Statistical analysis on branching characteristics of positive streamer discharges in N2–O2 mixtures

Streamers are fast-propagating ionization channels that can usually branch and form complex tree-like structures in dielectric media. In this paper, we perform experiments on positive streamers in different N2–O2 mixtures under varying conditions including voltage, pressure, and electrode geometry, with at least 125 discharge images captured for each condition. We present a statistical analysis on streamer branching characteristics from 3D models that are reconstructed by stereoscopic stroboscopic images and our dedicated semi-automatic 3D reconstruction method. We found that by varying the concentration of O2, the morphology and branching characteristics are greatly changed. Specifically, the average branching angle decrease significantly from 90∘ in air to 66∘ in 1% O2, suggesting that photoionization plays an important role in streamer branching. The branching angles in our work are generally larger than previously reported results due to the resolved 3D structures of discharges by our method. A linear relation between the streamer diameter ratio and the branching direction difference of two daughter branches is found, which intersects the vertical axis almost at unity. It is also found that the average branching angles, streamer velocities and diameters increase as the voltage increases. This is again attributed to stronger photoionization effect under higher voltages. The velocities and diameters are similar at different pressures but at the same reduced electric field. The average branching angle decreases from 90∘ at 133 mbar to 79∘ at 200 mbar. This suggests that stochastic fluctuations become dominant over photoionization effect at higher pressures.


Introduction
Streamers are fast-propagating ionization channels that appear in dielectric media when the applied voltage is larger than the breakdown threshold [1].As the first stage of gas breakdown, streamers play an important role in lightning inception and high voltage equipment.Due to their non-equilibrium nature, they also have a wide range of application in ozone generation [2], material surface modification [3,4], and plasma assisted combustion [5], etc.Therefore, streamer discharges have been extensively studied by experiments and simulations both for understanding the theory and exploring their applications.
Depending on their polarity, streamers can be classified into positive and negative streamers.Negative streamers propagate in the direction of the electron drift, while positive streamers propagate in the opposite direction.Therefore, a source of free electrons in front of the streamer channel is required for a positive streamer to support its development.In this paper, we focus on positive streamers because they are easier to generate and propagate in N 2 -O 2 mixtures.A streamer usually branches into separate channels during its development and then forms complex tree-like structures.Wang et al [6] used a 3D fluid model including photoionization as stochastic process to simulate branching of positive streamers, which shows a good agreement with experimental results, indicating that branching is triggered by the stochastic fluctuation of photoionization.In their work, the photoionization coefficient was also varied to show the high sensitivity of branching on the amount of photoionization.Yi and Williams [7] studied the formation and propagation of streamers in N 2 and N 2 -O 2 mixtures in a 13 cm point-plane gap.They found that the addition of O 2 increases the propagating velocities and diameters of streamers, and attributed this to photoionization, as the rate of photoionization is directly related to the O 2 concentration.However, Nijdam et al [8] argued that the O 2 concentration has little effect on the propagating velocity and minimal diameter in their experiment.They suggest that it could result from lower applied voltages and different electrode configurations and they also assume that above a certain threshold, the exact amount of photoionization has little effect on streamer properties.Photoionization also has a strong influence on the morphology of streamer discharges.Chen et al [9] investigated branching characteristics in N 2 -O 2 mixtures at 100 mbar in a point-plane gap and found that the branching angle in 0.01% O 2 is less than that in 20% O 2 .
Among the above mentioned investigations, imaging is the most intuitive way to study streamers.Branching characteristics including branching angles, streamer diameters, and velocities are extracted from discharge images.This implies a strong dependence on the imaging and processing techniques.Traditional imaging method only takes 2-dimensional images of a 3-dimensional phenomenon and this may lead to inaccurate or even wrong interpretation of the discharge properties and the physics behind it.Nijdam et al [10] developed a stereoscopic set-up to resolve the 3D structure of discharges in air.Ichiki et al [11] captured underwater streamer discharges from three directions to reduce the uncertainty in depth.Based on stereo-photography, Dijcks et al [12] further developed a semi-automatic post-processing technique to reconstruct the streamers into 3D representations from images that are both stereoscopic as well as stroboscopic.The feasibility of this method was tested by reconstructed discharges in air and then compared with simulation results, which shows good agreement.This new method gives us now the opportunity to investigate the effect of photoionization on branching by studying N 2 -O 2 mixtures using stereo-photography to resolve the 3D structures of such discharges.
In this work, we show experiments on positive streamers in different N 2 -O 2 mixtures under varying conditions including voltage, pressure, and electrode geometry.Stereoscopic and stroboscopic images are captured for different conditions.Tree-like structures of the discharges are reconstructed by our semi-automatic method, and the full 3D spatial and temporal development of the discharges is resolved.From these 3D models, we present a statistical analysis on the branching characteristics of discharges in N 2 -O 2 mixtures.In the section 2, the experimental set-up used to obtain the stereoscopic stroboscopic images of streamer discharges is explained, and the method for 3D reconstruction is briefly illustrated.Section 3 presents the general morphology and characteristic statistics of at least 125 (up to 500) discharges for different conditions.The effects of photoionization, pressure, voltage, and electrode geometry on branching are discussed.Finally, the paper is summarized and concluded in section 4. All data is available in the online Zenodo repository.

Experimental set-up and method
As is discussed above, the gas composition, which for this work is mainly defined by the concentration of O 2 in N 2 , affects the streamer properties significantly.Therefore, a vacuum vessel specially designed for the high purity of the working gases is used in this work.Details of the vacuum vessel were described in our previous paper [8].During an experiment, the gas inside the vessel is flushed with a constant flow rate that scales with the pressure such that the gas is replenished roughly every 25 min.We use synthetic air and N 2 that are provided by the supplier with impurity levels below 1 ppm.As other mixtures than air and pure nitrogen, we use N 2 with 1%, 2%, and 5% O 2 volumetric fractions, which are mixed by mass flow controllers (MFCs, Brooks Instrument, SLA5800 + Model 0254).For different sets of measurements (gas mixtures, electrode geometries, etc), the vessel is evacuated to a pressure below 10 −6 mbar before new gas is fed into it in order to ensure gas quality.
A schematic overview of the stereoscopic measurement setup is shown in figure 1. Inside the vessel, the lower plate has a radius of 450 mm and is grounded through a 50 Ω shunt resistor.The upper electrode is powered by high voltage pulses through a glass feedthrough.Generally, in this work the highvoltage electrode is a 40 mm round plate with a protruding tungsten needle (12 mm protrusion length, 1 mm diameter, and  In this study, streamer discharges are generated by repetitive positive pulses.A schematic diagram of the electric circuit is shown in figure 2(a).The pulsed voltage is generated by a solid-state push-pull switch (Behlke HTS 301-10-GSM), which discharges a capacitor C 1 that is charged by a DC power supply (Spellman SL150, 30 kV).A typical waveform is shown in figure 2(b), with a zoomed-in view of the rising edge.The pulse width is fixed at 5 µs to ensure that every pulse can generate a gap-crossing discharge.The voltage varies between 7 and 15 kV in different conditions with a rise rate of about 0.12 kV ns −1 .The high voltage pulses and the gating of the ICCD camera are synchronized by an arbitrary waveform generator (Keysight 33600A) and a digital delay generator (Highland Technology Model P400).During the experiments, the repetitive frequency of the voltage pulses was fixed at 20 Hz, while the frame rate of the CCD was 2 fps (frame per second) which was limited by its readout speed.
The gap voltage was measured by a 1000:1 high-voltage probe (NorthStar PVM-4, with 80 MHz bandwidth).The current was measured by a shunt resistor with a resistance of 50 Ω.All electrical signals were monitored and recorded by a digital oscilloscope (LeCroy HDO6104A) at a sampling rate of 10 GS s −1 .
With this set-up, images are captured that are both stereoscopic and stroboscopic, as illustrated in figure 3(a).We used a similar stereoscopic set-up as in [12], also shown in figure 1.The ICCD camera (LaVision PicoStar HR) is operated in stroboscopic mode with its gate rate controlled at up to 50 MHz.The actual gate rate is varied according to the discharge conditions (gas, pressure, etc).This enables us to obtain two 2D stroboscopic views with 32 • viewing angles of the same discharge on one camera.
During processing, firstly, such a stereoscopic stroboscopic image is pre-processed by a Gaussian blur filter and then clustered into different layers based on the intensity in order to detect each blob in both views.The 3D position of each blob is then determined by applying a triangulation method on the views.Finally, the full 3D tree structure of the discharge is reconstructed based on a shortest-path tree algorithm that can account for branching, as shown in figure 3(b).Details of this 3D reconstruction technique can be found in [12].
For mixtures with 1%, 2%, and 5% O 2 , a measurement series of 125 images is taken, respectively; while for air, 500 images (corresponds to 4 measurement series) are taken because of a lower branching probability.Although not all the streamers can be reconstructed by our method due to the complexity of their branching structure and bad luck with viewing angles, stereo-photography still provides us with much more detailed and accurate information of the discharge structures than normal 2D imaging [10,13].Take figure 3(a) as an example, it can be seen in the right view that one branch is missing as the thicker channel covers the thinner one coincidentally.Also, the 2nd branching event is hidden because the two daughter branches are overlapped at their early stage in the right view.
Streamer characteristics (streamer diameters, branching angles, and propagating velocities, etc) are extracted and analyzed based on the reconstructed 3D models as well as the captured 2D images.In the reconstruction method, each branching event is fitted on a unique plane and the father and daughter branches are fitted with a circular fit, as shown in figure 4. The branching angle θ is defined as the angle between two daughter branches.The angles between the father and two daughter branches are denoted as γ max,min , with the subscript indicating the larger and smaller ones, respectively.Note that the use of our circular fit method leads to larger branching angles than fitting of straight lines [12].The streamer diameter is determined as the full width half maximum of the light intensity perpendicular to the streamer propagation direction of each blob from the discharge image.Due to the resolution of the camera, diameters measured with at least 10 pixels in the image are required to ensure the accuracy, which means that diameters measured below 2 mm cannot be fully trusted in our case.The streamer propagation velocity is assumed to be constant between two branching events and is calculated as the ratio of the Euclidean distance between detected dots in 3D and the gating period.

Streamer characteristics in varying N 2 -O 2 mixtures
An overview of streamer discharges in different N 2 -O 2 mixtures at 133 mbar with 9 kV pulses is shown in figure 5.As can be seen, the general morphology of streamers in pure N 2 and 1% O 2 in N 2 is very similar.In these two mixtures, some of the branches have smaller sizes and stop propagating before they can reach the cathode plate even though the voltage pulse is long enough.These are called arrested streamers [14].This phenomenon implies that the electric field enhancement at the streamer head is not strong enough to sustain it to propagate because of lower density of free electrons.Li et al studied the behavior of stagnating streamer with an axisymetric fluid model and found that the electric field and electron density decay after the streamer stops [15].Additionally, feather-like structures can also be observed, which appear as many little sharp protrusions extending from the blobs in the captured  image.Due to their complexity and the limited resolution of the camera, such feather-like structures are filtered out by the Gaussian blur filter during the reconstruction.In comparison, streamers in 5% O 2 and air always reach the bottom electrode, even for the thinner channels of unequal branches.The discharge channels also become smoother and the feather-like structures fade away for increasing O 2 concentration.Nijdam et al [8] argue that if the O 2 concentration is high enough to have a photoionization length that is within the region where the electric field exceeds the critical breakdown threshold (E ⩾ E k ) and the amount of photoionization is high, a large density of electrons, and thereby avalanches, is created close to the streamer head.Due to the high density of these avalanches, they will merge together and blur out the stochastic fluctuations, resulting in a smoother streamer morphology and less branching.The morphology of 2% O 2 is more or less a transition case between the two extremes in which the blobs are smoother than those in pure N 2 and arrested streamers still exist.Furthermore, it is also clearly visible that the number of branches decreases as O 2 concentration increases in N 2 -O 2 mixtures, which is also reported by previous experiments [16] and tested by models with an adapted photoionization coefficient [6].The average number of branches and branching probability for different mixtures are summarized in table 1.More images are taken in air because the branching probability is significantly lower than in other mixtures.The label 'too complex' includes discharges with so many branches that they start to overlap with each other, and discharges with three daughter branches emerging from one father branch (3-5 cases per settings) which leads to ill-defined branching angles θ.
The branching probability and average number of branches both increase with an decreasing O 2 concentration.The average number of branches in air is below 2, suggesting that in most branched cases the discharges only branch once.This number increases to 3.9 for 2% O 2 and 4.8 for 1% O 2 .If the complex cases are taken into account, the branching probability and average number of branches for air and 5% O 2 will increase slightly, as the percentages of complex cases are not high (∼10%).However, the average number of branches for 2% and 1% O 2 will be much larger, due to the large fraction of complex cases (∼50%).
To acquire quantitative knowledge on streamer branching, statistical analysis of the branching characteristics has been performed.The branching angle is one of the most obvious quantifiable parameters for branching events.The branching angle shows how far the daughter branches are repelled by each other, while the branching direction indicates the deviation of the daughter branches from the original direction.It should be noted that only the cases with two daughter branches are considered here.
Figure 6 shows the distribution of branching angles and branching directions for four different N 2 -O 2 mixtures at 133 mbar with an applied pulsed voltage of 9 kV.The angles in discharges in pure N 2 could not be calculated because there are too many branches in the discharges which makes them impossible to reconstruct by our method.All of the distributions approximately show bell shapes and have average branching angles of 90 • , 81 • , 69 • , and 66 • for air, 5%, 2%, and 1% O 2 , respectively.Again, as discussed above, the complex cases will also affect the average branching angles.We expect the average branching angles to become smaller if those complex cases are taken into account.As a consequence of the same background electric field (as shown in figure 13(a), the discharges approaching the cathode follow the field lines approximately and occupy a similar volume even in different mixtures.For the complex cases, if more branches emerge in such a confined volume, it is not surprising that the average branching angle will become smaller.Although the discharges in pure N 2 are not reconstructed, figure 6 still shows a clear trend that the average branching angle decreases with decreasing O 2 concentration.Actually, it can be noticed in figure 5 that at the branching position (which cannot be accurately determined because of the limited gating frequency of the ICCD camera), the repulsion between two daughter branches looks more apparent in mixtures with higher O 2 concentration than in mixtures with less O 2 .For example, in 5% O 2 , even the thicker branch bents a little bit after the branching, while in N 2 the branches appear quite straight immediately after their separation, this is also reflected in the decreasing average angle γ.Compared with the 2D angles measured by Chen et al [9], which have average values of 53 • for air and 42 • for 0.01% O 2 at 100 mbar in a point-plane gap, the results we obtain here are remarkably larger.Although different electrode geometries are used, the main reason is that the 2D projection ignores information in depth of the view and therefore results in smaller measured angles.
As mentioned above, the branching direction γ quantifies the deviation of daughter branches from the original direction.Luque et al [17] simulated interaction between adjacent streamers and found competing mechanisms between electrostatic repulsion and attraction originated from non-local photoionization.Their results show that two adjacent streamers can repel or attract each other by varying the O 2 concentration and pressure.When the ionization in the space between streamers is not strong enough in gas mixtures with lower O 2 concentrations or at higher pressures, streamers diverge due to the electrostatic repulsion between them.Xiong and Kushner [18] investigated the branching mechanisms numerically with a model of statistical photon transport and photoionization.Their model suggests that branching is related to the breaking of the space charge layer and the direction of propagation can alter abruptly if the stable, symmetric space charge layer is interfered by statistical seeding of electrons.Figure 7 shows a scatter plot of the diameter ratio between two daughter branches against the difference between the corresponding angles in 133 mbar air under 9 kV pulses.A linear relation with a negative slope of −0.007 is found, which suggests that a thinner channel will be repelled further away by a thicker one.This is also what can be expected based on observing single images.The fit line intersects the vertical axis almost at 1, indicating that two equal daughter branches should have similar sizes and branching directions against the father one.This is to be expected based on symmetry considerations.For other mixtures, the linear relation is similar, see figure A1.The fitted slopes are approximately the same and the intersects at vertical axis are again at unity.
Figure 8 shows a scatter plot of average streamer diameters and propagating velocities for each individual branch in measurements in four different N 2 -O 2 mixtures.There is a positive correlation between velocity and diameter; a thicker streamer propagates faster.Briels et al studied positive and negative streamers in ambient air at 1 bar and made an empirical fit for velocity v and diameter d, which is v = 0.5d 2 mm −1 ns −1 for both polarities [19].Although the relation does not fit with our results, most likely due to the different pressure, the positive correlation retains.Figure 8 also shows that the velocities and diameters in higher O 2 concentration mixtures have a larger spread in diameter and velocity than lower concentrations.Ono and Oda reported that the average streamer velocity is 5 × 10 5 m/s in air and ∼2 × 10 5 m/s in pure N 2 at 18 kV [16].They also found that the streamer diameter is around 0.2-0.4mm while it increases up to 1.2 mm in air.The diameters we find here lay in the range of around 1 to mm for all mixtures.values are consistent with former results if the reduced diameter (multiplied by a factor δ, the ratio of the gas number density to its atmospheric value) is used.Briels et al focused on the minimal streamer diameters and found that the product of d min and p is constant and the values are p • d min = 0.20 ± 0.02 mm bar in air and p • d min = 0.12 ± 0.02 mm bar in N 2 [20].Nijdam et al further improved the gas purity and found lower values but similar trends for different mixtures [8].In this work, although we do not look for minimal streamers, no reduced diameter smaller than the p • d min determined in [8] is found.

Effects of voltage
Previous studies have shown that the applied voltage has a great influence on streamer structures and properties.Discharge images in 133 mbar air for different applied voltages are shown in figure 9.For these three conditions, the camera gating frequency and gating time are fixed at 40 MHz and 8 ns, respectively.Therefore, a brief estimation on streamer characteristics can already be obtained.At first glance, the streamer velocity is higher and the streamer diameter is larger with a higher applied voltage.At 10.5 kV, the streamer diameter is around 2-3 times larger than at 7.5 kV.The velocity at 10.5 kV is almost twice as the one at 7.5 kV by simply counting the number of dots below the brightest part in the images.In this paper, we focus on statistical distribution of streamer characteristics.Similar to figure 8, a scatter plot of streamer diameters and velocities for different voltages in air at 133 mbar is shown in figure A2.The streamer velocities in air at 7.5 kV mostly lay in the range of 1-3 × 10 5 m/s, with an average value of 1.9 × 10 5 m/s; while for 10.5 kV, they spread from 1 × 10 5 m/s up to 5 × 10 5 m/s, with an average of 3.3 × 10 5 m/s.The average diameters of all branches for these three voltages are 2.2 mm, 2.9 mm, and 3.5 mm, respectively.This confirms that a higher voltage leads to larger streamer diameters and higher velocities and is consistent with results reported by other researchers.In [21], Van Veldhuizen et al used a similar electrode configuration with a short gap of 25 mm and a protruding pin of 5 mm length.They found that at 12.5 kV, the velocity first decreases from 3 × 10 5 m/s near the anode to 1.4 × 10 5 m/s at the middle, and increases to 1.6 × 10 5 m/s at the cathode.They measured that the velocity increases to 1.2 × 10 6 m/s and that it is nearly constant across the gap at 25 kV.We do not specifically look for the velocity change in different stages of streamer propagation.However, a consistent example can also be seen in figure 3(b), which shows that the streamer propagates faster near the anode than at the middle and near the cathode.Chen et al measured the streamer diameters in 100 mbar synthetic air and the average diameter at 20 kV is about two times larger than the one at 10 kV [9].By using 3D fluid model with stochastic ionization process, Wang et al also showed the gap bridging time is shorter with a larger voltage and reaches good agreement with experimental results [6].It is straightforward to explain this trend because more energy is fed into the discharge for a larger voltage, the streamer can have a thicker ionized channel and propagate faster.Furthermore, as a result of a higher net charge density, the repulsion between two daughter branches is larger, leading to a larger branching angle.The average branching angles for 7.5 kV, 9.0 kV, and 10.5 kV are 87 • , 90 • , 96 • , respectively (see figure A3), which is within our expectation.This results suggest that branching characteristics of discharges in air are quite sensitive to applied voltage, which determines the ionization degree, affected by the photoionzation effect.

Effects of pressure
According to the similarity law, electrons will gain the same energy between collision processes under same reduced electric field E/N, so discharges should exhibit similar behavior.However, these scaling laws also have their limits.To study the effect of pressure, we examine discharges in air at 133, 167, and 200 mbar and keep voltage over pressure (V/p), and hence the external E/N fixed.Example discharge images are shown in figure 10.The discharges at three different pressures look similar, except that there are a bit more branches at a higher pressure.
Five-hundred images of discharges for each pressure/ voltage combination are captured and reconstructed with our method, with all cases summarized in table 2. It is found that the branching probabilities are 59%, 46%, and 68% for 133, 167, and 200 mbar, respectively.The average numbers of branches for 133 and 167 mbar are similar, which are 1.7 and 1.6, respectively; while for 200 mbar it increases slightly to 2.1.Even taking the complex cases into account, these indices will be similar for 133 and 167 mbar, but further increase for 200 mbar.It implies that the pressure has a more complicated effect on streamer branching.
Figure 11 shows a scatter plot of velocities against reduced diameters for the different pressure/voltage combinations with color filled markers, in comparison with previews experimental and simulation results with hollow markers.It should be noted that different definitions of diameter are used in literature.Usually optical diameters are measured in experiments, while the electrodynamic radius either defined by the peak electric field or the space charge layer are used in simulations, which can result in a factor of around 1-2 differences [22][23][24].The data points for different pressures in this work are located approximately in the same region, supporting the idea that velocity is independent of pressure while discharge size scales inversely with pressure.Naidis highlighted in an analytical model that the streamer velocity and diameter are correlated to the peak electric field E h at the streamer head [25].And it is found that previously reported velocitydiameter data for positive streamers in air are placed between the lines of E h /δ = 120 kV cm −1 and 180 kVcm −1 [25,26].Most of our data points also lay in the same region.Li et al [15] and Bouwman [27,28] simulated steady streamers in air with an axisymmetric model and an 1.5D model.They found that faster streamers can propagate in a lower background field and a small change in background field can accelerate or decelerate the streamers.Figure 11 shows that most of our experimental data lies above the results of Li and Bouwman if they are extrapolated to our conditions, indicating that streamers in our experiments propagate faster and should likely be classified as accelerating streamers.Only a small portion of streamer branches in this work propagates in a quasi-steady way.
Though the velocities and diameters remain similar at different pressure, the average branching angle decreases as the pressure increases.The average branching angles are 90 • , 86 • , and 80 • for 133, 167, and 200 mbar, respectively (see figure A4).We expect that the distribution of branching angles will shift downwards if the complex cases are included, because more branches in the same background field leads to smaller branching angles.Nijdam et al measured branching angles for positive streamer discharges in air at 200, 565, and 1000 mbar in a 14 cm point-plane gap by stereo-photography and found average values between 46 • and 39 • [10].There, the variation in branching angle for different pressures was found to be insignificant.However, the trend that an increase in pressure leads to a decrease in mean branching angle is obvious in our results using the newly developed method.That is because the electron density scales inversely with gas number density for similar discharges.At a higher pressure, with a lower number of electrons, the individual avalanches created by photoionzation are not strong enough to overlap with each other, and stochastic fluctuations start to dominate and also increase the probability of branching [32].A scatter plot of velocities against reduced diameter, together with previews experimental and simulation results reported by others [15,19,[27][28][29][30][31].Note that different definitions of diameter are used in these studies.The lines for different values of E h /δ are given in [25], where E h is the maximum electric field at the streamer head.

Effects of electrode geometry
The effect of electrode geometry on streamer discharge morphology has been widely investigated over the past years.Van Veldhuizen and Rutgers [21] studied the branching behavior of streamers in air in a point-wire electrode and a protrusionplane electrode.They found that ten times more branches are observed in the former case and attributed it to the impedance difference of the pulsed power circuit.We study the effects of electrode geometries by performing experiments in three different kinds of gaps, including protrusion-to-plate electrodes, with separation of two plates of 10 cm and 11 cm, and a pin-toplate geometry.The distance between the pin and the grounded plate is fixed at 88 mm.Discharge images using the three geometries are shown in figure 12.The discharge morphology in figures 12(a) and (b) does not show significant differences: the streamers ignite from the tip, propagate to about half of the gap, and then branch.On the contrary, in figure 12(c) for the point-plane geometry, an inception cloud is formed and multiple branches emerge from the cloud directly.Furthermore, the discharge occupies a much larger volume.As for statistical distribution, For electrode c, the discharges are too complex to reconstruct because many branches overlap with each other in the two views of the images.If we use the 2D angle of the inception cloud as an estimation, the angle is about 120 • or even larger.The branching angles at the lower half of the gap are expected to be smaller as the branches approximately overlap with the background electric field lines near the cathode.
The difference in streamer morphology between the geometries is attributed to the background electric field profile for these geometries.Figure 13 shows the electric field distribution simulated in COMSOL Multiphysics for all three electrode geometries.The electric field is always highest near the tip.Furthermore, the electric field lines are similar in the lower half region.Streamer discharges start from the tip and approximately follow the field lines when approaching the ground plate.Therefore, differences appear in the upper half region.Most electric field lines emerge from the upper plate of electrode a and b; while for electrode c the field lines extend outward from the pin and have a larger horizontal field component than in electrode a and b.This horizontal field component causes the large difference of discharge morphology between electrode c and the other two.

Conclusions and outlook
We perform experiments on positive streamers in different N 2 -O 2 mixtures.The 3D structures of the discharges are reconstructed by our semi-automatic method.A statistical analysis on the branching characteristics of at least 125 (up to 500) discharges per condition is presented.The effects of voltage, pressure, and electrode geometry are studied.The following conclusions are found: • Photoionization plays an important role in the propagation and branching of discharges in N 2 -O 2 mixtures.By varying the concentration of O 2 , the morphology of the discharges is greatly changed.The branching probability and the average number of branches increases as the concentration of O 2 decreases.• The distributions of branching angles are nearly bell shapes.
The average branching angle drastically decrease from 90 • in air to 66 • in 1% O 2 , which also results from the photoionization effect.It should be noted that the branching angles in our work are generally larger than the ones reported before, because the 3D structures of the discharges are resolved and we use circular instead of linear fitting of branches.Besides, the streamer velocities and diameters in mixtures with higher O 2 concentrations spread over a larger range and the average values are slightly higher than with lower concentrations.• The relation between diameter ratio and branching direction difference of two daughter branches is found to be linear with a negative slope, suggesting that the branching direction is governed by the repulsion between two branches.The fit lines in different mixtures all intersect the vertical axis at nearly unity, which is to be expected for symmetrical branching.
• An increased voltage results in increased branching angles, streamer velocities and diameters.It results from the higher energy input to the discharge and thus a larger ionization degree in the discharge channel.By changing the electrode geometry, it is also found that the background electric field profile has a great influence on the discharge morphology.• The velocities and diameters are similar at different pressures, while the average branching angle decreases from 90 • at 133 mbar to 79 • at 200 mbar.This suggests that at higher pressures the stochastic fluctuations become dominant over the smoothing photoionization effect.
Our experiments can be used as a large data set for the verification and validation of simulation models, which can further illustrate the mechanism of branching in positive streamers.
In future work, it would be interesting to apply our method on negative streamers to study their branching behavior.However, negative streamers usually require higher voltages to initiate and to branch.

Figure 1 .
Figure 1.Overview of the stereoscopic measurement set-up with a schematic drawing of the two imaging paths.A section of the vessel wall is cut out to show the layout inside.The black metal board is used to reduce light reflection from the window behind it.The detailed electrode configuration is illustrated on the top-right corner.All the dimensions are given in mm in this figure.

Figure 2 .
Figure 2. (a) Schematic diagram of the electrical circuit with a sketch of the stereo-photography set-up.(b) A typical voltage waveform with a zoomed-in view of the rising edge.

Figure 3 .
Figure 3. (a) Experimental image with Uapp = 13.5 kV, p = 200 mbar, camera gate of 8 ns and repetition rate of 40 MHz; (b) corresponding reconstructed 3D model.The colour scale indicates the velocity of the streamer, while the segment width is a measure of the average streamer diameter.An animation of the development of this streamer is available in supplementary file.

Figure 4 .
Figure 4. (a) 3D reconstructed trajectories of discharge shown in figure 3, with a branch plane of one branching event in red.The reconstructed points are projected onto the fitted u-v plane.(b) circular fitted curves of the father and daughter branches on the branch plane.

Figure 5 .
Figure 5. Overview of streamer discharges morphology for all N 2 -O 2 mixtures at 133 mbar with 9 kV pulses.The white arrow at the right side indicates the distance between the needle tip and the bottom plate.The brightest spots near the tip are due to secondary streamers which occur after the primary streamers have crossed the gap.

Figure 6 .
Figure 6.Distribution of branching angles for four different N 2 -O 2 mixtures at 133 mbar with 9 kV pulses.The left column shows the branching angle θ and the right column shows the branching direction of two daughter branches.Four rows represent four different mixtures, respectively.The dashed lines indicate the average values for each set of data.

Figure 7 .
Figure 7.The relationship between diameter ratio and branching direction difference of two daughter branches in air at 133 mbar with 9 kV applied voltage.A linear fit with 95% confidence bounds is also shown.

Figure 8 .
Figure 8.Average diameter and velocity plot for each branches in four different mixtures.The pressure is 133 bar and the applied voltage is 9 kV.

Figure 9 .
Figure 9. Discharge images in 133 mbar air under different voltages.

Figure 10 .
Figure 10.Discharge images in air at different pressures.

Figure 12 .
Figure 12.Discharge images in air for three electrode geometries, i.e. protrusion-to-plate electrode with D the distance of two plates (left and middle), and pin-to-plate electrode (right).The top electrodes are colored in gray.The distance between the pin and the bottom plate is fixed as 88 mm.In the right-hand side image, the right view of the original stereoscopic image is overlapping with the left view which means that the channels emerging from the right-hand side can be ignored.

Figure 13 .
Figure 13.Electric field distribution of three different electrode geometries, namely, (a) protrusion-to-plate electrodes with 12 mm pin length and 10 cm separation of two plates; (b) protrusion-to-plate electrodes with 22 mm pin length and 11 cm separation of two plates; and(c) pin-to-plate electrode.For each electrode configurations, the distance between the tip and the bottom plate is fixed as 88 mm.The electric field norm is normalized by the maximum field strength of (c) and the color scale is logarithmic.

Figure A1 .
Figure A1.The relationship between diameter ratio and branching direction difference of two daughter branches in different N 2 -O 2 mixtures.The voltage amplitude is 9 kV and the pressure is 133 mbar.

Figure A2 .
Figure A2.Average diameter and velocity plot for different voltages at 133 mbar in air.

Figure A3 .
Figure A3.Distribution of branching angles in air at 133 mbar for different voltages.

Figure
Figure Distribution of branching angles in air at different pressures with same E/N.

Table 1 .
The number of cases that are processed for branching and non-branching streamers and the corresponding branching statistics in four N 2 -O 2 mixtures.The too complex cases are excluded from the branching probability and average number of branches calculations.Non-branched Branched Too complex Probability <N branches >

Table 2 .
The number of cases that are processed for branching and non-branching streamers and the corresponding branching statistics for different pressures at fixed V/p.The too complex cases are excluded from the branching probability and average number of branches calculations.Non-branched Branched Too complex Probability <N branches > Figure 11.