Spatial and temporal dynamics of single nanosecond discharges in air with water droplets

Discharges generated in water or water-containing media have great potential for various technological applications. However, a fundamental understanding of plasma–liquid interactions, particularly the ignition and propagation of a discharge in a gap containing liquid droplets, is lacking. This study investigates the electrical characteristics and the spatial-temporal dynamics of nanosecond discharges in air containing one or two millimetric droplets of deionized water. Analysis of the effects of voltage amplitude (V a) and pulse width on the discharge mode shows that at low V a, the discharges are run in streamer mode; however, at high V a, a streamer-to-spark transition is observed. Although the droplet size (diameter between 2 and 4 mm) does not significantly influence the discharge dynamics, its position with respect to the gap (on- or off-axis) has a strong effect. Time-resolved imaging of three droplet configurations (one on-axis droplet, one off-axis droplet, and two on-axis droplets) was used to unveil the ignition and propagation dynamics of streamers and sparks at nanosecond time scale. The findings are of interest and contribute to a better understanding of` the plasma–droplet interactions, which is crucial for the development and optimization of plasma-based applications.

Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. application, a profound understanding of plasma-liquid interactions is needed to optimize the process [10]. Therefore, extensive research efforts have been put towards analyzing the physical and chemical phenomena occurring at the plasmaliquid interface, particularly during the last decade.
For obvious reasons, water is the most investigated medium in plasma-liquid systems, and various configurations of discharge in the presence of water have been proposed (e.g. direct generation in water, in gas in contact with water, in water with gaseous bubbles, and in gas with water droplets [11]). For all of these configurations, the interactions occurring at the plasma-water interface have been heavily researched [10], and the related literature can be classified based on electrical excitation (DC [12], AC [13], and pulsed [14] discharges). Due to their simplicity, DC discharges in contact with water are among the most studied configurations. The available literature demonstrates discharges induce pattern formation at the water surface and is strongly dependent on the polarity (anode or cathode) and properties (composition, acidity, conductivity, etc) of water [15][16][17][18]. For instance, Bruggeman et al [18] report that the plasma is filamentary at the water surface when its is cathode and is rather homogeneous when water is anode. Moreover, according to Kovach et al [19], the plasma pattern of DC discharges depends on the nature of the aqueous solution (13 different solutions were utilized). Like DC discharges, the plasma behavior of AC discharges at the water surface varies depending on water polarity. During the half-period when water is cathode, the plasma is filamentary; however, when water is anode, well-defined discharge patterns are observed [20]. In AC mode, the plasma behavior is also dependent on the experimental conditions, such as repetition rate, gap distance, solution composition, and instantaneous injected power [20]. Finally, pulsed discharges in contact with water produce highly organized filamentary patterns at the water surface when it is cathode, and a homogeneous and diffuse plasma is observed when water is anode [21].
To investigate the propagation of discharges in the presence of a miniaturized plasma-water interface, plasmas were generated in water with gaseous bubbles (1) and in gas with water droplets (2). The numerous experimental and simulation studies conducted on the former configuration (1) reveal that the presence of gas bubbles facilitates discharge occurrence due to the low density of the gas compared to the liquid, as well as the discontinuity of dielectric permittivity at the gas-water interface [22]. The discharge characteristics depend on the size, composition, and number of bubbles in the gap [23][24][25]. Meanwhile, the latter configuration (2) results in the generation of a plasma-spray and is typically used to deposit thin films [26], synthesize nanoparticles [27], and study the ignition of lightning [28]. In this case too, the droplet size has a significant effect on discharge ignition and propagation [29]. Based on a recent modelling study conducted by Meyer et al [30] on the dynamics and properties of He RF-glow discharge in contact with a water microdroplet, the authors reported the formation of a sheath at the droplet surface, and they also investigated the influence of power (1-15 W), droplet size (40-80 µm), dielectric permittivity (1-80), and electrical conductivity (10 −7 -10 −1 S cm −1 ) on the sheath dynamics and the charge distribution around the droplet.
Recently, important progress has been made in studying the dynamics of a discharge generated in gaseous medium with millimeter-scale droplets. Specifically, the influence of discharge processing on the properties of the liquid droplets has been assessed by analyzing the hydrodynamic, physical, and chemical phenomena (e.g. internal flows, plasma capillary phenomena, droplets drying under the effect of discharge, and residues left on substrates) occurring in the droplets [31]. In a recent study investigating the propagation dynamics of single discharges produced in an air gap containing a suspended water droplet with a diameter of 3.5 mm and an electrical conductivity of 50 mS cm −1 , Zhao et al [32] show that the discharge is initiated in the air gap between the anode and the droplet, then propagates over the droplet. Although the authors mention that the discharges produced in their study are nanosecond discharges, the acquired discharge current flows for as long as 2 µs, and the discharge emission is observed for 30 µs. The authors explain that in the period of 3-10 µs, the water droplet acts as capacitor, and the emission is attributed to the discharge of accumulated electrons and ions. Meanwhile, in the period of 10-30 µs, the emission is produced by the positive charges remaining on the water droplet. Konina et al [33] simulate the propagation of a negative and positive surface ionization wave at the surface of ∼0.5 mm water droplet. In both cases, the propagation dynamics is relatively fast and crosses the droplet within ∼3 ns. Interestingly, the positive wave propagates close to the droplet-air interface, while the negative one initially propagates close to the interface but later detaches and propagates away from the interface due to the polarization of the droplet. At the same moment, a positive ionization wave is launched in the opposite direction towards the droplet and another one is launched from the droplet's apex towards the cathode.
In this study, we investigate the electrical characteristics, as well as the spatial and temporal dynamics of discharge emission in an air gap containing one or two droplets of deionized water. The total discharge emission and its temporal evolution are analyzed using integrated (1 µs exposure) and timeresolved (2 ns exposure) ICCD imaging, respectively. The analyses are conducted under variable applied voltage conditions (voltage amplitude (V a ) and pulse width) and droplet characteristics (size, number, and position of the droplet(s) relative to the electrodes). Although it remains a fundamental study, the findings may contribute to the development and optimization of applications that are based on plasma-liquid interactions, such as water processing, material synthesis, and others.

Experimental setup
As shown in figure 1(a), the experimental setup is composed of two copper (Cu) electrodes (2 mm diameter, 99.99% purity; Goodfellow) that are horizontally mounted on a Teflon base. The electrode tips are polished to a pin angle of ∼30 • using an electrode sharpener grinder angle controller, and the distance between them is kept at 5 mm. Using a syringe, a droplet of deionized water (electrical conductivity of ∼3 µS cm −1 ) is deposited on the Teflon surface. Note that the nature of the base influences the behavior of the droplet at the surface (contact angle) as well as the discharge behavior. Herein, only the results obtained with Teflon are presented, as the corresponding discharges are more reproducible. Droplets of various sizes between 2.2 and 4 mm are tested, and their position with respect to the electrode axis is also varied (on-or off-axis). The droplet shape is almost hemispherical, and its influence on discharge behavior is not addressed here. To avoid changes in the droplet shape (elongation in the E-field direction) or composition, the droplets are replaced after about 100 discharges. One of the two electrodes is connected to a pulsed negative polarity power source (NSP 120-20-N-500-TG-H; Eagle Harbor Technologies) that supplies a voltage with adjustable amplitude (from −1 to −20 kV) and pulse width (from 50 to 500 ns). Herein, discharges are generated under negative polarity conditions, as they offer better control of the plasma-droplet interactions. Indeed, under positive polarity conditions, streamers propagating at the Teflon's base are observed (not shown here), in addition to the discharge propagating along the droplet's surface. In this study, the discharges are generated at the pulse repetition rate of 1 Hz (single discharge).
A commercial camera (Fujifilm x-s10 model, exposure time of 1 s) is used to monitor the behavior of single (spark) discharges in the interelectrode space. Note that without droplet(s) in the gap, the discharges are either failed or streamers. A typical image of a spark discharge in presented in figure 1(b). To further investigate the temporal dynamics, an ICCD camera (PIMAX-4: 1024 EMB, Princeton Instruments) is mounted vertically above the droplet and used to monitor the plasma emission at the water surface. The camera is equipped with an RB-type intensifier that covers the wavelength range of 200-850 nm with a quantum efficiency between 2% and 15%, depending on the wavelength. The dimension of the captured zone is 10 mm × 10 mm, and the ICCD integration time is set either to 2 ns (for time-resolved imaging) or 1 µs (for timeaveraged imaging). A delay generator (Quantum Composers Plus 9518 Pulse Generator) is used to synchronize the ICCD camera with the voltage pulse. Finally, a high-voltage probe (P6015A, ×1000; Tektronix) and a current monitor (6585, 0.5 V/A; Pearson) are used to measure the voltage and current waveforms of the discharge generated between the two electrodes, respectively. The waveforms of each discharge, as well as the camera synchronization signal, are displayed on an oscilloscope (MSO54, 2 GHz, 6.25 GS/s; Tektronix).

Electrical characteristics and time-integrated images
Each conducted discharge was characterized electrically, as well as by ICCD imaging (1 µs exposure). Depending on V a , failed (i.e. non-occurred), streamer, or spark (always preceded by a streamer) discharges were observed. Figure 2 shows the  probabilities of different discharge modes at different V a values. For instance, at V a = −10 kV, only 30% of discharges are successful (70% are failed discharges), and all of them are streamers. At V a = −11 kV, the percentage of successful (occurred) discharges increases to ∼50% (the other 50% are failed discharges), and still, only streamers are observed. In contrast, at V a = −12 kV, all three modes are observed, with 5% failed discharges, ∼40% streamers, and ∼55% sparks. All discharges are successful at V a = −13 and −14 kV, and the majority of them (>90%) is in spark mode. Finally, at V a ⩾ −15 kV, 100% spark discharges are obtained.
Streamer and spark discharges may be distinguished based on the discharge emission (intensity and morphology), as well as the electrical characteristics. Figure 3 shows the emission of typical discharges under different V a conditions. At V a = −10 and −11 kV, only streamer discharges are observed, and a concentrated emission between the electrodes and the droplet poles is identified. Near the cathode (electrode on the left), the emission intensity is higher than that near the grounded electrode, and it covers a larger region of the droplet. Moreover, low-intensity filaments propagating at the top of the droplet and connecting both electrodes are detected. As indicated in the images, the current peak values (see also figure 4 for typical waveforms) corresponding to the V a conditions of −10 and −11 kV are ∼0.3 and 0.7 A, respectively. Two sets of images recorded at V a = −12 kV are presented in figure 3, one for the streamer and one for the spark. As shown in these images, the behavior of the streamer generated at −12 kV is similar to that of the streamers generated at lower voltages; however, the emission intensity and current peak value (∼0.8 A) are relatively higher. Meanwhile, the emission of the spark discharge consists of a single filament that connects both electrodes and exhibits an inhomogeneous distribution of intensity along the gap: high in the electrode-droplet gap and low at the top of the droplet. The current peak value is ∼5 A. At higher voltage values, all discharges are run in spark mode. Nevertheless, two emission profiles are distinguished, as shown in figure 3. The first one is characterized by high emission intensity in the electrode-droplet gap and low intensity at the top of the droplet, whereas the second one exhibits high intensity along the whole emission channel connecting the two electrodes and propagating at the top of the droplet. Notably, the current peak of the former discharge (non-continuous emission profile) is ∼2 A less than that of the latter discharge (continuous emission profile). For instance, at V a = −13 kV, the peak values of the continuous and non-continuous emission discharges are ∼8 and 6 A, respectively. These values increase to ∼19 and 17 A at V a = −20 kV. The identified discharge modes, streamer or spark (with continuous and non-continuous emission profiles) may be also distinguished based on electrical measurements. Figure 4(a) shows the current and voltage waveforms of the streamer discharge at V a = −11 kV. Based on the current waveform, the discharge starts at ∼170 ns, when a series of current spikes in the order of several hundreds of mA are measured. The spikes' intensity decreases with time, and they vanish at t > 250 ns. As for the voltage waveform, it exhibits no significant variation (no voltage drop), which further confirms that the discharge mode is streamer. The waveforms observed in figure 4(b) present characteristics of a typical spark discharge with non-continuous emission profile (V a = −20 kV). The current waveform clearly shows the formation of a streamer discharge between 25 and 50 ns, as current spikes of few amperes are detected. Beyond 50 ns, the discharge transits to spark that is characterized by a current peak of ∼17 A and a voltage drops to ∼0. The spark current flows during ∼50 ns and is composed of multiple peaks that may be related to different stages of propagation dynamics. Finally, the waveforms presented in figure 4(c) correspond to a discharge with continuous emission profile (V a = −20 kV), and they are similar to those observed in figure 4(b). However, the continuous-emission discharge exhibits a relatively lower current of the streamer and higher current of the spark (by few amperes).
To determine the criterion needed to have a streamer-spark transition, the injected charge (by integration the current over time) of the two discharge modes (Q Streamer and Q Spark ) and the temporal delay between them were measured at different V a conditions. The obtained results (figure 5) demonstrate that both, Q Streamer and Q Spark increase with increasing V a . For instance, in the V a range of −10 to −20 kV, Q Streamer and Q Spark increase from ∼4 to 40 nC and from ∼150 to 650 nC, respectively. In contrast, the delay between the two discharge modes decreases as V a is increased. For instance, at lower V a (e.g. −12 kV), the delay varies significantly between 175 and 315 ns, while at higher V a , the spark starts a few nanoseconds after the streamer. Figures 5(b) and (c) present the voltage-current characteristics of only streamer and streamer that transited to spark at V a = −12 kV, respectively. Notably, the Q Streamer values of measured in the two cases are very close (∼10 and 10.2 nC), and in both cases, streamer discharge happens almost at the same moment in the pulse, i.e. at close voltage values. This indicates that the transition from streamer to spark discharge is not related to a specific criterion of Q Streamer (e.g. charge accumulation at the water droplet) or E-field. However, we believe that the transition may be governed by a criterion comprising both parameters. Indeed, at low V a , streamers only are observed due to small Q Streamer and E-field values; however, at high V a , the two values are relatively high, resulting in the transition of streamer to spark. Considering that both, only streamers and sparks are observed at V a = −12 kV, and that their probabilities are comparable, this condition may be regarded as a limit or a threshold. This means that if Q Streamer ≲ 10 nC and the voltage < −11 kV (note that for V a = −12 kV, the actual measured voltage is −11 kV), streamer-to-spark transition will not occur. Beyond these values, the occurred discharges will always be streamers that transit to sparks.
The influence of the high voltage pulse width on discharge behavior was also assessed. The obtained results suggest that in the range of 150-500 ns, this parameter has no significant effect on the streamer discharges observed at low V a (not shown); however, it influences the spark discharges (with continuous and non-continuous profiles). Figure 6 shows the behavior of the non-continuous emission of a spark discharge at V a = −15 or −20 kV and various pulse widths. As shown in the figure, similar emission profiles are observed at different pulse width conditions, with high-intensity emission between the electrode and the droplet, and low-intensity emission at the top of the droplet. However, when the pulse width is significantly increased, both, the emission intensity and the dimension of the emission zone near the electrode increase. To quantify the effect of pulse width on the emission zone, the diameter of a circle (D) surrounding the emission near the electrode (as shown in figure 6(a) near the cathode) has been determined, and its variation as a function of pulse width is analyzed at V a conditions of −15 and −20 kV (figure 6(c)). Interestingly, at V a = −15 kV and short pulse width (150 ns), the D values corresponding to discharge emission near the cathode and ground electrodes are similar (∼0.3 mm). Both values increase with increasing pulse width; however, the rate of increase is greater at the cathode than at the grounded electrode, resulting in a larger difference between the two D-values. Similar trends are observed at V a = −20 kV, but the D values measured at this condition are consistently higher than those determined at V a = −15 kV. Moreover, the rate of increase in D near the cathode is comparable to that detected near the grounded electrode. Figure 6(c) presents typical electrical characteristics of the discharges generated at V a = −15 or −20 kV and variable pulse widths between 150 and 500 ns. As shown in the figure, for a given V a , the current peak (∼12 A at −15 kV and ∼17 A at −20 kV) is rather independent of the pulse width. However, one identifies a DC current (∼7 A at −15 kV and ∼12 A at −20 kV) that continues to flow during the pulse duration at longer pulse widths.

Influence of the droplet position, size, and number on the discharge
The effect of droplet/electrode configuration on discharge behavior was tested at −13 kV V a and 300 ns pulse width. Figure 7(a) shows the emission corresponding to a 'standard' configuration, wherein a 4 mm-diameter water droplet is centered on-axis between the electrodes. Clearly, the spark discharge generated at this condition propagates at the top of the droplet. When the droplet's position is shifted relative to the interelectrode axis (a part of it remains in the interelectrode zone), the plasma channel of the spark propagates along the upper boundary of the droplet, as shown in figure 7(b). Moreover, two short, low-intensity channels directed towards the droplet center may be distinguished at the first and last points of contact with the droplet (denoted A and B). Figure 7(c) depicts the emission profile of water droplet that is significantly shifted relative to the interelectrode axis (completely outside the interelectrode zone; the distance between the electrodes' axis and the droplet is ∼0.5 mm), and it shows that the spark channel is directed towards the droplet and propagates at its boundary before joining the grounded electrode. When the droplet is shifted by more than 1 mm, streamer discharges occur between the two metal electrodes and do not touch the droplet. For distances between 0.5 and 1 mm, sparks or streamers may be obtained. The shift in propagation observed in the case of off-axis droplets is attributed to the disturbance of E-field lines in the presence of a water droplet, whose dielectric permittivity (ε ∼ 80) is much higher than that of the ambient air (ε ∼ 1) around it. This behavior can be explained by a simple E-field simulation (see figure 8). Notably, the two short, low-intensity channels at the first and last points of contact with the droplet (A and B) are also visible in figure 7(c). Although the E-field simulation is static, the dynamic field at the streamer head is also expected to be high, and it controls the propagation at the droplet-air interface, as well as the transition to a spark. Figures 7(d)-(f) show the behavior of discharge emission across an on-axis droplet with a diameter of 2.2, 2.5, or 3 mm, respectively. Compared to the 4 mm droplet, no significant differences are observed. The emissions across off-axis droplets with various sizes are shown in figures 7(g)-(i), and their behaviors are obviously similar to that of the 4 mm off-axis droplet. Figures 7(j) and (k) show the emissions across a 4 mm onaxis droplet that is shifted towards the cathode and the anode, respectively. Both emissions are similar to that obtained with a 4 mm on-axis droplet centered between the two electrodes.  Finally, the emission across two 1.5 mm on-axis droplets centered between the electrodes was analyzed. With this configuration, a mixture of streamers (∼10%) and sparks (∼90%) is produced, the latter always preceded by a streamer. As shown in figure 7(l), the spark emission connects the two electrodes and propagates in the electrode-droplet regions, as well as between the two droplets. Similar behavior is observed for the streamer emission; however, the emission intensity is much smaller, as illustrated in figure 7(m). The voltage and current waveforms of the discharges produced with two water droplets are similar to those obtained with the one droplet configurations, with the current peak values of the streamer and spark being ∼1-2 and 8-10 A, respectively.
The observed discharge emission is strongly related to the E-field distribution. Although this latter evolves with time due to the space charge field of the discharge channels, its estimated value before breakdown may help understand the observed discharge behaviors. Therefore, a 2D simulation of the E-field distribution was conducted using the Comsol Multiphysics software. The Laplace equation (∆V = 0, where V is the electrical potential) was solved for conditions (electrode geometry, water droplet of ε = 80 in air of ε = 1) similar to those applied in the experiments. As shown in figure 8(a), in the case of a 4 mm on-axis droplet, the simulation predicts a distribution with high E-field between the electrodes and the droplet poles, which corresponds to the regions with high emission intensity (e.g. figure 7(a)). Similarly, in the case of 4 mm off-axis droplets, the highest E-field regions are between the electrodes and the droplet, but they are oriented obliquely, as shown in figures 8(b) and (c). This is consistent with the discharge emission observed in figures 7(b) and (c). The Efield distributions obtained with 3 and 2 mm on-axis droplets (figures 8(d) and (e), respectively) are comparable to that predicted for the 4 mm droplet configuration; however, the E-field values estimated in the latter case are lower (the highest Efield values predicted for the 4, 3, and 2 mm on-axis droplet configurations are 10 × 10 7 , 8 × 10 7 , and 7 × 10 7 V m −1 respectively). As shown in figure 8(f), the E-field distribution of the emission with two 1.5 mm on-axis droplets exhibits high intensity regions between the droplet and the electrodes, as well as between the two droplets. This distribution is well correlated with the emission profile presented in figure 7(m). Time-resolved imaging of the emissions obtained with selected configurations (namely those presented in figures 7(a), (c) and (l)) was conducted and is discussed in section 3.2.

Time-resolved imaging
In this section, we discuss the discharge propagation dynamics based on time-resolved images recorded using the ICCD camera. Three configurations were tested: (i) 4 mm centered, on-axis droplet; (ii) 4 mm centered, off-axis droplet; (iii) two 1.5 mm centered, on-axis droplets. The voltage conditions (V a = −13 kV, pulse width of 300 ns) were chosen such that the discharges achieved were streamers that transited to sparks after ∼100-150 ns. Figure 9(a) shows the currentvoltage characteristics of a typical discharge generated under these conditions. The integrated images of the streamer only (from 25 to 125 ns) and the spark only (from 175 to 675 ns) are presented in figure 9(b). Based on these images, the temporal dynamics of the streamer and the spark at the exposure time of 2 ns are discussed.  Figure 10 shows the temporal evolution of a streamer emission (2 ns exposure time) in the presence of a 4 mm on-axis droplet centered between the two electrodes. The first emission detected is shown in the top-left image, and the corresponding time indicating the first stage of streamer ignition is labelled t St (which is shorter than 2 ns). Clearly, the discharge ignites near the cathode pin and rapidly reaches the droplet. Two nanoseconds later, one observes emission near the grounded electrode, which is also connected to the droplet. Moreover, the emission close to the cathode seems to be diffuse-like, whereas that close to the grounded electrode is less diffuse. To analyze the propagation of the two ionization fronts, images were recorded at 4, 7, and 10 ns after t St . The corresponding images show that the ionizations fronts are directed towards each other, and that the one generated near the cathode (negative ionization front) is more homogeneous than the one initiated close to the grounded electrode (positive ionization front). Moreover, the propagation of individual streamers associated with the latter is more rapid than that of the negative ionization front streamers. The difference in propagation velocity is attributed to variations in the nature and mobility of species (electron and ions) generated at the positive and negative ionization fronts, and it has also been observed in the case of discharges in air in contact with water [34]. At 12 and 14 ns, the positive streamers encounter the negative ones, and a relatively stronger emission is observed due to recombination. An emission is detected in the gap at 20-40 ns, and it is probably attributed to the propagation of a second ionization wave from the cathode to the anode. In this time range, low-intensity filaments are observed near the electrodes (e.g. at 34 and 40 ns), possibly due to the memory effect induced by the initial stage of propagation of positive streamers. Between 50 and 70 ns, localized emission propagating from the grounded electrode to the cathode is observed, and at 75 ns, a single filament connecting both electrodes and propagating at the top of the droplet is identified. Beyond 75 ns, the discontinuous emission between the electrodes becomes more and more intense (e.g. at 96 ns), then it becomes continuous Finally, the emission becomes more continuous and more intense, such the image at 105 ns. The last image collected at 110 ns was recorded using a lower camera gain (10 instead of 10 000) in order to avoid saturation when imaging the spark phase.
The temporal evolution of the spark emission is presented in figure 11. The first image (top-left) corresponds to 5 ns before the spark, whereas the second one (at t Sp , which is shorter than 2 ns) corresponds to the first emission detected. Both images show a single filament of non-homogeneous emission connecting the two electrodes and propagating on top of the bubble. At 2 ns after t Sp , the emission becomes rather homogeneous, more so at 5 ns. Between 12 and 24 ns, the continuity is lost, and interestingly, the low intensity region becomes localized at the top of the droplet. This behavior cannot be readily explained, and further quantitative investigation is needed. At 29 ns, the emission becomes asymmetric, with highest intensity near the grounded electrode and lowest intensity at the top of the droplet. Beyond this time, the global intensity of the emission gradually decreases until extinction is reached at ∼240 ns. Figure 12 shows the temporal evolution of a streamer discharge generated in the presence of a 4 mm off-axis droplet centered between the electrodes. The initial image recorded at t St (<2 ns) corresponds to streamer ignition at the cathode, and it shows localized emission near the cathode tip. At 2 ns after t Sp , the discharge propagates obliquely (downward) towards the surface of the water droplet, and at 4, 7, and 10 ns, the emission seems to be more diffuse and less intense. At 14 ns, the emission intensity between the cathode and the droplet increases, and an emission between the ground and the droplet is also detected. As in the previous case of on-axis water droplet, the emission on the cathode side is more diffused than that on the ground side. The intensity of the emission near the grounded electrode decreases at 18 and 25 ns, probably due to the fast propagation of positive streamers (as in the previous case). Meanwhile, the intensity near the cathode increases due to the recombination of positive and negative streamers. Between 30 and 100 ns, low emission intensity is detected, similar to the emission obtained in the range of 40-90 ns with an on-axis droplet. The localized discharge observed at the droplet's pole (e.g. at 84 ns) is similar to the one identified by Konina et al [33] and may be attributed to a surface ionization wave. As explained before, the decrease in emission intensity at this time may be attributed to repropagation from the cathode to the anode. In the range of 108-180 ns, the discharge emission propagates from the grounded electrode to the cathode, and two kinds of channels are identified: (1) channels connecting the electrode tip to the droplet surface at the shortest distance and (2) channels propagating almost horizontally across the top of the droplet to reach the cathode. These two channels may also be correlated with Konina et al's [33] simulations of negative surface ionization wave propagation at the droplet's surface. Indeed, a few nanoseconds after ignition, the authors observed detachment of the surface ionization wave from the droplet's surface. At this stage, the wave becomes a bulk ionization wave, and it is redirected towards the surface by the applied E-field. A Few nanoseconds later, a positive ionization wave is launched, and it propagates towards the droplet. Such different ionization waves may be compared to the two discharge channels from the ground side, despite the difference in time scales. Finally, at 188 ns, the streamer starts transition into a spark. The last image in the series is an integrated image of the streamer phase only. This image clearly shows the diffused-like behavior of the discharge near the cathode, the focused-like behavior near the ground, and the filamentary behavior at the droplet (ground side).
The temporal evolution of the spark emission obtained with an off-axis droplet configuration is shown in figure 13. The first and second images were captured at 3 ns before the spark appears and at t Sp , the time at which the first emission is detected (<2 ns), respectively. Both images show a direct and strong connection between the electrode tips and the droplet, and at t Sp , a thin zigzag filament propagating on the top surface of the droplet is also detected. The emission profile observed at 1 ns after t Sp is similar to that observed at t Sp ; however, the intensity of the emission is higher in the former case. Between 7 and 14 ns, the discharge filament becomes well defined and joins the two electrodes with preferential propagation at the droplet periphery. The highest intensity and homogeneity are reached at 15 ns. Two short, low-intensity channels propagating perpendicularly to the main filament at the droplet contact points are also noticeable at this time. Irregularities in the discharge filament appear beyond 15 ns, and the emission intensity gradually decreases until extinction is reached at ∼190 ns. The last image in the series is an integrated image of the spark phase only, and it clearly shows that the filament behavior resembles the emission profile observed at 15 ns.
The final configuration investigated herein is that of discharges generated in the presence of two 1.5 mm on-axis droplets centered between the electrodes. The ignition in this case is relatively faster than that obtained with other configurations. As shown in figure 14, at t St , three zones of emission are clearly observed: one between the cathode and the droplet, another between the two droplets, and another between the grounded electrode and the droplet. The highest intensity and largest emission region are detected near the cathode. A similar emission profile is observed at 2 ns after t St . However, at 4 ns, the emission becomes localized at the top of the droplets on the cathode side, as well as between the droplets. At 12 ns, the emission is primarily localized between the droplets, and between 14 and 30 ns, several emission regions are detected in the inter-electrode gap. In the range of 35-60 ns, the progressive formation of a single filament between the electrode tips and the droplet surfaces is clearly observed. The same filament is also observed at 65-80 ns; however, the emission intensity is much higher in this time range. The last image in the series constitutes an integrated image of the streamer phase only. This image clearly shows that the discharge behavior is diffused-like near the cathode, focused-like near the ground, and filamentary at the top of the droplets. In general, this two-droplets configuration may be compared to that of dielectric packed-bed reactors. These latter have been investigated experimentally and computationally by Engeling et al [35] and Kruszelnicki et al [36], among others. Discharge ignition near the anode, near the cathode, and in the inter-bed regions   has been observed. This latter ignition is due to the E-field intensification induced by the discontinuity of the dielectric permittivity.
Based on the temporal evolution of the spark phase ( figure 15), a thin filament is produced in the inter-electrode gap at 2 ns after t St . This filament is most intense in the region between the two droplets. With time, the intensity of the filament increases significantly, but it remains non-homogeneous over nearly the entire discharge lifetime. Beyond 30 ns, the emission intensity decreases over the interelectrode gap, until extinction is reached at ∼260 ns. The integrated image of the spark phase only (last image) shows that the filament exhibits strong emission between the two droplets.
Regardless of the electrode/droplet configuration, the timeresolved emission profiles show a rapid dynamic and clear transition between the different steps. In the case of a single centered on-axis droplet, the conventional characteristics of negative and positive streamers are observed. The two streamers recombine, and additional propagation phases are generated, probably due to charge accumulation at the droplet surface, as well as the high E-field (voltage drop has not been observed in the streamer phase). Intense emission is often observed tens of nanoseconds after spark ignition, and the intensity gradually decreases with time. Overall, the results provide insight into the dynamics (ignition and propagation) of discharges generated in the presence of a liquid droplet, with controlled conductivity and permittivity, between the electrodes.

Conclusion
This study investigates the electrical characteristics, as well as the spatial-temporal dynamics of nanosecond discharges in air with water droplets. Depending on the applied voltage, failed, streamer, and/or spark discharges may be obtained. The former is characterized by the absence of emission. Meanwhile, the streamer discharge is characterized by the emission of low-intensity filaments, with current spikes of ∼0.5-2 A. As for the spark discharge, it is always preceded by a streamer, and it is characterized by strong emission of a single filament, with a current peak in the order of tens of amperes. The influence of the high voltage pulse width on discharge behavior is also investigated, and the obtained results demonstrate that this parameter affects emission intensity and the injected charge (both increase with increasing pulse width). Although the discharge dynamics is not affected by the droplet size (diameter between 4 and 2 mm), the position of the droplet in the inter-electrode gap (on-or off-axis) has a significant effect. Finally, time-resolved nanosecond imaging is used to reveal the propagation dynamics of streamers and sparks generated in the presence of an on-axis droplet (4 mm), an off-axis droplet (4 mm), or two on-axis droplets (1.5 mm) centered between the electrodes. Overall, the findings reported herein complement the available data in the field of discharge dynamics in the presence of liquid media, and they can be applied in various applications, such as water processing, surface decontamination, or nanomaterial synthesis. Depending on the targeted application, plasma-droplet interactions can be controlled by adjusting the different experimental conditions, such the ones explored here (droplet size, position, number) or others that need to be investigated (voltage polarity, droplet composition and shape, substrate nature, etc.).

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).