A computational study of positive streamers interacting with dielectrics

The fluid model used here is of the drift-diffusion-reaction type with the local field approximation [30]. The model keeps track of the electron density n_e, the positive ion density ${n}_{\mathrm{i}}^{+}$ and the negative ion density ${n}_{\mathrm{i}}^{-}$, which involve in time as

$$\begin{equation}\frac{\partial {n}_{\mathrm{e}}}{\partial t}=-\nabla \cdot {\mathbf{\Gamma }}_{\mathrm{e}}+{S}_{\mathrm{i}}-{S}_{\mathrm{a}}+{S}_{\mathrm{p}\mathrm{i}}+{S}_{\mathrm{s}\mathrm{e}},\end{equation} \tag{ 1 }$$

$$\begin{align}\hfill {\mathbf{\Gamma }}_{\mathrm{e}}& =-{n}_{\mathrm{e}}{\mu }_{\mathrm{e}}\mathbf{E}-{D}_{\mathrm{e}}\nabla {n}_{\mathrm{e}},\hfill \\ \hfill \frac{\partial {{n}_{\mathrm{i}}}^{+}}{\partial t}& =-\nabla \cdot {\mathbf{\Gamma }}_{\mathrm{i}}^{+}+{S}_{\mathrm{i}}+{S}_{\mathrm{p}\mathrm{i}},\hfill \end{align} \tag{ 2 }$$

$$\begin{align}\hfill {\mathbf{\Gamma }}_{\mathrm{i}}^{+}& ={{n}_{\mathrm{i}}}^{+}{\mu }_{\mathrm{i}}^{+}\mathbf{E},\hfill \\ \hfill \frac{\partial {{n}_{\mathrm{i}}}^{-}}{\partial t}& =-\nabla \cdot {\mathbf{\Gamma }}_{\mathrm{i}}^{-}+{S}_{\mathrm{a}},\hfill \end{align} \tag{ 3 }$$

$$\begin{equation*}{\mathbf{\Gamma }}_{\mathrm{i}}^{-}=-{{n}_{\mathrm{i}}}^{-}{\mu }_{\mathrm{i}}^{-}\mathbf{E},\end{equation*}$$

Here, fluxes are indicated by a Γ, μ_e is the electron mobility, D_e the electron diffusion coefficient, E the electric field, and ${\mu }_{\mathrm{i}}^{{\pm}}$ the positive/negative ion mobilities. Furthermore, several source terms are present. The electron impact ionization and electron attachment terms are given by S_i = αμ_e|E|n_e and S_a = ημ_e|E|n_e, respectively, where α and η are the ionization and attachment coefficients. The production of photoelectrons from photoionization is included with the term S_pi. Secondary electron emission due to the impact of both ions and photons is accounted for by the term S_se, defined below.

The local field approximation is used, so that μ_e, D_e, α and η are functions of the local electric field strength. Electron transport and reaction coefficients for air (1 bar, 300  K) were generated with Monte Carlo particle swarm simulations (see e.g. [31]), using Phelps' cross sections [32]. The positive ion mobility ${\mu }_{\mathrm{i}}^{+}=3{\times}1{0}^{-4} {\mathrm{m}}^{2} {\mathrm{V}}^{-1} {\mathrm{s}}^{-1}$ is here considered to be constant, but in section 3.5 it is varied to investigate its effect on surface discharges. For simplicity, the negative ion mobility is set to zero (${\mu }_{\mathrm{i}}^{-}=0$) throughout the paper.

We assume that electrons and ions attach to the surface when they flow onto a dielectric. They do not move or react on the surface, but secondary electron emission from the surface is taken into account. For the impact of positive ions, a SEE (secondary electron emission) coefficient γ_i is used. When a photon hits a dielectric surface, we assume that the photon is absorbed. A SEE photoemission coefficient γ_pe is used to determine the photoemission flux, see section 2.3. The effect of these SEE coefficients is studied in section 3.4, elsewhere they are set to zero. The secondary electron emission source term S_se in equation (1) is non-zero only in cells adjacent to the dielectric surface. In these cells, it is given by

$$\begin{equation}{S}_{\mathrm{s}\mathrm{e}}=-\nabla \cdot \left({\tilde {\mathbf{\Gamma }}}_{\mathrm{p}\mathrm{e}}-{\gamma }_{\mathrm{i}}{\tilde {\mathbf{\Gamma }}}_{\mathrm{i}}^{+}\right),\end{equation} \tag{ 4 }$$

where ${\tilde {{\Gamma}}}_{\mathrm{i}}^{+}$ is flux of positive ions onto the surface, and ${\tilde {{\Gamma}}}_{\mathrm{p}\mathrm{e}}$ is the photoemission flux coming from the surface. By definition, both these fluxes are non-zero only at the surface.

Secondary emission leaves behind positive surface charge on the dielectric. Therefore, the surface charge density σ_s changes in time as

$$\begin{equation}{\partial }_{\mathrm{t}}{\sigma }_{\mathrm{s}}=-e\left({\tilde {{\Gamma}}}_{\mathrm{e}}+{\tilde {{\Gamma}}}_{\mathrm{i}}^{-}\right)+e\left(1+{\gamma }_{\mathrm{i}}\right){\tilde {{\Gamma}}}_{\mathrm{i}}^{+}+e{\tilde {{\Gamma}}}_{\mathrm{p}\mathrm{e}},\end{equation} \tag{ 5 }$$

where e is the elementary charge and the other terms correspond to the fluxes onto the dielectric surface: ${\tilde {{\Gamma}}}_{\mathrm{e}}$ for electrons, ${\tilde {{\Gamma}}}_{\mathrm{i}}^{-}$ for negative ions, ${\tilde {{\Gamma}}}_{\mathrm{i}}^{+}$ for positive ions, and ${\tilde {{\Gamma}}}_{\mathrm{p}\mathrm{e}}$ for photons. We study positive streamers, which means that electrons generally move away from dielectrics. Therefore, an accurate description of the electron flux towards the surface [33] is not required here.

The electric field E is calculated by first solving Poisson's equation for the electric potential φ:

$$\begin{equation}\nabla \cdot \left(\varepsilon \nabla \varphi \right)=-\left(\rho +{\delta }_{\mathrm{s}}{\sigma }_{\mathrm{s}}\right),\end{equation} \tag{ 6 }$$

where ɛ is the dielectric permittivity, ρ is the volume charge density, and δ_s maps the surface charge σ_s on the gas–dielectric interface to the grid cells adjacent to the dielectric. Afterwards, the electric field is computed as

$$\begin{equation}\mathbf{E}=-\nabla \varphi .\end{equation} \tag{ 7 }$$

At the dielectric interface, we ensure that the normal component of the electric field satisfies the classic jump condition

$$\begin{equation}{\varepsilon }_{2}{E}_{2}-{\varepsilon }_{1}{E}_{1}={\sigma }_{\mathrm{s}},\end{equation} \tag{ 8 }$$

where ɛ₁ and ɛ₁ denote the permittivities on both sides of the interface, and E₁ and E₂ the electric field components normal to the interface.

Details about the numerical implementation, which is compatible with adaptive mesh refinement, will be presented in a forthcoming paper.

Positive streamer discharges need a source of free electrons ahead of them in order to propagate. Photons can generate such free electrons through photoionization in the gas or photoemission from a dielectric surface. In N₂–O₂ mixtures, non-local photoionization can take place when an excited nitrogen molecule emits a UV photon in the 98 to 102.5 nm range, which has enough energy to ionize an oxygen molecule. Photoionization often plays an important role in electrical discharges, see e.g. [34, 35].

The role of photoemission in surface discharges is less well understood. Photons can be emitted from several excited states. The probability of photoemission not only depends on the photon energy, but also on the surface properties [10]. For simplicity, we consider only two types of photons in this paper: high-energy photons, which can generate photoionization and photoemission, and low-energy photons, which can only contribute to photoemission and are not absorbed in the gas. These processes are illustrated in figure 1.

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Photoionization and photoemission are here modelled with a Monte Carlo (MC) method. We use the same Monte Carlo photoionization model as described in [36] and chapter 11 of [37]. The idea is to approximate the photoionization source term S_pi and the photon flux onto the dielectric ${\tilde {{\Gamma}}}_{\mathrm{p}\mathrm{e}}$ by randomly sampling discrete photons. For computational efficiency, these terms are updated every Δt_γ = 10Δt, where Δt is the time step used for solving equation (1).

First, the number of ionizing (high-energy) photons produced per grid cell during a time Δt_γ is determined. We use Zheleznyak's model [38], in which the number of ionizing photons is proportional to the number of impact ionization events. The corresponding proportionality factor is set to ξp_q/(p + p_q), where ξ = 0.05 is a numerical factor, p = 1 bar is the gas pressure and p_q = 40  mbar is the collisional quenching pressure. We remark that ξ should in principle depend on the electric field [38], but that it is here approximated by a constant, as in [36]. Per cell, a random number is drawn to determine how many photons are generated, see [37].

Since simulations are here performed in 2D, the discrete photons do not correspond to (single) physical photons. Instead, the total photon number N_photons is fixed, so that the MC method always uses 10⁵ photons. The weight factor w of these photons is given by w = ∫S_γdV/N_photons, where

$$\begin{equation}\int {S}_{\gamma }\mathrm{d}V=\xi {p}_{\mathrm{q}}/\left(p+{p}_{\mathrm{q}}\right)\int {S}_{\mathrm{i}}\mathrm{d}V,\end{equation} \tag{ 9 }$$

is the volume-integrated production rate of ionizing photons.

For simplicity, the number of produced low-energy photons is assumed to be equal to the number of high-energy photons. As the low-energy photons only contribute to photoemission, their effect can be controlled through the corresponding photoemission coefficient.

Second, an isotropically distributed direction is sampled for each photon. Afterwards, absorption lengths are determined. The absorption length of high-energy photons is sampled from the absorption function for air, see e.g. [36]. Low-energy photons are not absorbed by the gas, so their absorption length is set to a large value, making sure they always end up outside the computational domain.

Third, we determine which photons hit a dielectric surface, and where they do so. These photons are absorbed by the surface, where they contribute to the local photoemission flux ${\tilde {{\Gamma}}}_{\mathrm{p}\mathrm{e}}$. For the low-energy and high-energy photons, photoemission coefficients γ_peL and γ_peH are used, respectively. If a surface cell of area ΔA is hit by n_L and n_H low-energy and high-energy photons, then

$$\begin{equation}{\tilde {{\Gamma}}}_{\mathrm{p}\mathrm{e}}=\left({\gamma }_{\mathit{peL}}{n}_{\mathrm{L}}+{\gamma }_{\mathit{peH}}{n}_{\mathrm{H}}\right)w/{\Delta}A.\end{equation} \tag{ 10 }$$

The effect of photoemission is investigated in section 3.4; elsewhere in the paper photoemission is not taken into account (so that γ_peL = γ_peH = 0).

Fourth, the remaining high-energy photons that are absorbed in the gas contribute to photoionization source term S_pi. If n_γ photons are absorbed in a grid cell with volume ΔV, then

$$\begin{equation}{S}_{\mathrm{p}\mathrm{i}}={n}_{\gamma }w/{\Delta}V.\end{equation} \tag{ 11 }$$

Low-energy photons and high-energy photons that are absorbed outside the computational domain have no effect.

We use a parallel-plate electrode geometry with a flat dielectric in between, as shown in figure 2. The computational domain measures (40 mm)², and a dielectric is present on the left side with a width of 10  mm. The dielectric permittivity is set to ɛ = 2, but in section 3.3 it is varied to investigate its effect on surface discharges.

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As a gas, artificial air (80% N₂ and 20% O₂) at 1 bar and 300 K is used. For the electric potential, Dirichlet boundary conditions are applied at the upper and lower boundaries, and Neumann zero boundary conditions on the left and right side. A voltage of 100  kV is applied. In section 3.2, this voltage is varied. The background densities of electrons and positive ions are set to 10¹⁰ m⁻³ [39].

To start a discharge, the background field has to be locally enhanced. We do this by placing an ionized seed of about 2  mm long with a radius of about 0.4 mm. The electron and positive ion density are 5 × 10¹⁸ m⁻³ at the center, and they decay at distances above d = 0.2  mm with a so-called smoothstep profile: 1 − 3x² + 2x³ up to x = 1, where x = (d − 0.2  mm)/0.2  mm. When the electrons from a seed drift upwards, the electric field at the bottom of the seed is enhanced so that a positive streamer can form.

Previous experiments have revealed that dielectrics attract positive streamers, see e.g. [4, 5]. This attraction is also present in our numerical model. Figure 3 shows the evolution of the electron density for an initial ionized seed placed 1  mm away from the dielectric. It can be seen that the streamer starts to grow in air and then gradually develops towards the dielectric. After connecting with the dielectric, the streamer propagates down over its surface. The evolution shown here is in qualitative agreement with the discharge photographs in [4].

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To study the streamer–dielectric attraction, we have placed the initial ionized seed at different distances from the dielectric. Figures 4(a)–(d) show the electron density for seeds placed at 0.5  mm, 1  mm, 2  mm and 5  mm from the dielectric. For comparison, the electron density for a streamer in bulk gas is also shown in figure 4(e). In this simulation the dielectric was removed, so that the whole computational domain contained gas, and the initial seed was placed at x = 20  mm. Figure 4(a) shows the electron density at 15  ns, for the other cases, which develop more slowly, results at 20  ns are shown. The closer the streamer is located to the dielectric, the stronger the attraction to the dielectric becomes. It can also be seen that a nearby dielectric increases the streamer's velocity in the gas, and that streamers here propagate faster on the surface than in the gas.

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3.1.1. Attraction to the dielectric

As photoemission is disabled here (see table 1), the attraction of the streamer to the dielectric is purely electrostatic. The net charge in the streamer head polarizes the dielectric, which increases the electric field between the streamer and the dielectric. This effect is illustrated in figure 5, which shows the horizontal electric field (E_x) around the streamer heads. With a dielectric present, |E_x| increases on the dielectric side, and it is reduced on the other side. The closer the streamer is to a dielectric, the stronger this effect becomes. Eventually, the streamer will turn into a surface streamer. As shown in figure 4, such a surface streamer is thinner and quite asymmetric compared to a gas streamer.

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In past research, the attraction of streamers to dielectrics could often be explained by the enhanced static field between pointed electrodes and dielectrics, see e.g. [4, 6]. In contrast, the attraction to the dielectric is here due to the space charge from the streamer itself, since in our plate-to-plate geometry, the electric field has no horizontal component before a discharge is present. Assuming that streamers propagate approximately along electric field lines [40], we can therefore state that streamers lead themselves to the dielectric: their space charge modifies the background field and causes a horizontal field component that attracts them to the dielectric.

Surface charge on the dielectric could also play a role in attracting streamer discharges, by modifying the background electric field. However, there is negligible surface charge here, as secondary electron emission is disabled in this section and electrons move away from the dielectric. Positive ions do move towards the dielectric, but there are initially few of them near the dielectric, and they drift with a relatively low mobility, so that they hardly contribute to the surface charge.

3.1.2. Effect on streamer velocity

Several experimental studies have found that surface discharges are faster than bulk gas streamers [5, 8, 41]. Their increased velocity was attributed to increased ionization rates near the dielectric, accumulated negative charge and electron emission from the dielectric surface. Electron emission from the dielectric is not included here (it is in section 3.4), but we still find that the surface streamers are significantly faster. Figure 6 shows the streamer velocity and its maximal electric field for the case d = 1  mm. Note that both the velocity and the maximal electric field increase when the surface streamer forms, at around y = 35  mm. Even though the higher field is mostly in the horizontal direction, see figure 5, it still contributes to a faster vertical growth.

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As can be seen in figure 4, the electron density inside a surface streamer (∼10²⁰ to 10²¹  m⁻³) is higher than in a gas streamer (∼10¹⁹  m⁻³) when the discharge conditions are otherwise similar. This was also observed in (for example) [23]. We remark that besides the presence of a dielectric, the electron density inside a streamer discharge also depends on other factors, such as the applied voltage, the streamer radius, the gas composition, the amount of photoionization etc. Also note that we use a Cartesian 2D model, in which there is less field enhancement around the streamer heads than in full 3D, reducing the electron density in both surface and gas streamers.

There seem to be several related effects that lead to the increased surface streamer velocity. The strong electric field between a surface streamers and a dielectric pulls surface streamers towards the dielectric. This reduces their radius (as shown in figure 4), and results in an asymmetric streamer head shape, which also leads to stronger electric field enhancement. The result is that the ionization rate is increased, that the streamer has a higher degree of ionization, and that it propagates faster. This behavior is quite distinct from gas streamers, which typically propagate faster when they have a larger radius [42].

3.1.3. Cathode sheath

As shown in figure 7, the surface streamer 'hovers' over the dielectric surface without fully connecting to it. This phenomenon was also observed in simulations of dielectric barrier discharges [12, 43–45], and it only occurs for positive streamers. The reason is that positive streamers grow from incoming electron avalanches, but such avalanches require sufficient distance before they reach ionization levels comparable to the discharge. Positive streamers can therefore not immediately connect to the dielectric surface. Due to the net charge in the streamer head, a very high electric field is present in the narrow gap between streamer and dielectric. The effect of secondary electron emission on these dynamics is studied in section 3.4, and the role of the positive ion mobility is investigated in section 3.5.

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We remark that the maximum electric field of a positive surface streamer can rapidly rise to very high values, as shown in figures 6(a) and 7. These high-field areas typically contain a low electron density, but they still pose a problem for plasma fluid simulations. In the present fluid simulations, the transport and reaction coefficients (e.g., the ionization coefficient or the electron mobility) are functions of the local electric field strength. These coefficients are tabulated up to a certain maximum electric field, which is here 35  kV mm⁻¹; for higher fields, we use the tabulated value at 35  kV mm⁻¹. More generally, the validity of the local field approximation is questionable when there are such high electric fields (and corresponding strong gradients). For future studies in such ultra-high electric fields, particle-in-cell simulations could therefore be more suitable, as was also observed in [20]. Finally, we remark that a potential physical limitation for this maximum electric field is field emission of electrons from the surface.

To study the effect of the applied voltage on surface discharges, we have performed simulations for applied voltages of 92  kV, 100  kV and 112  kV, which correspond to background electric fields of 2.3  kV mm⁻¹, 2.5  kV mm⁻¹ and 2.8  kV mm⁻¹, respectively. In all cases, the initial seed was located at 0.5  mm from the dielectric, and the evolution up to 15  ns was simulated. Figure 8 shows the maximum electric field versus time, and figure 9 shows the electron density for three cases at 7  ns and 9  ns. Both figures reveal the following stages in the streamer's development:

• (I)  
  The inception stage, in which the maximum electric field is from 0 to about 9  kV mm⁻¹ in our setup and the streamer is hardly propagating, as shown in figure 9(a).
• (II)  
  The gas-propagation stage, in which streamers propagate in the gas with a maximum electric field below 12.5  kV mm⁻¹. This stage is visible in figures 9(a) and (b).
• (III)  
  The transition stage from a gas streamer to a surface streamer, in which the maximum electric field increases sharply. The streamer also loses its rounded head shape, as shown in figures 9(b) and (c).
• (IV)  
  The surface propagation stage. The growth of the maximum electric field is slowing down, and the streamer propagates along the dielectric, as shown in figure 9(c).

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Figure 8 shows that when the voltage is changed, the streamers still exhibit similar behavior in these four stages. The main difference is that the inception stage becomes shorter. For background fields of 2.3  kV mm⁻¹, 2.5  kV mm⁻¹ and 2.8  kV mm⁻¹, the inception stages last 8.5  ns, 5.5  ns and 3.75  ns, respectively. The second stage also becomes slightly shorter for a higher applied voltage.

Figure 10 shows the streamer velocities versus their vertical location for the three applied voltages. As expected, a higher background electric field leads to a higher streamer velocity for streamers of the same length, in agreement with the experimental results of [9].

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The relative permittivity ɛ of dielectric materials varies over a wide range. To study how ɛ affects surface discharges, we have performed simulations with ɛ set to 2, 3 and 5. As before, the initial seed was placed at 0.5  mm from the dielectric, and simulations were performed up to 15  ns.

The maximum electric field versus time for the three permittivities is shown in figure 11. The main difference we observe is that the second and third stages are shorter for a higher permittivity. A higher ɛ means the dielectric polarizes more strongly, which leads to a stronger attraction of streamers to the dielectric. Streamers therefore start the surface propagation stage earlier, and their maximum electric field increases more rapidly. Note that their maximum electric field is also higher during the surface propagation stage. In summary, we can conclude that a higher permittivity leads to a faster transition into a surface streamer, and a higher maximum field during the surface propagation stage.

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Figure 12 shows the electron density for these three cases when all the streamers are at y = 36  mm. It can clearly be seen that the streamers attach more rapidly to the dielectric with a higher permittivity. Notice also that the surface streamer's radius is smaller with a higher dielectric permittivity, a result of the stronger electrostatic attraction.

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Figure 13 shows the velocity versus the streamer's vertical position for the different ɛ_r. The permittivity has only a small effect on the streamer's velocity, in agreement with the experimental observations of [6]. In contrast, a negative correlation between the permittivity and the streamer velocity was found in [9]. The discrepancy could come from the different geometry that was used, in which multiple surface and gas streamers propagated next to a cylindrical dielectric.

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Electron emission from dielectrics may influence streamer velocities [46] and affect the high electric field in the dielectric–plasma gap [47]. In this section, we study how secondary electron emission affects surface streamers in our computational geometry. Both ion-induced secondary emission (ISEE) and photo-emission are considered.

3.4.1. Ion-induced secondary electron emission

3.4.2. Photoemission

The photo-emission coefficient γ_pe for typical dielectric materials varies between 10⁻⁴ to 10⁻¹ for photon energies of 5–20 eV [51, 52]. This yield can be higher if the material contains stains or defects, or when it is negatively charged [10, 48]. The measurement of γ_pe of dielectrics in air is often quite challenging [53]. We here use several values for γ_pe to demonstrate how photoemission affects positive streamers.

As discussed in section 2.3, we consider low-energy and high-energy photons, with the main distinction that high-energy photons can be absorbed in the gas. The following four cases are considered for the photoemission coefficients γ_peL and γ_peH for low-energy and high-energy photons:

• (a)  
  γ_peH = 0, γ_peL = 0.
• (b)  
  γ_peH = 0.5, γ_peL = 0.
• (c)  
  γ_peH = 0, γ_peL = 0.5.
• (d)  
  γ_peH = 0.5, γ_peL = 0.5.

In the simulations, streamers start near the dielectric, with the seed placed at either 0.5  mm or 1  mm from the dielectric.

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As shown in figure 15, photoemission by low-energy photons helps to start a discharge near a dielectric. At 5.5  ns, the γ_peL = 0.5 cases show the streamer already bending towards the dielectric with a sharp tip, due to photoemission. However, we remark that the inception time for these four cases is the same when the seeds are placed at 1  mm away from the dielectric. The secondary electrons from the dielectric then need more time to reach the streamer, and the streamers have already started due to the photoionization they generate. We conclude that photoemission can be important for discharges close to dielectrics and for discharges in gases with less photoionization than air.

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Figure 16(a) shows the electron density distribution for the above four cases at 11  ns. The streamers with γ_peL = 0.5 are longer than the other two, since they start earlier. Photoemission also causes them to attach to the dielectric more rapidly. Another difference is that the narrow gap between streamer and dielectric is smaller with more photoemission. This happens because photoemission provides seed electrons in the gap, which allows the streamer to get closer to the dielectric. To see this more clearly, the electron density distributions at y = 36.4  mm [the dashed line in figure 16(a)] are shown for these four cases in figure 16(b). Without photoemission, the electron density has a wider profile with a lower maximum, and it is located farther from the dielectric. When photoemission is included, the effect of the low-energy photons (i.e., γ_peL = 0.5) is most important here.

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Figure 17 shows the streamer velocities versus their vertical position for all four cases. The photoemission coefficients have only a small effect on the velocity, which is a little higher with γ_peL = 0.5. We think this is somewhat surprising. A possible explanation is that photoemission mostly leads to growth towards the dielectric, whereas photoionization in the gas contributes most of the free electrons that cause growth parallel to the dielectric. Another effect in the simulations presented here is that high-energy photons contribute less to a streamer's growth very close to a dielectric. There are two reasons for this. First, these photons are absorbed at shorter distances if they hit a dielectric. Second, their photoemission coefficient is here less than one (γ_peH = 0.5), whereas in the gas they always lead to photoionization.

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The positive ion mobility ${\mu }_{\mathrm{i}}^{+}$ can affect surface streamers in two ways. First, a higher ion mobility increases the amount of ion-induced secondary electron emission (ISEE). However, since ISEE was found to play a negligible role in section 3.4.1, its role is not further studied here, and we set the ISEE yield to zero (i.e., γ_i = 0).

A second effect is that a higher ion mobility increases the conductivity of the discharge, in particular in regions where the ion density is high compared to the electron density. For positive surface streamer discharges, such a region is present in the streamer–dielectric gap. This gap typically contains a high electric field, especially close to the streamer head, see section 3.1.3. Electrons rapidly drift away from the surface, whereas positive ions move from the high-density discharge region towards the surface, as illustrated in figure 18(a).

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To investigate how the positive ion mobility (${\mu }_{\mathrm{i}}^{+}$) affects the decay of the high electric field in the streamer–dielectric gap, we have performed simulations with positive ion mobilities of 0, 1 × 10⁻⁴  m² V⁻¹ s⁻¹, 5 × 10⁻⁴  m² V⁻¹ s⁻¹ and 1 × 10⁻³  m² V⁻¹ s⁻¹, using a seed placed 0.5  mm from the dielectric. For these simulations, we have recorded E_x in the middle of the gap at the point indicated in figure 18(a). The recorded fields are shown versus time in figure 18(b). The maximum electric field occurs when the streamer heads pass by the observation point indicated in figure 18(a). The decay of the peak in E_x is faster for higher ${\mu }_{\mathrm{i}}^{+}$, which is most clearly visible for the ${\mu }_{\mathrm{i}}^{+}=5{\times}1{0}^{-4} {\mathrm{m}}^{2} {\mathrm{V}}^{-1} {\mathrm{s}}^{-1}$ and ${\mu }_{\mathrm{i}}^{+}=1{\times}1{0}^{-3} {\mathrm{m}}^{2} {\mathrm{V}}^{-1} {\mathrm{s}}^{-1}$ cases. Note that the field also decays when the ions are immobile. This mainly happens because the amount of net space charge is lower behind the streamer head, but electron avalanches in the gap (due to e.g. photoionization) also contribute.