#
Modelling of Positive Streamers in SF_{6} Gas under Non-Uniform Electric Field Conditions: Effect of Electronegativity on Streamer Discharges

^{1}

^{2}

^{*}

## Abstract

**:**

_{6}in electrical insulation and fast-switching applications cannot be overemphasized. This is due to its excellent dielectric properties and high breakdown voltage, which are especially important for practical applications such as gas-insulated switchgears and pulsed power switches where pressurized SF

_{6}is used. Breakdown in the gas occurs via streamer–leader transition; however, this transition is difficult to quantify numerically at atmospheric pressure because of the electronegativity of the gas. In the present work, streamer discharges in SF

_{6}gas at pressures of 10 and 100 kPa were studied using a plasma fluid model implementation. Analysis of the electric field in the streamer body, streamer velocity, diameter, and the effect of the high electronegativity of the gas on streamer parameters are presented for positive polarity in a point-to-plane geometry. The streamers in SF

_{6}for non-uniform background fields are compared to those in air, which have already been studied extensively in the literature.

## 1. Introduction

_{6}gas is used for different HV applications, including gas-insulated switchgear (GIS, circuit breakers, switches) and gas-insulated transmission lines (GIL), because of its excellent dielectric and arc quenching properties [1,2]. Reference data relating to the breakdown properties of the gas in these applications have been extensively investigated and presented in the literature. Notably, it has also been widely adopted in fast-breaking plasma-closing switches for pulsed power applications because the switching element exerts a strong influence on the rising time and amplitude of the pulse [3,4]. These switches generally work under high voltage and current conditions and are prone to breakdown and discharge processes both thermal and non-thermal because of the extreme operational conditions and thus require a careful consideration of all discharge phases [5] (and references there-in after).

_{6}is required. For the present contribution, the main focus is placed on the numerical modelling of fast-transient plasma discharge processes—streamers, their evolution in SF

_{6}gas, and how that compares with a reference gas. A comparative numerical study has been carried out on the plasma discharge behaviour in air and SF

_{6}in conditions relating to pulsed power applications: short separation distances, high instantaneous voltage application and non-uniform configuration, to minimize the time delay for discharge formation. The upper bounds of pressure, especially for SF

_{6}and its influence on discharge behaviour, has also been analyzed, highlighting major physical concepts and some challenges in modelling of streamer discharges at 1 bar in the gas.

_{6}under non-uniform fields is available, highlighting the complexity of such simulations and justifying the aims and objectives of the current study. The insights obtained from the comprehensive behaviour of an electronegative gas under high voltage stress, the streamer initiation, and propagation physics will help in identifying new gas mixtures for use in electro-technical applications.

_{6}is given in Section 2. Section 3 provides a brief description of the mathematical model with the swarm parameters used in the present study. The obtained results for streamers in air and SF

_{6}at 10 and 100 kPa are reported and discussed in Section 4, and the work is concluded in Section 5.

## 2. Review of Numerical Simulations of Streamers in SF_{6}

_{6}develops through formation and propagation of a stepped leader (streamer-leader transition) [6,7,8,9]. This is further highlighted by Chalmers et al. in [10], where it was found that the development of leaders in a point to plane SF

_{6}gap at 500 kPa produced a streamer during the leader step development. In the recent work by Bujotzek et al. in [11], positive and negative streamer radii and their propagation lengths have been obtained at gas pressures between 50 and 100 kPa using strong and weak non-uniform background electric fields.

_{6}and SF

_{6}gas mixtures.

_{6}and highlighted the very high value of the electron attachment coefficient leading to a rapid formation of negative ions in the gas at atmospheric conditions. Since corona discharge formation is a time-dependent process, the effect of the high values of the attachment coefficient leading to the creation of negative ions was taken into account by evaluating the characteristic attachment time and subsequently the rate of change of electron density due to the attachment process.

_{6}at atmospheric pressure in a uniform field was studied in [14]. Similarly, the continuity equations were used in [14], but no photoionization was included. Instead, a background ionization was used in this model, which was implemented by introducing background charged particles (electrons and positive ions) uniformly distributed throughout the gap with the number density of 10

^{10}–10

^{14}m

^{−3}. This provided the benefit of understanding the dependence of the streamer propagation on the ionization density in front of it. A Flux Corrected Transport (FCT) technique coupled with a finite difference method for discretization was used in the solving of the higher order equations. The Poisson’s equation, which was used for obtaining the electric field, was solved using a fast Fourier transform in the z-direction and a cubic spline interpolation in the r-direction.

_{6}, were also observed.

_{6}with air at 0.1 kPa, with N

_{2}at 100 kPa and 200 kPa. For the 1D implementation, the swarm parameters of SF

_{6}and the gas mixtures were computed by the Monte Carlo Simulation (MCS) method where only the collisions between electrons and neutral particles were considered. The implementation of the continuity equations, however, duplicates the work done in [15]. The 2D admixture approach developed in [17] for negative streamers was implemented in COMSOL™ Multiphysics for a non-uniform gap. The convection-diffusion equations were coupled with the Poisson’s equation, and the influence of photoionization was omitted in this work in order to simplify the model presented [17]. Further information on the evolution of the negative streamer, the effect of the mixed gas ratio, and effect of electrode shape has been provided. In [18], the fluid model was coupled with 47 chemical reactions, and the reason for streamer decay in SF

_{6}/N

_{2}mixtures with high and low SF

_{6}content was assessed.

## 3. Simulation Model

^{−3}); µ is the mobility of the charged species (cm

^{2}/V·s); $D$ is the diffusion coefficient (cm

^{2}/s); $\overrightarrow{E}$ is the electric field (kV/cm); $t$ is the time (s); $\alpha $is the Townsend’s ionization coefficient (1/cm); $\eta $is the attachment coefficient (1/cm); and $\beta $is the respective recombination coefficient (cm

^{3}/s). ${S}_{b}$ is the background ionization source used to model the photoionization processes. The logarithmic values of the charged species were used. This is useful when steep gradients are involved in the transported scalar quantity and reduces the need for artificial diffusion terms while ensuring that density of species always remain positive. Validation of the logarithmic implementation of the drift diffusion equations can be found in [19]. The convention to use a background ionization term as opposed to a full photoionization model is already established in literature notably in [21] where the importance of photoionization and background ionization in air and N

_{2}-O

_{2}mixtures for pulsed repetitive discharges were investigated; in [22] for N

_{2}-O

_{2}mixtures comparing different streamer codes and also used as a convention for gases where the photoionization process is not well established [14,23].

_{6}and air. Air has been included in this analysis as a reference gas since streamer behaviour in air is well documented and understood. As shown in Figure 1, the critical reduced electric field of SF

_{6}is ~362 Td as opposed to ~120 Td for air. This is the first evidence of the high attachment rate in the highly electronegative gas. The full list of parameters for the gases can be found in Appendix C and Appendix D.

^{−9}and 10

^{−15}, respectively. The steps taken by the solver were automatically adapted within these bounds at each iteration to find a suitable solution. An output step time of 0.1 ns was used for post processing of the time dependent simulation.

## 4. Results and Discussion

_{6}gas; the needle electrode was stressed with a constant positive voltage. Development of streamers have been studied at the ambient gas temperature of 293 K and pressure values of 10 kPa and 100 kPa corresponding to the number density values of 0.2472 × 10

^{25}m

^{−3}and 2.472 × 10

^{25}m

^{−3}. A background ionization in the present model is taken into account by the charge density term in Equation (1), ${S}_{b}$ = 10

^{23}m

^{−3}s

^{−1}. No Gaussian concentration of charge species was introduced for the initiation of the streamer. The gap between the point and the plane electrode was 5 mm.

#### 4.1. Atmospheric Pressure (100 kPa)

_{2}and air at atmospheric pressure have been extensively studied both numerically and experimentally, however very little information on the development of streamers in highly electronegative gases such as SF

_{6}is available. To adequately understand the effect of the electronegativity on the streamer initiation and propagation process, a comparison has been made between the inception and propagation phases in air and SF

_{6}for an applied voltage of 20 kV with a maximum electric field of 610 kV/cm at the tip of the needle electrode. With a 5 mm gap, the nominal average electric field in the computational domain was obtained by dividing the applied voltage, V, by the gap distance, d, ${E}_{av}=\raisebox{1ex}{$V$}\!\left/ \!\raisebox{-1ex}{$d$}\right.$, is ${E}_{av}=40$ kV/cm. The high applied voltage and the non-uniform electrode configuration ensures that the electric field in the domain, specifically near the tip of the needle, is greater than the dielectric breakdown field for air. The average electric field ${E}_{av},$is 40 kV/cm, which is equal to or higher than the stable electric field required for streamer or leader propagation in SF

_{6}which is between 30–40 kV/cm [29,30].

^{−3}while the point 20 represents 10

^{20}m

^{−3}. The equivalent surface plots for SF

_{6}are shown in Figure 8, Figure 9 and Figure 10.

^{20}m

^{−3}. The different discharge phases are highlighted by the electric field diagram in Figure 6 where the electric field achieves its peak values at the earlier times steps, then reduces when the electrostatic coupling between the streamer head and anode reduces and increases again as the streamer approaches the cathode. The head of the streamer in the stable propagation mode has an electric field of ~150 kV/cm. A comparison between the plots for the number densities of cations and anions at 2 ns (Figure 7) show the higher concentration of positive ions to negative ions. The rate of production of positive ions is equivalent to the rate of production of electrons as both of these parameters are governed by the ionization coefficient. The rate of production of negative ions, on the other hand, is defined by the attachment coefficient which is lower in a weak electronegative gas such as air. This accounts for the difference in the ion densities.

_{6}, at the early time steps when the discharge initiates, the electric field is strengthened by the applied voltage and, consequently, the rate of ionization far exceeds the attachment rate. As the streamer propagates, the electrostatic coupling between the streamer head and the needle electrode reduces. This causes a strong affinity for attachment resulting in a high concentration of negative ions in the bulk of the streamer (See Figure 10). In the electron density profile diagram shown in Figure 8 (in logarithmic scale), a discontinuity can be observed between the head and the tail of the streamer resulting from the high level of attachment in the electronegative gas. In the streamer head however, due to high electric field and high ionization rate, the electron density is high and has almost the same order of magnitude as the positive ion density. The corresponding electric field plots in Figure 9 show the enhancement of the electric field in the streamer head. A thinner streamer head with the electric field concentrated in the tip is observed as the attachment process dominates in the bulk of the streamer.

_{6}, the streamer stops propagating at about 1 mm away from the needle electrode. This can be due to the limitations of the computational model as it was shown experimentally that in SF

_{6}at 100 kPa and 1 cm, a leader discharge is formed, (See Figure 11); thus, the continuity equations used in the present work cannot accurately describe the complete breakdown process. In the case of leaders, the heating of the gas should also be taken into account.

#### 4.2. Sub-Atmospheric Pressure (10 kPa)

_{6}at the lower pressure of 10 kPa and for an applied step voltage of 5 kV with a Laplacian electric field at the tip of the needle electrode of 155 kV/cm. Similar to streamers in air at atmospheric pressure (100 kPa), the plasma front in air at 10 kPa has a radius of 1.6 mm and was observed to cross the complete inter-electrode space (5 mm gap). The streamer radius was obtained by measuring the radial extension of the electric field in the head of the streamer. Figure 12 shows the line diagram of the electron density profile in its logarithmic form at various time steps and the corresponding electric field distribution. At steady state propagation of the streamer with an average velocity of 1.3 mm/ns, the electric field in the head of the streamer is 23 kV/cm. The electron density values attained are an order of magnitude lower than that obtained for the streamers in air at 100 kPa. Nevertheless, the general behaviour of the streamers at both pressures is the same. The flat profile of the electron density in the streamer body signifies a steady conduction path between the point electrode and the streamer head. Comparing the positive and negative ion densities, the higher ionization rate as compared to the attachment rate is highlighted in Figure 13 where the line diagrams of the ion densities are presented in a logarithmic form for identical time steps. This correlates well with results obtained in air at 100 kPa, where a similar order of magnitude difference (2) between the positive ion density and the negative ion density was realized (see Figure 7).

_{6}at pressure of 10 kPa, streamers that behave similarly to those in air were attained in the 5 mm gap when the applied voltage was 5 kV. The 1D line diagrams for the log of the electron density profiles and the electric field are shown in Figure 14. In Figure 15, the corresponding ion densities are presented. It was also confirmed experimentally that at gas pressure of 10 kPa, streamer discharges are obtained (see Figure 16) as opposed to leaders observed at atmospheric pressure.

_{6}at atmospheric pressure. The high reduced electric field is required to maintain the higher ionization effect in the electronegative gas. The computed streamer velocity and radius are 0.96 mm/ns and 0.8 mm, respectively.

#### 4.3. Influence of Voltage

_{6}at a pressure of 10 kPa has been studied. The needle electrode was stressed with 5 kV and 10 kV, and the gap distance was 5 mm. As expected, the streamer forms early and travels faster with an increased applied field demonstrating a streamer velocity of 4.17 mm/ns for the applied voltage of 10 kV as opposed to 0.96 mm/ns seen in the preceding section for 5 kV. Additionally, it is interesting to discuss the electron density in the streamer body. As the voltage increases, resulting in an increase in the electric field, the rate of ionization far exceeds the rate of attachment, thus higher electron densities are realized in the streamer channel. The shape of the electron density profile resembles the flat profile characteristic of streamers in air (see Figure 12). This is shown in Figure 17 where the line diagrams of the positive ion density (blue dash-point lines) and electron density (solid gray lines) are plotted on the same figure for 5 kV (left) and 10 kV (right). The streamer head in this case is thicker at lower voltages than at higher ones.

#### 4.4. Comparison with Experimental Results

^{+}is provided for both strong and weak non-uniform fields at 50 kPa and 100 kPa. For strong non-uniform electric fields, the radius of the streamer is in the range between 41 ± 6 µm and 53 ± 6 µm at 50 kPa, and 17 ± 3 µm and 22 ± 3 µm at 100 kPa. In weak non-uniform electric fields, the radius of the streamer ranged between 48 ± 7 µm and 59 ± 5 µm at 50 kPa, and 21 ± 6 µm and 28 ± 5 µm at 100 kPa. Using Equation (6) and C

^{+}= 2 (m·Pa) the expected radius of streamer at 50 kPa was 40 µm and 20 µm at 100 kPa. Taking only the average maximum values, the percentage error in strong non-uniform fields is 32.5% at 50 kPa and 10% at 100 kPa. For weak non-uniform fields, the percentage error is 47% at 50 kPa and 40% at 100 kPa for the average maximum radii. This is summarized in Appendix E.

_{6}at 10 kPa, a similar error analysis has been conducted. For proportionality factor C

^{+}= 2 (m·Pa), the expected streamer radius is 0.2 mm and for C

^{+}= 5 (m·Pa), a radius of 0.5 mm is estimated. Both values are less than the 0.8 mm (for 5 kV stress) obtained in the current simulation yielding a 30% or 60% deviation depending on the proportionality factor. Even the experimental results demonstrate an increase in the deviation as the pressure decreases, which could be a possible explanation for the difference. Another possible reason for this discrepancy could be the background ionization term used in the model instead of photoionization. However, this investigation is beyond the scope of the present work as very limited information is available on the photoionization mechanisms in SF

_{6}. For more comparable values, the applied voltage can be varied. This is, however, a trade-off between the streamer velocity and its radius.

## 5. Conclusions

_{6}at atmospheric pressure are difficult to model numerically because of the stepped leader transition leading to the complete breakdown. In the present work, the streamer initiation is observed however the high attachment rate in the streamer channel exceeding the ionization rate leads to the reduction and eventual interruption of the conduction channel due to decreasing electron density. When this occurred, the streamer radius decreases, it halts, and an increase in the electric field in the streamer head is observed. This could be a result of the limitations of the current computational model. At low pressures, however, the streamer discharge can be characterized using the drift-diffusion partial differential equations. This modelling has been conducted for a pressure of 10 kPa with varying voltages, and the quantitative parameters have been compared to that of air at the same pressure. The results show that the electronegativity of SF

_{6}is particularly important for the cathode directed streamer discharges as it affects the electron density in the streamer channel with both increasing and decreasing gas densities. The number densities of ions in the streamer channel are approximately equivalent with a deviation occurring in the streamer head as shown in the spatial diagrams. It was found that streamers in SF

_{6}are characterized by smaller radii in comparison to air. The streamer radius in air obtained in the present work shows a twofold increase as compared with that in SF

_{6}. However, the velocities of streamer propagating in air and SF

_{6}at 10 kPa are comparable with each other; ~1 mm/ns for a 5 kV applied voltage. The results presented in this paper highlight the influence of increased electronegativity on streamer discharges in gases and provide a foundation for gas mixtures with SF

_{6}reducing streamer initiation and propagation probabilities. In future studies, the possibility of using the fluid approach to quantify the streamer to leader transition and propagation of leaders in SF

_{6}gas will be explored. Also, the influence of field utilization factor, η on streamer dynamics in SF

_{6}will be studied.

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Influence of Background Ionization Seed in Air

^{15}m

^{−3}s

^{−1}and 10

^{23}m

^{−3}s

^{−1}at atmospheric pressure. Figure A1 shows the 2D surface plot of the electric field at axial position 2 mm for both ${S}_{b}$ values.

^{15}m

^{−3}s

^{−1}having a maximum electric field of 154 kV/cm and background ionization level of 10

^{23}m

^{−3}s

^{−1}having a maximum electric field of 129 kV/cm in the head of the streamer. The ease of propagation in the higher background ionization is further highlighted by the average velocity of the streamer, 1.2 mm/ns as compared to 1.11 mm/ns for the lower background ionization. The radius, however, is ~0.02 mm smaller with the lower background ionization.

**Figure A1.**2D surface plot of electric field for ${S}_{b}={10}^{15}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$ (

**left**) and ${S}_{b}={10}^{23}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$ (

**right**).

## Appendix B. Model Verification

^{15}m

^{−3}.

**Figure A2.**Electron density variation (

**left**) and electric field distribution (

**right**) on streamer axis at times t = 1, 2, and 3 ns for COMSOL ™ model (solid line) and reference work by PRHE team, Laplace lab [33] (dotted line). No background ionization and no adaptive mesh used.

**Figure A3.**Electron density variation (

**left**) and electric field distribution (

**right**) on streamer axis at times t = 1, 2, and 3 ns for COMSOL™ model (solid line) and reference work by PRHE team, Laplace lab [33] (dotted line). Adaptive mesh used in COMSOL™ model.

## Appendix C. Cross Sectional Species and Swarm Parameters of SF_{6} Used in Simulations

Reaction Type | Cross Sectional Reactions |
---|---|

Electron Impact Ionization | ${\mathrm{e}}^{-}+{\mathrm{SF}}_{6}\to {\mathrm{SF}}_{6}^{+}+2{\mathrm{e}}^{-}$ |

Electron Attachment | ${\mathrm{e}}^{-}+{\mathrm{SF}}_{6}\to {\mathrm{SF}}_{6}^{-}$ |

Electron—Ion Recombination | ${\mathrm{e}}^{-}+{\mathrm{SF}}_{6}^{+}\to {\mathrm{SF}}_{6}$ |

Ion—Ion Recombination | ${\mathrm{SF}}_{6}^{-}+{\mathrm{SF}}_{6}^{+}\to {\mathrm{SF}}_{6}+{\mathrm{SF}}_{6}$ |

Parameter | Equation | Reference |
---|---|---|

$\mathrm{Reduced}\mathrm{Electron}\mathrm{Diffusivity},{D}_{e}N$ (1/m s) | $\left(3.553\xb7{10}^{-2}(|E]/N{)}^{0.2424}\right)\times N$ $\mathrm{for}\frac{E}{N}650\mathrm{Td}$ | [24,27] |

$\mathrm{Reduced}\mathrm{Electron}\mathrm{Mobility},{\mu}_{e}N$ (1/(m V s)) | $\left(1.027\xb7{10}^{19}(|E]/N{)}^{0.7424}\right)\times \frac{N}{E}$ $\mathrm{for}10\frac{E}{N}2000\mathrm{Td}$ | [24] |

$\mathrm{Positive}\mathrm{Ion}\mathrm{Mobility},{\mu}_{p}$ (m^{2}/(V.s))) | $6.0\xb7{10}^{-5}$ $\mathrm{For}\frac{E}{N}120\mathrm{Td}$ $1.216\xb7{10}^{-5}\xb7\mathrm{ln}(|E]/N)+5.89\xb7{10}^{-4}$ $\mathrm{For}120\le \frac{E}{N}350\mathrm{Td}$ $-1.897\xb7{10}^{-5}\xb7\mathrm{ln}(|E]/N)-7.346\xb7{10}^{-4}$ $\mathrm{For}\frac{E}{N}\le 350\mathrm{Td}$ | [24] |

$\mathrm{Negative}\mathrm{Ion}\mathrm{Mobility},{\mu}_{n}$ (m^{2}/(V.s))) | $1.69\xb7{10}^{32}(|E]/N{)}^{2}+5.3\xb7{10}^{-5}$ $\mathrm{For}\frac{E}{N}\le 500\mathrm{Td}$ | [24] |

$\mathrm{Reduced}\mathrm{Electron}\mathrm{Impact}\mathrm{Ionization},\frac{\alpha}{N}$ (m^{2}) | $\left(3.4473\xb7{10}^{34}(|E]/N{)}^{2.985}\right)$ $\mathrm{for}\frac{E}{N}\le 460\mathrm{Td}$ $\left(11.269\xb7(|E]/N{)}^{1.159}\right)$ $\mathrm{for}\frac{E}{N}460\mathrm{Td}$ | [24] |

$\mathrm{Reduced}\mathrm{Electron}\mathrm{Attachment},\frac{\eta}{N}$ (m^{2}) | $\begin{array}{c}2.0463\xb7{10}^{-20}-0.25379(|E]/N)+1.4705\xb7{10}^{18}(|E]/N{)}^{2}-3.0078\\ \xb7{10}^{36}(|E]/N{)}^{3}\end{array}$ $\mathrm{For}50\frac{E}{N}\le 200\mathrm{Td}$ $7\xb7{10}^{-21}\mathrm{exp}(-2.25\xb7{10}^{18}(|E]/N))$ $\mathrm{for}\frac{E}{N}200\mathrm{Td}$ | [24] |

Reduced Effective Ionization (m^{2}) | $\frac{\alpha -\eta}{N}$ | |

$\mathrm{Electron}\u2014\mathrm{Ion}\mathrm{Recombination},{\beta}_{ep}$ (m^{3}/s) | 1e13 | [34] |

$\mathrm{Ion}\u2014\mathrm{Ion}\mathrm{Recombination},{\beta}_{pn}$ (m^{3}/s) | 1e12 | [35] |

## Appendix D. Cross Sectional Species and Swarm Parameters of Air Used in Simulations

Reaction Type | Cross Sectional Reactions |
---|---|

Electron Impact Ionization | ${\mathrm{e}}^{-}+\mathrm{A}\to {\mathrm{A}}^{+}+2{\mathrm{e}}^{-}$ |

Electron Attachment | ${e}^{-}+A\to {A}^{-}$ |

Electron—Ion Recombination | ${e}^{-}+{A}^{+}\to A$ |

Ion—Ion Recombination | ${\mathrm{A}}^{-}+{\mathrm{A}}^{+}\to \mathrm{A}+\mathrm{A}$ |

Parameter | Equation | Reference |
---|---|---|

$\mathrm{Reduced}\mathrm{Electron}\mathrm{Diffusivity},{D}_{e}N$ (1/cm s) | $\left(0.3341\xb7{10}^{9}(|E]/N{)}^{0.54069}\right)\xb7{\mu}_{e}N$ | [25] |

$\mathrm{Reduced}\mathrm{Electron}\mathrm{Mobility},{\mu}_{e}N$ (1/(cm V s)) | $-\left(\left|E\right|/E\right)\xb7({10}^{5.5236702+0.7822439\xb7log10\left(\left|E\right|/N\right)})\times \frac{N}{E}$ $\mathrm{for}9.8\mathrm{Td}\le \frac{E}{N}\le 1000\mathrm{Td}$ $-\left(\left|E\right|/E\right)\ast ({10}^{5.8692884+0.4375671\xb7log10\left(\left|E\right|/N\right)})\times \frac{N}{E}$ $\mathrm{for}\frac{E}{N}9.8\mathrm{Td}$ | [26] |

$\mathrm{Positive}\mathrm{Ion}\mathrm{Mobility},{\mu}_{p}$ (m^{2}/(V.s))) | $2\mathrm{e}-4$ | [36] |

$\mathrm{Negative}\mathrm{Ion}\mathrm{Mobility},{\mu}_{n}$ (m^{2}/(V.s))) | $2.2\mathrm{e}-4$ | [36] |

$\mathrm{Reduced}\mathrm{Electron}\mathrm{Impact}\mathrm{Ionization},\frac{\alpha}{N}$ (cm^{2}) | $2\xb7{10}^{-16}\xb7\mathrm{exp}\left(\frac{-7.248\ast {10}^{-15}}{\left|E\right|/N}\right)$ $\mathrm{for}150\mathrm{Td}\frac{E}{N}$ $6.619\xb7{10}^{-17}\xb7\mathrm{exp}\left(\frac{-5.593\ast {10}^{-15}}{\left|E\right|/N}\right)$ $\mathrm{for}\frac{E}{N}\le 150\mathrm{Td}$ | [25] |

$\mathrm{Reduced}\mathrm{Electron}\mathrm{Attachment},\frac{\eta}{N}$ (cm^{2}) | $\begin{array}{c}6.56041\xb7{10}^{-19}-1.45181\\ \xb7{10}^{-21}(E/N)+1.45951\xb7{10}^{-24}{\left(E/N\right)}^{2}-5.69565\xb7{10}^{-28}{\left(E/N\right)}^{3}\end{array}$ $\mathrm{for}600\mathrm{Td}\le \frac{E}{N}1000\mathrm{Td}$ $6.23261\xb7{10}^{-19}-1.17646\xb7{10}^{-21}(E/N)+7.51103\xb7{10}^{-25}{\left(E/N\right)}^{2}$ $\mathrm{for}170\mathrm{Td}\le \frac{E}{N}600\mathrm{Td}$ $-3.611\xb7{10}^{-19}+1.01192\xb7{10}^{-20}(E/N)-3.17875\xb7{10}^{-23}{\left(E/N\right)}^{2}$ $\mathrm{for}69\mathrm{Td}\le \frac{E}{N}170\mathrm{Td}$ $3.10976\xb7{10}^{-19}-9.41213\xb7{10}^{-21}(E/N)+1.09693\xb7{10}^{-22}{\left(E/N\right)}^{2}$ $\mathrm{for}23\mathrm{Td}\le \frac{E}{N}69\mathrm{Td}$ $\begin{array}{c}1.2409\xb7{10}^{-19}+8.9497\\ \xb7{10}^{-18}exp(-|E|/N/1.0931)+1.3216\xb7{10}^{-18}{\left(-\left|E\right|/N/6.05148\right)}^{2}\end{array}$ $\mathrm{for}1\mathrm{Td}\le \frac{E}{N}23\mathrm{Td}$ | [26] |

$\mathrm{Electron}\u2014\mathrm{Ion}\mathrm{Recombination},{\beta}_{ep}$ (m^{3}/s) | $2\mathrm{e}-13$ | [25] |

$\mathrm{Ion}\u2014\mathrm{Ion}\mathrm{Recombination},{\beta}_{pn}$ (m^{3}/s) | $2\mathrm{e}-13$ | [25] |

## Appendix E

**Table A5.**Summary of Experimental Radii Corresponding to Different Proportionality Factors in SF

_{6}at 50 and 100 kPa.

${\mathit{C}}^{+}$ | Electric Field Strength | Pressure | Radius Experimental | Radius Computed | Reference |
---|---|---|---|---|---|

5 m Pa | 100 kPa | 50 µm | [8] | ||

2 m Pa | Strong non-uniform electric field | 50 kPa | 41 ± 6 µm → 53 ± 6 µm | 40 µm | [11,32] |

100 kPa | 17 ± 3 µm → 22 ± 3 | 20 µm | |||

Weak non-uniform electric field | 50 kPa | 48 ± 7 µm → 59 ± 5 µm | 40 µm | ||

100 kPa | 21 ± 6 µm → 28 ± 5 µm | 20 µm |

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**Figure 5.**2D surface plot of electron density profile at times 0.5, 1, and 2 ns in air at atmospheric pressure. Gap distance 5 mm, S

_{b}= 10

^{23}m

^{−3}s

^{−1}. Scaling parameter: 20 on logarithmic plot represents a density value of 10

^{20}m

^{−3}.

**Figure 6.**Surface plot of electric field distribution (kV/cm) corresponding to the electron density profile in Figure 5.

**Figure 7.**Surface plot of positive ion density profile (

**left**) and negative ion density profile (

**right**) at time 2 ns in air at atmospheric pressure. Scaling parameter: 20 on logarithmic plot represents a density value of 10

^{20}m

^{−3}.

**Figure 8.**2D surface plot of electron density profile at times 0.5, 1 and 2 ns in SF

_{6}at atmospheric pressure. Gap distance 5 mm, S

_{b}= 10

^{23}m

^{−3}s

^{−1}. Scaling parameter: 20 on logarithmic plot represents a density value of 10

^{20}m

^{−3}.

**Figure 9.**Surface plot of electric field distribution (kV/cm) corresponding to the electron density profile in Figure 8.

**Figure 10.**Surface plot of positive ion density profile (

**left**) and negative ion density profile (

**right**) at time 2 ns in SF

_{6}at atmospheric pressure. Scaling parameter: 20 on logarithmic plot represents a density value of 10

^{20}m

^{−3}.

**Figure 11.**Photo (experiments) [31] of leader discharges in SF

_{6}at 1 bar in point to plane configuration. Needle radius of curvature = 0.2 mm, gap distance = 50 mm. Applied voltage = 107 kV.

**Figure 12.**Line diagrams along the symmetry axis of the log of the Electron Density profile (

**left**) and its corresponding Electric Field distribution (

**right**) for a streamer propagating in a 5 mm air gap at a pressure of 10 kPa. Applied voltage of 5 kV. Scaling parameter for density profiles: 20 on logarithmic plot represents a density value of 10

^{20}m

^{−3}.

**Figure 13.**Line diagrams along the symmetry axis of the log of the Positive Ion Density profile (

**left**) and the Negative Ion Density profile (

**right**) for a streamer propagating in a 5 mm air gap at a pressure of 10 kPa. Applied voltage of 5 kV. Scaling parameter for density profiles: 20 on logarithmic plot represents a density value of 10

^{20}m

^{−3}.

**Figure 14.**Line diagrams along the symmetry axis of the log of the Electron Density profile (

**left**) and its corresponding Electric Field distribution (

**right**) for a streamer propagating in a 5 mm SF

_{6}gap at a pressure of 10 kPa. Applied voltage of 5 kV. Scaling parameter for density profiles: 20 on logarithmic plot represents a density value of 10

^{20}m

^{−3}.

**Figure 15.**Line diagrams along the symmetry axis of the log of the Positive Ion Density profile (

**left**) and the Negative Ion Density profile (

**right**) for a streamer propagating in a 5 mm SF

_{6}gap at a pressure of 10 kPa. Applied voltage of 5 kV. Scaling parameter for density profiles: 20 on logarithmic plot represents a density value of 10

^{20}m

^{−3}.

**Figure 16.**Photo (experiments) [31] of streamer discharges in SF

_{6}at 0.1 bar in point to plane configuration. Needle radius of curvature = 0.2 mm, gap distance = 50 mm. Applied voltage = 37 kV.

**Figure 17.**Line diagram of the positive ion density (blue dash-point lines) and electron density (solid gray lines) in SF

_{6}at 10 kPa when a voltage of 5 kV (

**left**) and 10 kV (

**right**) is applied. Scaling parameter for density profiles: 20 on logarithmic plot represents a density value of 10

^{20}m

^{−3}.

**Figure 18.**1D line graph of the electric field for a streamer in SF

_{6}at 10 kPa when a voltage of 10 kV is applied.

**Figure 19.**Line diagram along the symmetry axis of the log of the Electron Density profile for an applied voltage of 2 kV at pressure 10 kPa. Scaling parameter for density profiles: 20 on logarithmic plot represents a density value of 10

^{20}m

^{−3}.

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**MDPI and ACS Style**

Boakye-Mensah, F.; Bonifaci, N.; Hanna, R.; Niyonzima, I.; Timoshkin, I.
Modelling of Positive Streamers in SF_{6} Gas under Non-Uniform Electric Field Conditions: Effect of Electronegativity on Streamer Discharges. *J* **2022**, *5*, 255-276.
https://doi.org/10.3390/j5020018

**AMA Style**

Boakye-Mensah F, Bonifaci N, Hanna R, Niyonzima I, Timoshkin I.
Modelling of Positive Streamers in SF_{6} Gas under Non-Uniform Electric Field Conditions: Effect of Electronegativity on Streamer Discharges. *J*. 2022; 5(2):255-276.
https://doi.org/10.3390/j5020018

**Chicago/Turabian Style**

Boakye-Mensah, Francis, Nelly Bonifaci, Rachelle Hanna, Innocent Niyonzima, and Igor Timoshkin.
2022. "Modelling of Positive Streamers in SF_{6} Gas under Non-Uniform Electric Field Conditions: Effect of Electronegativity on Streamer Discharges" *J* 5, no. 2: 255-276.
https://doi.org/10.3390/j5020018