4}^{+}$. This statement is confirmed by the present simulations, where the densities of the other positive ions quickly decrease behind the ionization front.

For m = 1, the electrons created in the ionization front attach to oxygen mainly through three-body attachment, hence the ${\mathrm{O}}_{2}^{-}$ density increases behind the moving front in about the same manner as the electron density decreases. Dissociative attachment creating O⁻ has no substantial contribution at standard temperature and pressure. As ${\mathrm{O}}_{3}^{-}$ is produced in collisions of O₂ with O⁻, its concentration stays low as well. It is quite a stable ion in air. There is a rapid trail of positive ion conversion within the streamer head, from ${\mathrm{N}}_{2}^{+}$ to ${\mathrm{N}}_{4}^{+}$ to ${\mathrm{O}}_{2}^{+}$. This is followed by the fast conversion of ${\mathrm{O}}_{2}^{+}$ into ${\mathrm{O}}_{4}^{+}$ in the streamer channel.

For m = 17 and 26, the ionization and attachment reaction rates k₁ to k₄ are enhanced by a factor m in the region below the breakdown field. This concerns the complete streamer channel. Accordingly, the electron density decreases rapidly behind the ionization front, while the ${\mathrm{O}}_{2}^{-}$ density increases in about the same manner as the electron density decreases. For both values of m, an electrically neutral streamer channel is formed, consisting mainly of ${\mathrm{O}}_{4}^{+}$ and ${\mathrm{O}}_{2}^{-}$ ions while all other ion densities are at least an order of magnitude smaller. Note that in figure 7, the yellow line that represents ${\mathrm{O}}_{2}^{-}$ mostly coincides with the green line that represents ${\mathrm{O}}_{4}^{+}$ for m = 17 and 26. As ion mobility is much smaller than electron mobility and neglected in the present model, this streamer channel cannot screen the electric field anymore and the field returns to the background value.

Since electrons are quickly attaching and disappearing behind the streamer head for large values of m, the electric current due to the electrons is significantly decreasing along the streamer channel. Thus, ionic conductivity could possibly become relevant, and we investigate that here. We performed streamer simulations which included ion motion through the drift term in equation (8) with the finite ion mobility given in section 2.1.

The electronic conductivity was calculated with σ_e = en_e μ_e and ${\sigma }_{\mathrm{i}}={\sum }_{i}\;e\left[{Z}_{i}\right]{\mu }_{i}$ was used for calculations of the ionic conductivity. From the conductivities, the current densities j_e and j_i, for the electrons and the ions respectively, were calculated using j_e = σ_e E and j_i = σ_i E. Figure 8 presents the electronic and ionic current densities on the streamer axis for cases m = 1, 17, and 26 when the heads have arrived at z = 20 mm as in the previous section.

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For m = 1, the electronic current density is nearly 3 orders of magnitude higher than the ionic current density, rendering the effect of the ions to the total current negligible. For m = 17 and m = 26, at about z = 34 mm and 26 mm behind the ionization front respectively, the ionic current density starts to get larger than the electronic current density. This ionic current density is of the order of 1 kA m⁻² for all values of m considered. Here, the densities of ${\mathrm{O}}_{2}^{-}$ and ${\mathrm{O}}_{4}^{+}$ are dominating in the increased attachment cases. Due to approximate charge neutrality in the streamer channel, the electric currents of positive and negative ions contribute about equally. The ionic current densities do not have a significant effect on the streamer behavior.

We have simulated and analyzed single positive streamers with photoionization in a constant electric field below the breakdown value. To study systematically the effect of electron attachment and of the subsequent loss of channel conductivity on the streamer dynamics, we performed simulations in artificial air (N₂:O₂ = 80:20) at standard temperature and pressure, and then we modified the effective ionization coefficient such that electron attachment is increased in the region below breakdown, while we kept all other parameters the same. Our main conclusions are:

• (a)  
  A streamer head can keep propagating even if the ionized channel behind it loses its conductivity due to rapid electron attachment.
• (b)  
  Depending on parameters, the streamer can be accelerating or decelerating. Between these parameter regimes, the streamer can propagate uniformly, i.e. with unchanged velocity and spatial structure. In this case, the electrically isolated streamer head carries a fixed amount of positive electric charge, and the electric field behind it returns to its background value.
• (c)  
  It is remarkable that this uniform translation is dynamically stable, at least for the duration of our simulations. This illustrates that the streamer head is a coherent structure like a solitary wave, created by the nonlinear interaction between ionization reaction, electron motion, and local electric fields.
• (d)  
  If attachment is too strong in a given electric field, the streamer radius and velocity decrease while the electric field and the charge carrier density increase rapidly. This dynamics is reminiscent of the stagnation dynamics described in [35], but in that reference the streamer channel stayed conductive.

We remark that the electrically isolated streamer heads found here should not be confused with the glowing heads of propagating streamers as the glow only indicates the regions with a strong ionization reaction and not the conducting regions in a streamer. These should not be confused either with the isolated head model [40, 41], which ignores the existence of the streamer channel.

The behavior of the streamer in the m = 26 case could be related to an older definition of the streamer stability field. Before the stability field was used in relation to the minimal voltage needed for a streamer to travel a certain distance, it was defined as the homogeneous electric field where a streamer propagates without any changes in velocity and shape [42–44]. In the m = 26 case, the streamer can apparently propagate indefinitely in a field of 15 kV cm⁻¹ in a stable manner. It could be claimed that 15 kV cm⁻¹ is the stability field of the m = 26 streamer following its original definition. For larger m, the streamer length is limited.

Finally, the relation between radius, velocity, and maximal electric field at the streamer head, and of electron density and electric field behind the streamer ionization front should be analyzed further. The charge balance between different parts of the streamer requires further analysis as well. In this context, model reductions for uniformly translating streamers given in [45, 46] and reviewed in [3] should be checked carefully, both on the underlying assumptions and calculations, and in comparison with simulations. These questions will be addressed in future papers, in particular, in view of deriving reduced models for streamer trees [47].