Discharge in air in contact with water: influence of electrical conductivity on the characteristics and the propagation dynamics of the discharge

Due to the high reactivity and the non-thermal properties of streamer discharges, they are applied in various fields, such as water treatment and medicine. Streamer discharges are usually produced in the gas phase before interacting with a liquid or solid surface. Although the dynamics of a streamer discharge in gases is well described, its propagation at liquid surfaces remains poorly understood. In this study, we investigate the influence of water electrical conductivity (σ), between 2 and 1000 µS cm−1, on the characteristics and propagation dynamics of pulsed positive DC nanosecond discharges with the solution serving as a cathode. σ strongly influences τ r (the dielectric relaxation time), and two discharge modes may be obtained, depending on whether τ r is shorter or longer than the delay to achieve breakdown (τ pulse). This latter can be indirectly modified by adjusting the voltage amplitude (V a). In the case of V a = 14 kV, the breakdown voltage (V bd) at low σ is lower than that measured at high σ, probably because τ pulse < τ r and > τ r, respectively. In the case of V a = 20 kV, V bd decreases slightly with σ, probably because of the decrease of the resistivity of the global electrical circuit as τ pulse ∼ τ r for high σ. In addition to the electrical characterization, the dynamics of the discharge at the solution’s surface is investigated using 1 ns-time-resolved imaging. Its morphology was found to evolve from a disc to a ring before it splits into highly organized plasma dots (streamers’ head). The number (N dots) and propagation velocity of plasma dots are determined as a function of σ. At V a = 14 kV, N dots does not vary significantly with σ despite the increase of V bd; this latter likely compensates the neutralization of charge accumulated at the surface by ions in solution. In the case of V a = 20 kV, N dots decreases with σ, and it can be related to a decrease of accumulated charge at the water surface. Finally, based on the electrical measurements, we found that the charge per plasma dot (Q dot) increases with σ, which does not correlate with the imaging results that show a short length of propagation at high σ. Then, considering the plasma dot mobility at low σ and the instantaneous propagation velocities at high σ, a more realistic Q dot is measured.


Introduction
The unique physical and chemical properties of streamer discharges make them highly attractive for applications. For example, the thermodynamic non-equilibrium state of streamer discharges allows their integration in thermosensitive applications, especially in medicine, where reactive plasmas at room temperature are required [1,2]. One of the important properties that streamer discharges exhibit is the high E-field at the ionization front which originates from charge separation [3,4]. This space-charge field controls both the breakdown of the medium and the dynamics of the ionization front through the initiation of secondary electron avalanches [5].
The majority of streamers discharges are usually produced by applying high voltage to a sharp electrode (i.e. a pin) in a dielectric medium (e.g. air or water) [5]. Although it can be sustained freely in the medium, streamers usually come in contact with a surface whose nature (dielectric, conductive, solid, liquid, etc) depends on the targeted application [6,7]. Therefore, understanding streamer propagation in the medium (air or liquid) and at the surface of an object (solid or liquid) is necessary to further develop and optimize applications. For instance, when generated in air to process a liquid surface, the streamer faces two steps of propagation: in air (i.e. vertical propagation) and at the liquid surface (i.e. horizontal propagation) [8]. The transition from vertical-to-horizontal propagation is not well understood. However, it is accepted that when the streamer approaches the surface, the production rate of electron avalanches ahead of it is highly reduced [9]. Hence, charged species can accumulate at the surface until the space charge field becomes high enough to induce ignition of streamers that propagate radially, i.e. at the surface [10]. In this context, the properties of the surface play a crucial role on the streamer characteristics and its propagation dynamics [11].
In general, a surface can be solid, liquid, conductive, dielectric, etc. During the last several years, interaction between a streamer and solid surfaces was extensively studied, experimentally and by simulations [12,13]. In the case of liquid surfaces, plasma-liquid interaction is rather less understood, even though similarities between solid and liquid surfaces were reported under specific discharge conditions [14,15]. In the context of applications, water may exhibit different conductivities that range from few µS cm −1 (i.e. distilled water) to tens of mS cm −1 or more [16]. Most of the studies are performed on solution with low conductivity (e.g. distilled water), but many applications involve plasma processing of solutions with high conductivity, such as biological liquids [17,18]. In all cases, it is expected that propagation of streamer at the surface of water solution will be affected by the conductivity. The simple and basic reason is that both the species composing the streamer (e.g. electrons and ions) and the E-field at its front interact with the charged species in the solution. Such interactions could influence back the properties of the streamer, its propagation dynamics, and therefore the induced plasma-chemistry [8,15,19].
In general, a solution with a dielectric permittivity ε and an electrical conductivity σ exhibits a characteristic time (also called polarization-or Maxwell-time) τ r [20] that is proportional to ε and inversely proportional to σ (equation (1)). For instance, in the case of distilled water with ε = 80 and σ ∼ 2 µS cm −1 , τ r is ∼3.5 µs. In the case of solution with high σ, τ r decreases significantly (for example, it is several nanoseconds at σ of few mS cm −1 ) [21]. The order of magnitude of these temporal values (ns to µs) are of great interest as they match the characteristic time of streamer propagation [22].
One of the common observations that researchers have reported when σ is augmented is the decrease of the streamer propagation length. This trend was observed in the case of streamers immersed in water solution as well as at the water surface. For instance, Zhu et al [23] have investigated the propagation of streamers in water solution and observed a decrease in breakdown voltage, steamer length, and streamer formation time delay when σ is augmented from few hundreds of µS cm −1 to several mS cm −1 .
Studies that addressed the formation of plasma at the surface of solutions, by AC or DC discharges, are numerous, e.g. [24][25][26][27][28]. Such studies have demonstrated the formation of emission patterns at surface. The influence of the experimental conditions, including electrical conductivity, on pattern morphology [29,30] as well as the production rate of reactive species [16] have been reported. The majority of the available studies show the patterns using cameras with long exposure time (e.g. ms), which hinders the temporal dynamics of the discharge; this latter occurs at the nanosecond time scale. Although several mechanisms of patterns formation are proposed, their validity remain proper to a set of experimental conditions, particularly to the discharge period. For instance, Joule heating seems to have a crucial role in the pattern formation, due to water evaporation and electron attachment mechanisms [31]. Also, electrostatic-based interactions due to the accumulation of charges at water surface and production of space field may play a role in the pattern formation [32]. We believe that the former (Joule heating) dominates at long time scale (∼µs-ms), while the latter (electrostatics) prevails at short time scale (∼ns). In this context, time-resolved diagnostics become essential to further investigate the behavior of plasma at solutions' surface.
Studies that address the temporal dynamics of discharges at water surface are very scarce. Our group has recently addressed the dynamics of streamers at the surface of distilled water in AC [33] as well as in nanosecond pulsed discharges [22,34,35]. In this latter condition, the temporal dynamics of the discharge at the surface of distilled water, from ignition to extinguishing, has been provided with a time resolution of 1 ns. It has been shown that the discharge emission at water surface evolves rapidly, as it takes the shape of a disc-like during the first few nanoseconds, then it evolves to a ring-like structure, before splitting into highly organized plasma dots propagating at a velocity of a few hundred of km s −1 . Such a dynamics suggests the dominance of fast mechanisms such as electrostatic-based ones. In this context, the properties of the solution, namely its electrical conductivity, may influence the pattern formation mechanisms due to the presence of ions near the ionization front, as well as the induced chemistry in plasma and in liquid. Although there is a lack of experimental data on the formation and propagation of streamers at solution's surface with nanosecond time resolution, some simulation studies have highlighted the main behavior. For instance, Akishev et al [15] have performed a 2D simulation study and investigated the influence of σ (5 or 600 µS cm −1 ) on the propagation of streamer at the surface of dielectric material with ε = 81 (similar to that of water). They showed that the increase of σ has led to a decrease of the streamer propagation length (from ∼3.5 to 2 mm), in the surface charge (from ∼150 to 100 nC cm −2 ), and in the E-field intensity at the streamer head (from ∼4 × 10 4 -3 × 10 4 V cm −1 ). All these modifications are due to the presence of ions in the solution, which contributes to the neutralization/compensation of species with opposite charge in the streamer [19]. In this context and as mentioned earlier, the relaxation time (that depends on ε and σ) plays a major role in the discharge dynamics.
In this study, our goal is to report the discharge electrical characteristics and dynamics (by nanosecond time-resolved imaging) at the surface of a solution with σ between 2 and 1000 µS cm −1 . It is expected that the presence of ions in solution may influence the space charge formation and, therefore, the characteristics of the discharge, such as the propagation velocity, streamers' number, deposited charge, etc. In addition to σ, the influence of the gap distances (<1 mm) and the amplitude of the positive high voltage are reported.

Experimental setup
The experimental setup is schematically shown in figure 1. We used a nanosecond positive pulsed power supply (NSP 120-20-P-500-TG-H, Eagle Harbor Technologies) to ignite an electrical discharge in air in contact with water solution. The electrical conductivity (σ) of this latter has been adjusted between 2 and 1000 µS cm −1 using potassium chloride. Note that these values correspond to those measured by a conventional conductivity meter, and the dependence of σ on the E-filed (magnitude and frequency) is not considered. Similarly, we consider that the dielectric solution of water is ∼80, and its dependence on the E-field (magnitude and frequency) is not considered too. The pulse plateau period was fixed at 100 ns, whereas the magnitude of the applied voltage (V a ) was varied between 8 and 20 kV. The anode is made of a tungsten rod (diameter of 1 mm) with its tip mechanically polished to obtain an aperture angle of ∼30 • . This electrode is mounted on a micrometer positioning system allowing a fine control of the distance between the anode tip and the solution surface (i.e. the gap distance, d) from 10 to 1000 µm. The cathode is made of a stainless-steel rod (4 mm diameter) placed at the bottom of a cylindrical Teflon cell (diameter of 67 mm and height of 5.7 mm) filled with 20 ml of solution.
The voltage and current waveforms of the discharges were measured using a high-voltage probe (P6015A, Tektronix) and a current monitor (6585, Pearson), respectively. The waveforms were visualized and recorded using an oscilloscope (MSO54, 2 GHz, 6.25 GS s −1 ).
An ICCD camera (PIMAX-4: 1024 EMB, Princeton Instruments) mounted vertically above the water cell was used to monitor the behavior of the plasma emission at the solution surface. The camera is equipped with an RBtype 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 was 8 mm × 8 mm, and the ICCD exposure times used here are either 1, 5, or 50 ns. A delay generator (Quantum Composers Plus 9518 Pulse Generator) was used to adjust the delay between the ICCD camera and the voltage pulse.  Figure 2 shows typical voltage and current waveforms of discharges in air in contact with water at different conductivities (σ = 2, 300, 650, and 1000 µS cm −1 ); these waveforms were obtained for discharges at d = 400 µm and V a = 14 and 20 kV. We note that experiments were performed at other distances (between 10 and 1000 µm), and no significant difference was found for σ ⩾ 300 µS cm −1 cases (the dependence for σ = 2 µS cm −1 has been reported in [34]). First, we note that σ influences the voltage plateau value as the latter decreases when σ is increased. Also, σ influences the falling period of the pulse with a more rapid fall at higher σ. These effects are related to the increase of the ion concentration in the solution at higher σ, which modifies the global electrical circuit. Zooms on specific regions of the voltage and current waveforms (figures 2(b) and (d)) clearly show the breakdown event. This moment is indeed characterized by a slight voltage drop (noted ∆V in figure 2(b)) and a current peak followed by oscillations (similar to a conventional RLC circuit probably due to the presence of parasitic component in the circuit). The electrical data acquired for numerous (thousands) discharges were analyzed using a homemade developed algorithm. The data of interest are the breakdown voltage (V bd ), voltage drop (∆V), discharge delay (τ pulse ), and injected charge (Q =´|i(t)|dt), and they are reported for the discharges under various conditions of d, σ, and V a . Figure 3 shows the variation of V bd as a function of σ for different gap distances (d = 200, 400, and 600 µm) at two applied voltages: V a = 14 and 20 kV. Regardless the d value, one observes two distinct behaviors depending on V a . In fact, we performed the measurement for other V a values between 8 and 20 kV, and the two just mentioned behaviors occur for V a ⩽ 14 kV and V a ⩾ 18 kV, respectively. Between 14 and 18 kV, a transition-like behavior is observed. Therefore, to limit the number of figures, only these two typical cases of V a (14 and 20 kV) are shown here. For instance, at d = 200 µm and V a = 14 kV, we find that V bd (average value) is ∼7.5 kV at σ = 2 µS cm −1 , and it increases with σ to reach a plateau at ∼9.5 kV for σ ⩾ 650 µS cm −1 . Although the reported values at longer distances (figures 3(b) and (c)) are slightly higher from those reported at d = 200 µm, the observed trend remains similar. At V a = 20 kV, the trend is reversed. Indeed, higher V bd is observed at low σ. For instance, at d = 200 µm and σ = 2 µS cm −1 , one measures V bd of ∼9 kV that decreases to ∼7.2 kV at σ = 1000 µS cm −1 . At longer distances, similar trends were also observed. Figure 4 shows the variation of the voltage drop (∆V) as a function of σ for different gap distances (d = 200, 400, and 600 µm) and for two applied voltages: V a = 14 and 20 kV. Here also, two different behaviors were identified depending on the applied voltage. At V a = 14 kV and the three gap distances, ∆V ∼ 0.5 kV at σ = 2 µS cm −1 , and it increases with σ to reach a plateau of ∼1.5-2 kV at σ ⩾ 650 µS cm −1 . At V a = 20 kV, the behavior of ∆V is completely different from that at 14 kV. Indeed, for the three gap distances, the variation of ∆V as a function of σ can be neglected and statistically varies between 0.5 and 1.0 kV. Comparing the two-V a cases, it is interestingly to note that ∆V are comparable for low σ values, but for high σ values, they are higher at low V a . This measurement is of interest as it can be related to the modification of the discharge mode with σ; clearly such modification is more pronounced at V a = 14 kV than at V a = 20 kV. This finding concurs with the behavior of V bd .

Electrical characterization
These trends can be explained by considering two factors, the first one is related to the characteristics of the high-voltage pulse and the second one to the characteristics of the solution. In fact, the high voltage pulse is characterized by its rising period τ pulse . As a breakdown will eventually happen after where ε r and ε 0 are the relative dielectric permittivity of water (∼80) and the dielectric permittivity of vacuum (∼8.85 × 10 −12 F m −1 ), respectively. The values of τ r for different σ values are also shown in table 1. The decrease of τ r with σ means that the reorganization of ions in solution at low σ takes time relatively longer than that at high σ in response to the applied field. In this context and at V a = 14 kV and σ = 2 µS cm −1 , which corresponds to τ pulse < τ r , we measured the lowest V bd . At σ = 300 µS cm −1 , which corresponds to τ pulse ∼ τ r , we measured V bd higher than that at σ = 2 µS cm −1 . At higher σ (650 and 1000 µS cm −1 ), τ pulse is almost twice of τ r , and V bd is higher than that measured in the case of 300 µS cm −1 . This result can be well explained when comparing τ pulse and τ r . Indeed, at σ = 2 µS cm −1 , τ r is longer, which means that the solution did not reorganize before breakdown. At higher σ, the fact that τ pulse > τ r indicates that the ions in solution move in response to the E-field, which results in an increase of the voltage needed to induce breakdown. At V a = 20 kV, τ pulse is shorter than τ r at low σ (2 and 300 µS cm −1 , table 1). Therefore, the solution did not reorganize before breakdown. At higher σ (650 and 1000 µS cm −1 ), τ pulse ∼ τ r , which indicates that the ions could move in response to the E-field. However, the slight decrease of V bd (by ∼1 kV) with σ can be related to a decrease of the resistivity of the global electrical circuit. Hence, it is feasible to assume that the breakdowns obtained at V a = 20 kV are equivalent, which also further supported by the behavior of ∆V, but the streamer dynamics following the breakdown depends on σ, as clarified hereafter.
The injected charge was also calculated for the different discharge conditions. The results under different conditions of d and V a are similar. Therefore, we show in figure 5 the variation of injected charge as a function of σ for one typical case (at V a = 14 kV and d = 400 µm). The charge was calculated during two temporal periods: during the first peak of the discharge current (∼10 ns) to account only the first stage of streamer ignition and propagation (Q peak ) [14], and during the whole pulse to consider all the events that may appear later in the pulse (Q total ). It is noted that at σ = 2 µS cm −1 , Q peak and Q total are relatively close (∼40-60 nC). At σ = 300 µS cm −1 , the charge values become different (∼150 vs. ∼400 nC) depending on the integration interval. The difference becomes more and more significant at higher σ values. Such a difference between Q peak and Q total is probably due to the contribution of the conduction current that become important at higher σ; this point will be discussed later.

ICCD imaging of the discharge
In this section, we show the features of discharge emission at the solution surface acquired using ICCD imaging technique. Three exposure times of the ICCD were used: 50, 5, and 1 ns. The former provides the overall discharge morphology, while the latter show the temporal dynamics of the streamer. Figure 6 shows the 50 ns-integrated images of the discharge under different conditions of σ and V a (d was fixed at 200 µm). The pattern of plasma emission clearly depends on both V a and σ. For instance, at σ = 2 µS cm −1 and for both V a (14 and 20 kV), the emission pattern is a disc-like (diameter of ∼2 mm) connected to filaments of 2-3 mm that are distributed evenly around the disc. At V a = 14 kV and higher σ (650 and 1000 µS cm −1 ), the emission intensity decreases significantly, but it remains possible to distinguish the presence of the disc as well as the organized filaments that have relatively shorter length. At V a = 20 kV and higher σ, the same remarks remain valid, but the intensity of the disc is much higher than that of the filaments. The dependence of the emission on V a and σ concurs with the results of electrical characteristics. As discussed earlier, at V a = 14 kV, τ pulse < τ r at σ = 2 µS cm −1 and τ pulse > τ r at σ = 650 and 1000 µS cm −1 . Therefore, it is expected that the discharge emission at low σ is much higher than that at high σ, because in the latter case the presence of ions in solution can neutralize the charge accumulation on its surface. In the case of V a = 20 kV, the discharge emission of the disc-like structure remains comparable for different σ cases, but the length and intensity of the organized filaments are significantly reduced. In this case, as τ pulse ∼ τ r , except in the case of σ = 2 µS cm −1 , the ions in the solution do not move under the action of the E-field, i.e. the major part of the energy is spent to produce the discharge that is characterized by a rather high emission. Concerning the organized filaments, as they appear several nanoseconds later, and their propagation is expected to result from charge accumulation at the solution surface and from the produced space charge field, the charged species in solution do have time to reorganize and neutralize the charge accumulated at solution's surface leading to a shorter propagation length.
To further investigate the temporal dynamics of the discharge, 1 ns-integrated images were performed under different conditions. The results are shown in figure 7. Overall, the evolution of the discharge emission for the selected conditions of V a and σ is very similar. Indeed, the emission starts with a disc-like emission that expands and evolves toward a ringlike structure. Finally, because of the destabilization caused   by radial electronic avalanches [36], the ring splits into plasma dots, which are the head of the streamers. Moreover, the time at which the transition from disc to ring (∼3-4 ns) and from ring to dots (∼4-5 ns) occur does not significantly depend on V a and σ. This finding is of interest as it demonstrates that despite the differences in discharge conditions, the dynamics of streamers at the water surface remains comparable. This further supports the assumption that streamers are propagating under the action of space charge field created by charge separation at the streamer's head. However, the total emission lifetime (i.e. propagation length) depends on V a and σ. Indeed, the higher is V a and the smaller is σ, the longer is the propagation length. This finding further shows that streamers do interact with the solution and are sensitive to its properties (here electrical conductivity) while propagating. Therefore, it seems that the propagation length is shorter at higher σ due to the reduction of the E-field at water surface because of the presence of charged species in the solution.
The acquired temporal resolved images (thousands) are then processed using an algorithm that we developed to achieve two goals, namely report the instantaneous radial position of plasma dots and their number (N dots ). The developed algorithm is detailed in [34]. Briefly it consists to (i) plot the images in polar coordinates (r, θ), (ii) sum the pixels over θ to provide the plasma dot position at each time, and (iii) sum the pixels over r to provide a series of peaks, with their number corresponding to N dots . Figure 8 shows the temporal evolution of the radial position of the plasma dots for V a = 14 and 20 kV and for σ = 2, 650, and 1000 µS cm −1 . We note that at 14 kV the plasma emission remains visible up to ∼20 ns at σ = 2 µS cm −1 , but only up to ∼12 ns at higher σ. At V a = 20 kV, it remains visible up to ∼20 ns at σ = 2 µS cm −1 , while it disappears from ∼15 ns at higher σ.
In all cases, the data are well fitted using the mathematical function (A + Bexp(−t/τ ); A, B, and τ are constants and their values are provided in the figure). Although this specific mathematical function was chosen here to describe the data, other functions may be used as well. The main objective of using such a fit is indeed to describe the data by a simple enough function whose derivative provides the instantaneous speed of propagation. The initial velocity at σ = 2 µS cm −1 and V a =14 kV is smaller than that at higher σ (∼0.4 vs. 0.9-1.0 mm ns). In the case of V a = 20 kV, the influence of σ on the initial propagation velocity is not significant, but interestingly the measured values at high σ are lower than those measured in the case of V a = 14 kV (∼0.6-0.7 vs. 0.9-1.0 mm ns). This behavior can be understood by recalling the V bd values. Indeed, at V a = 14 kV and high σ, V bd is higher than that at V a = 20 kV (for instance, at d = 200 µm, we measured ∼10 vs. 7 kV, figure 3). Indeed, as the ions in the solution move, a higher breakdown voltage was needed to induce breakdown. In both V a cases, the velocity measured in high-σ cases decreases rather quickly as compared to the low-σ case. In fact, this rapid decrease illustrates the effectiveness of the ions in the solution to neutralize the space charge at the streamer head.
The number of plasma dots, determined by processing the images under the different experimental conditions are shown in figures 9(a) and (b) for V a = 14 and 20 kV, respectively; only the results at d = 200 µm are shown, as the trends are similar for other distances. In the figures, we show the statistical variation as well as the average with standard deviation. At V a = 14 kV, N dots was rather constant in average (∼15) despite the higher V bd measured at high σ. The plasma dots are likely formed by the destabilization of the circular ionization front [36]. In this context, it is expected that the higher is the accumulated charges at solution's surface, the more stable the circular ionization front, the longer the propagation length, and the higher the N dots . Therefore, at V a = 14 kV, the increase of V bd with σ is likely compensated by the neutralization of accumulated charge by ions in solution. At V a = 20 kV, N dots decreases quasi linearly from ∼20 to ∼14 when σ increases from 2 to 1000 µS cm −1 . This behavior can be related to a decrease of accumulated charge at the water surface. Although we have previously showed (table 1) that at V a = 20 kV and high σ, τ pulse ∼ τ r , which means that the charged species does not have time to reorganize before breakdown, it is worth noting that the dots are formed several nanoseconds after ignition, which a priori is enough to induce ions movement to neutralize the accumulated charges. Figure 9(c) depicts typical 5 ns-integrated ICCD images that show the plasma dots at V a = 14 and 20 kV for different σ. These images correspond to discharges that occurred at the same time delay of 10 ns after ignition.

Discussion
This Section aims to summarize the main experimental findings, while addressing the influence of σ on some properties of the discharges at the solution surface. Concerning the electrical characteristics, the results have clearly shown two behaviors depending on V a . In fact, the main parameters driving the discharge behavior are not the amplitude of the voltage (V a ) itself but rather τ pulse (defined as the time delay to achieve breakdown) and V bd . τ pulse is to be compared with τ r (relaxation time of the solution) which is inversely proportional to σ. For τ pulse > τ r , that corresponds to the case of V a = 14 kV and σ ⩾ 300 µS cm −1 , we observed an increase of V bd and of ∆V with σ. This means that the ions in solution respond to the E-field before breakdown occurs. Therefore, this phenomenon must be balanced by a higher voltage. In the opposite case, when τ pulse is shorter or comparable to τ r , corresponding to the case of V a = 20 kV and σ ⩾ 300 µS cm −1 , the ions barely respond the E-field before breakdown, and the slight decrease of V bd may be related to the decrease of the resistivity of the global electrical circuit. Akishev et al [8] have nicely addressed the physical description of such a situation. Indeed, for τ pulse < τ r , the conduction current in water does not play a significant role as compared to the displacement current in the establishment of the discharge. The solution can be thus considered as a dielectric medium. In this context, similarities have been reported between discharges with solution at low-σ and surface dielectric barrier discharges [20,37]. On the other hand, for τ pulse > τ r , the conductive current flowing in solution plays a dominant role in the discharge occurrence. The authors also indicated that for a relatively high solution conductivity, it becomes impossible to produce discharge as they are shunted by the conductive solution. Moreover, in a numerical simulation study [15], the propagation of streamer at the surface of a dielectric material (ε = 81) has been simulated under different conditions of electrical conductivities, including 5 and 600 µS cm −1 , with voltage characteristic time of τ pulse ∼ 5 ns; these conditions are close to our experimental conditions of V a = 20 kV. The authors simulated the voltage waveforms and observed a drop with values comparable to those measured here (∼0.5 kV). Temporal resolved ICCD images have shown the formation and propagation of plasma dots at the solution surface in each investigated condition. This is an interesting finding as it means that streamers continue to propagate even at a σ as high as 1000 µS cm −1 (in fact tests are also performed at 2500 µS cm −1 and plasma dots remained observable). Considering these plasma dots as a pattern, discussing their formation mechanism(s) is primordial. Because they are formed few nanoseconds after the symmetrical ring, plasma dots likely result from the destabilization of the circular ionization front [36]. As a preliminary answer that needs to be validated by simulation (under development), we believe that there is a strong role of the ignition of radial avalanches that may trig and develop instabilities in the ring; such instabilities induce distortion of the ring and its decomposition into dots. Such a mechanism has been highlighted by Xiong et al [36] in a simulation study of streamer propagation in air, where the authors mentioned that 'The emergence of individual streamers in the coaxial geometry results from the linear destabilization of the circular ionization front. This destabilization typically undergoes three stages: (1) initially stable circular ionization front, (2) destabilization and distortion of the ionization front and (3) breakaway of individual streamers. ' Although the plasma dot formation does not depend on the conditions (e.g. voltage magnitude and pulse width, gap distance, and solution conductivity), the propagation length as well as the emission intensity significantly depend on the conditions. In this context, as the propagation of the plasma dots, i.e. streamers' head, strongly depends on the space charge field produced by a single plasma dot, estimating this field is crucial. For this purpose, as performed in previous studies [18,27], we will utilize the results presented above: the injected charge (calculated based on the main peak, Q peak , as well as on the complete waveform, Q total ) and N dots . Knowing these quantities for each discharge condition, we calculate the charge per dot, Q dot , in the two cases, i.e. Q peak /N dots and Q total /N dots . The results presented in figure 10 show the data for one distance (200 µm) and two V a values: 14 and 20 kV. The results for other distances (d = 400 and 600 µm) are similar to those obtained for d = 200 µm, thus, they are not shown here. It is worthy to note that, (i) in all cases, Q dot increases quasilinearly with σ and (ii) at low σ, Q dot is similar regardless the scheme of calculation (Q peak /N dots or Q total /N dots ); however, at high σ, the values of Q dot calculated by Q total /N dots are higher than that calculated by Q peak /N dots .
The increase of Q dot with σ means that the space charge field produced by a single dot should be higher. Therefore, the plasma dot is expected to propagate farther at the water surface, but this contradicts the ICCD images that showed short propagation length at higher σ. In fact, this contradiction is due to the unrealistic high values of Q dot measured for higher σ values. This is because the measured current accounts for the conduction current in the solution as well as for the discharge current. Separating these two contributions may be helpful in the calculation of Q dot . Rumbach et al [38] have developed a stationary model that may allow distinguishing the two contributions, but it cannot be utilized in the conditions of this study because of the fast temporal dynamics. Therefore, to obtain more realistic values of Q dot , we consider the case of σ = 2 µS cm −1 as a reference (because the conductive current can be neglected at low σ). In this context, we previously studied the propagation of plasma dots at the surface of distilled water (σ = 2 µS cm −1 ) [34] and estimated a plasma dot mobility (µ dot ) of ∼1.5 cm 2 V −1 s −1 . Then, using µ dot and the instantaneous propagation velocity (v dot ) at high σ, the relation v dot (t)= µ dot E dot (t) is used to determine the temporal evolution of the space charge field produced by the plasma dot, E dot (t); it is the component parallel to the solution's surface. The results are presented in figures 11(a) and (b) for V a = 14 and 20 kV, respectively. Then, we used Gauss's law to evaluate Q dot (t) as well as the number of charges in a plasma dot (figures 11(c) and (d). Interestingly, this simple calculation indicates that at V a = 14 kV and at high σ, more charges are initially injected. This finding can be explained by the fact that because τ pulse > τ r at this condition, more charges are needed to balance the phenomenon of ions movement in the solution leading to E-field reduction at the solution surface. At V a = 20 kV and high σ, because τ pulse ∼ τ r , the ions movement in the solution can be neglected initially, and thus less charges are needed. However, due to a later interaction with the solution, Q dot at high σ decreased faster than that at low σ. These values as well their temporal evolution will be further confirmed by a simulation study.

Summary
In this paper, we performed an experimental study to investigate the characteristics of nanosecond discharges in air in contact with a solution at various electrical conductivities (σ). A large number of discharges were characterized electrically and by imaging. Depending on V a and σ, two discharge modes were found. For instance, in the case of V a = 14 kV, V bd increases with σ, while it slightly decreases in the case of V a = 20 kV. The behavior of the results can be well explained by comparing τ pulse (the delay to observe breakdown) to τ r (polarization-or Maxwell-time), which corresponds to the characteristic time to neutralize the space charge field due to the movement of ions in solution. The propagation of discharge at solution surface is also investigated using 1 ns-time-resolved ICCD images. In all conditions, we found that the discharge starts with a disc-like emission that expands and turn into a ring-like structure to finally break down into plasma dots. The images were processed to obtain the number of plasma dots (N dots ) and their propagation velocity. In the case of V a = 14 kV, we statistically found that N dots does not vary significantly with σ even though a higher V bd was measured; such higher V bd likely compensates the lowering of the E-field due to ions movement in solution. In the case of V a = 20 kV, although V bd was comparable, the decrease of N dots with σ is probably due charge neutralization by the movement of ions, as the dots are formed few nanoseconds after breakdown. On the other hand, we used the discharge current to calculate the injected charge and, knowing N dots , we found the charge per plasma dot, Q dot . The values of Q dot are found unrealistically high for larger σ values. This is because the measured current accounts for both the discharge current and the conduction one in the solution. To obtain more realistic values of Q dot , we considered the case of σ = 2 µS cm −1 as a reference, because in this condition, the conductive current can be neglected. Thus, using the plasma dot mobility and the instantaneous propagation velocity at different σ, we determined the E-field and finally Q dot . The findings reported here are of interest for many applications, particularly in plasma medicine where the plasma-solution interactions (charge transfer, E-field, etc) are considered as a cornerstone.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.