Physics of plasma jets and interaction with surfaces: review on modelling and experiments

Electric field diagnostics for non-thermal plasma jets can roughly be divided into methods using active spectroscopy, optical emission spectroscopy (OES), methods based on electro-optic effects and a combination of emission spectroscopy and various versions of collisional radiative models.

A frequent example of the latter is combining OES on the spectrum of nitrogen (although using other systems is also often done) and calculating the electric field in the system from the line intensity ratios of selected lines [24]. The ratio depends on the population density of the upper states of the associated lines, which can be computed in a collisional radiative model. Emission lines from different systems can also be used when working in gas mixtures. Finally, the lines used in this method are typically those originating from levels that are populated by direct electron impact. As the associated reaction rates thus depend on the electron temperature, which is a function of the reduced electric field, this method is also suitable for the determination of electron temperature [25]. The positive side of this approach is that information can be obtained using setups that are cheaper, simpler and easier to set up and to use than those of other methods. Moreover, pre-calculated relationships between intensity ratios and the reduced electric field can be used [26]. In addition, vast improvements in the quality of the results are attainable by improving the detection arm of the setup; a good example of extreme temporal accuracy in an erratic system is the use of cross-correlation spectroscopy in the detection arm of the setup, resulting in data that is otherwise challenging to obtain [27]. Still, the associated collisional radiative models of nitrogen or mixtures with nitrogen are complex and the sensitivity of the results to the conditions under which the discharge is measured and modelled should be central when comparing results [28–30].

An emission spectroscopy technique specific to jets in helium is the OES on the Stark-shifted forbidden He lines. In general OES probes the region of the discharge emitting light, which in case of kHz-driven jets is the head of the discharge, very closely linked to the ionization wave. Generally speaking, transitions exist that are forbidden by selection rules under unperturbed conditions. When a strong electric field is present these lines start appearing and shifting in the spectrum as a function of the electric field strength. A method based on the theory by Foster [31] has been later developed by, among others, Kuraica and his team [32–35]. It has been used extensively on He jets that are freely expanding or interacting with substrates [36–49]. That research has revealed that the gas flow speed, being approximately five orders of magnitude slower than the speed of the ionization wave constituting the kHz discharge, still has a fundamental role in the determination of the properties of the discharge. It has been shown that the electric field grows with the distance from the capillary, but also that this phenomenon depends critically on the gas mixing in the effluent of the jet, which is determined by the gas flow speed [43]. In the interaction with substrates it has been shown that the presence of the substrate causes an increase of the electric field just above the surface of the target [46, 47], through two effects: the redistribution of the equipotential lines at the interface and the narrowing of the He channel near the surface. However, if the gas flow is high enough to form a wide footprint on the interface, then the presence of the target does not increase the local electric field with respect to the freely expanding jet [46]. More details about the role of gas flow and targets on electric field profile observed experimentally will be given in light of modelling results in sections 3.3 and 3.4.

Active spectroscopy techniques most used on jets are CARS (coherent anti-Stokes Raman spectroscopy) based four-wave mixing [50, 51] and electric field induced second harmonic generation (E-FISH) [52, 53]. Of the two, E-FISH is simpler to implement. However, both methods have shown impressive results in electric field measurements, especially considering temporal and spatial resolution of the measurements [54, 55]. While in OES-based techniques the resolution is determined by the detection-arm of the setup, in this case the properties of the used laser pulses allow for significant improvements and sub-ns temporal resolution. In addition, active techniques offer the opportunity to measure across the entire structure of the discharge, not only the light-emitting part. However, both methods depend on the susceptibility of the gas under investigation. Consequently, both are predominantly used on nitrogen discharges and are difficult to use on helium. In addition, the influence of the laser beam on the discharge is not always negligible and should be a point of attention [56].

While the electric field in the gas phase is determined by the free charges in volume that make up the discharge and the Laplacian potential distribution, the introduction of surfaces exposed to a plasma leads to the consideration of the effect of surface charges on the electric field. As such, most used techniques for the measurement of the electric field inside dielectric targets are based on the Pockels effect. A relatively simple technique is based on the Sénarmont setup and is capable of measuring only one component of the electric field vector. It was developed for the interaction between streamers and dielectric surfaces [57–65] but has since also been applied to plasma jets [66–69]. This version of the setup is relatively easy to implement and, assuming an ideal configuration, the calculation of the electric field component in question is straightforward and independent of the type of gas. However, imperfections in the setup cause errors in the calculated electric field when the polarization state of the probing light beam is not fully characterized. A more complicated method involves the use of Mueller polarimetry in combination with the Pockels effect, where a complete set of information on the polarization state of the probing light is obtained, revealing all the components where the electric field information is stored [70–74]. In addition, this method allows for the use of imperfect, depolarizing targets, making it possible to measure the electric field in targets closer to application [74]. Furthermore, commercial probes that measure the electric field on the basis of the Pockels effect are available for use on non-thermal plasmas. In this case the Pockels element is not in direct contact with the plasma and the electric field has to be inferred from the relative position of the probe with respect to the plasma [75–77].

Except for the case when commercial probes are used at a distance from the plasma jet to measure only the surrounding electric field, when the plasma is in direct contact with the electro-optical crystal (whether for Pockels or Mueller polarimetry methods) the electric field detected is the one induced by charge interacting with and deposited on the dielectric. The results are, therefore, somewhat different than those obtained by techniques described previously. Indeed, the techniques based on OES or active gas-phase spectroscopy will result in the electric field profile almost exactly reflecting the dynamics of the integral emitted light from the discharge, meaning that the high electric field front will follow the same trajectory as a function of time as the emitted light, within the limits of the resolution of the measurement setup. In contrast, Pockels-based techniques show the electric field footprint inside the dielectric. The charge and the electric field footprint appear with the same dynamics as the light emitted from the surface discharge, but unlike the emitted light, they typically remain until polarity change from the voltage source takes place.

In addition, it should be noted that the discharge growing in the gas phase has a different electric field profile than that interacting with a target. Modelling results show a fast redistribution of charge and equipotential lines as soon as the discharge reaches a dielectric target. Therefore, the comparison of gas-phase data and the data measured on the surface, even if the gas phase data is measured just above the surface, is not straightforward. Pockels-based diagnostics for the determination of electric field inside a target and surface charge require specific electro-optical materials like the BSO (Bi₁₂SiO₂₀) crystal. This material has a dielectric constant of 56. Simulations have shown that when the dielectric constant of the target is greater than four the experienced electric field inside the target is governed by surface charges, while if it is lower the free volume charges play a role as well [78].

In an attempt to give an overview of the experimentally determined electric field data, results obtained with different techniques in the plasma (but not inside the target with electro-optic crystals) have been gathered in figure 1. On the left-hand side the data of the freely expanding jet is gathered [24, 35, 39, 43, 44, 47, 52, 79, 80], while in the graph on the right-hand side the jets are impinging on targets [36, 38, 47, 49, 53]. The x-axis of the left graph is scaled with the gas flow speed. The reason for this is that within a few cm from the end of the capillary, for the flow speeds typically used in plasma jets, this scaling is likely to reflect the gas composition for a given gas type. To demonstrate this, the graph in the inset on the left shows the gas composition by means of air fraction as a function of the distance from the nozzle scaled to the gas flow speed. The data has been taken from the work by Sobota et al (2016) [43], from the results of a simulation of gas mixing of He freely expanding in air. The simulation has been done for particular jet dimensions and flow speeds, thus the validity of the approach can only be inferred to the electric field results from works using the same jet. However, the results do show that close to the end of the capillary, the distance from the nozzle scaled with the gas flow speed does reflect the gas composition.

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As shown in a number of papers, the electric field strength grows with the distance from the nozzle. The left graph in figure 1 shows that the electric field strength is likely a function of gas composition, as almost all data gathers along a unique line, for both square pulse voltages and sinusoidal ones. This confirms the strong influence of air entrainment on the electric field value. For large gas velocities, the electric field value tends to be slightly higher, which can be explained by the fact that these data correspond to gas flow values at which turbulence starts appearing, leading to higher air admixture at a given position. The opposite trend is observed with the kHz argon jet for which the electric field has been measured by E-FISH [52], shown in green inverted triangles in figure 1(a). This can be caused by several effects, such as the Ar jet consisting of branching streamers, while the kHz He jets feature guided streamers. The random nature of streamers makes it difficult to collect all data from the same position in the ionization front. E-FISH as an active technique is able to collect data from all points in space, unlike the OES-based technique used for the most of the other data in this figure. The red square data in the top left corner of the figure are from [39]. An important addition to the setup in that work was a grounded (or biased) cylinder around the effluent of the jet, which influences the electric field profile, as was also shown in that article. The data denoted with violet circles and triangles are from [24]. Both have been obtained in the same jet. The triangles were measured with an electrical probe on which the jet is impacting at variable distances from the nozzle. The circles were obtained from N₂ and ${\mathrm{N}}_{2}^{+}$ line ratios. The values in this case have been divided by 5 in order to be able to appear on this figure. There could be several reasons for this discrepancy, for example the complexity behind the population mechanism of excited N₂ and ${\mathrm{N}}_{2}^{+}$ levels, especially in interaction with He [29, 30].

The graph on the right-hand side shows the electric field data when a target is present. Overall, the electric field strength shows the same behaviour as in the freely expanding jet, with the exception in the areas just next to the targets, where a sharp increase of the electric field strength is observed. This is caused by the redistribution of the electric field lines at the interface and the narrowing of the He channel at the interface, as will be detailed in section 3.4. Again the majority of data falls on the same line, showing clearly the dependence of the electric field strength on gas mixing. The data obtained by the E-FISH method [53], shown here in blue triangles, was obtained in an atypical jet system comprised of two surface dielectric barrier discharges joined together to form a large rectangular nozzle. This peculiar configuration allows to increase the interaction length between the helium plasma and the laser beam to increase the weak E-FISH signal induced by helium susceptibility. The discharge is ignited in a helium flow above liquid target. As the gas flow configuration is completely different than for other jets in this graph, the gas velocity has been multiplied by 20 in order to be able to visualize the data.

Electron properties, such as electron density and temperature or energy distribution can be measured or inferred by more methods than the electric field. For example, electron density can be estimated from current measurements [81], but it can also be measured by microwave interferometry [82] or microwave cavity resonance spectroscopy [83] or electrical probes, taking care that they do not alter the discharge. Still, just as is the case for the electric field, directly measuring electron properties in jets is not simple due to their highly transient nature, small size, significant jitter and to low electron densities. That is why trusted methods are borrowed from diagnostics on plasmas that are significantly different from plasma jets, like low-pressure steady-state plasmas.

One OES technique for the measurement of electron density is Stark broadening [82, 84, 85]. It can be applied to atomic lines of different gases like oxygen, nitrogen or hydrogen. The idea is that just like in any general type of collisional line broadening, the interaction between the system under observation (our probed atoms) and the perturber results in a shift in the atomic energy levels. In the case of the interaction of a radiating system and a charged particle, the interaction is of the Coulomb type and the resulting broadening of emitted spectral lines is a function of the density and the temperature of the perturber. The method, thus, relies on the electrostatic interaction between particles, but it also assumes that this field originates only from the charge in the close vicinity of the emitting atoms, a 'microfield' [86] in an otherwise neutral medium. This is crucial, as plasma jets in the kHz region comprise of propagating ionization waves with high electric fields in the head of the discharge, complicating the analysis. Low electron densities complicate the analysis further, as has been discussed in several works [84, 87].

Thomson scattering is an active technique for the determination of electron properties in plasma jets [48, 88–99]. Scattering light on a plasma results in Thomson scattering, Raman scattering (in case of a molecular gas) and Rayleigh scattering, the latter largely dominating the signal intensity. Methods exist to filter out the component belonging to Rayleigh scattering. This is necessary since it is a multitude of orders bigger in magnitude and therefore completely overshadows the other two components. Thomson-scattered signal belongs to the free electrons in the system, if we limit ourselves to non-thermal plasma jets, but appears in the same spectral range as the Raman-scattered signal. If the signal from Thomson scattering is intense enough, a fitting procedure can separate its contribution from the signal from Raman scattering. The intensity of the signal depends on the particle density and its shape on the electron energy distribution function (EEDF). In atomic plasmas it is therefore also possible to determine the EEDF, while in molecular plasmas a Maxwellian distribution is typically assumed. As the intensity has to be high enough, the minimum detectable electron density is around 10¹⁹ m⁻³, but the actual limit depends on the Thomson scattering setup. The method is difficult and expensive to set up, however, electron properties can be measured directly, regardless of the presence of high electric field in the front or other discharge parameters. There are several operational difficulties. For example, the cross section for Thomson scattering is very small, meaning that the integration times in measurements can be very long. One solution is an increase of the intensity of the scattered light, however, this can disturb the discharge. Another issue is that the jitter associated with all non-thermal atmospheric pressure plasmas in combination with a very thin ionization front with high density gradients in plasma jets results in measurements that probe the electrons over the entire volume of the ionization front and somewhat after it. The resulting measured electron properties are thus still not resolved over the head of the discharge and often the electron densities are examined in the channel behind the front. This has big consequences on the electron temperature, that is more likely to reflect the lower-energy electrons just after the head of the discharge instead of the high-energy electrons at its tip [96].

Figure 2 shows an overview of the electron density data from the literature. Data is taken for jets with different excitation sources and buffer gases, since there is few data available for kHz He jets and it is useful to compare it with other jet sources. The data spans over six orders of magnitude. There is quite some data available that is not axially resolved along the plume; in those cases in figure 2 the data is represented by values at x = 0, with minima and maxima connected by a straight line. Furthermore, the articles showing the radial profile of electron density are here represented by the maximum value only. Other papers provide the time evolution of electron density at a given position along the plasma jet. In this case, data are represented with a shaded vertical column at the corresponding position, with the minima and maxima points connected by a straight line. A variety of methods is used for the determination of the electron density and most of them are not direct measurements. Nevertheless, the densities belonging to Ar jets are on average higher than those in He jets, which had been stated by Hofmann et al (2011) [100]. Furthermore, the measurements in He show that the electron density profile follows the electric field profile, discussed in the previous subsection. Both the calculations of electron density from the electric field profile in Sretenović et al (2017) [44] and Thomson scattering measurements [91, 94, 96] give a rising trend of the electron density as the distance from the exit of the capillary increases. The only exception is the decaying trend of electron density measured in [101] with the broadband emission of neutral bremsstrahlung, which could be influenced by the volume of plasma actually emitting this radiation.

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Figure 2 shows a wide range of electron densities measured in He operated jets. To analyse the differences between the results, the maximum value of the electron density of a spatial or temporal series is compared between the various works. The lowest n_e,max values are reported by Karakas et al (2012) [102] and Kieft et al (2004) [103] with respectively 3 × 10¹⁰ cm⁻³ and 7 × 10¹⁰ cm⁻³. These values have been estimated using the detected current signal and by estimating the average cross section and average drift velocity. Where Kieft et al (2004) [103] examined a radio frequency-powered plasma needle without an extended plasma plume commonly seen in a plasma jet, Karakas et al (2012) [102] did investigate the plume of a He jet powered by 2 μs high voltage pulses of 5 kHz. The value of 3 × 10¹⁰ cm⁻³ is based on a current peak of 10 mA, a cross-section of 0.39 cm² and a drift velocity of 5.6 × 10⁴ m s⁻¹. Slightly higher values of electron density are shown in Sretenović et al (2017) [44] up to 1 × 10¹¹ cm⁻³, estimated through measured electric field values based on Stark shift of forbidden helium lines. This jet has an inner AC powered electrode with an outer grounded ring, rather than two external rings as in Karakas et al (2012) [102], as well as a slightly thinner inner tube diameter (2.5 mm compared to 3.0 mm) which affects the electron density, as will be shown in section 3.2.1.

Values of 5 × 10¹¹ cm⁻³ have been measured through Rayleigh microwave scattering by Lin and Keidar (2016) [104] for a freely expanding He jet with inner electrode. Additionally, it can be seen from photographs that their plume is much longer than shown by Sretenović et al (2017) [44]. This can be a result of the higher He flow used (5 slm compared to 1.5 slm) and/or lower operating frequency (16 kHz compared to 30 kHz), which can influence the amount of leftover charges present for the next HV cycle. A different AC operated plasma jet has been examined by Yambe et al (2015) [105], which estimates n_e,max = 1–5 × 10¹² cm⁻³, based on a measured current of 10 mA, a drift velocity of approximately 10 km s⁻¹ and a cross-section of 0.01 cm². The large difference with e.g. [102] comes from the estimation of the cross-section, which is much smaller in [105], since the investigated ionization wave travels in a thin ring rather than a 'bullet'-package through the plume.

Higher values have been detected using Thomson scattering by Hofmans et al (2020) [96] and Hübner et al (2014) [91], of 1 × 10¹³ and 2 × 10¹³ cm⁻³, respectively. These works have investigated pulsed helium jets instead of AC powered ones and the rise in electron density can also be noticed through an increase in detected current (180 mA [106]) and through a drift velocity of 8 × 10⁴ m s⁻¹ [48]. The same jet as in Hofmans et al (2020) [96] has been examined by Klarenaar et al (2018) [94] when impacting various surfaces like glass and metal. An increase of electron density towards 3 × 10¹³ cm⁻³ has been shown when interacting with glass (close to the surface) and towards 1 × 10¹⁴ cm⁻³ for a metal target at floating potential.

Recently, another laser based technique has been used by Lietz et al (2020) [107], called laser-collision-induced fluorescence (LCIF). A laser at 389 nm is used to excite He to the He(2³ S) metastable state which is then excited in the plasma by electron impact to He(3³ P), whose radiative relaxation is monitored to deduce the electron density. The data from [107] shown on figure 2 have been obtained with a pure He flow and a shroud gas of He containing 5% H₂O expanding in a vacuum chamber pumped down to 600 Torr. Despite this peculiar configuration, the measurement shows an increasing profile of electron density with distance from the nozzle similarly to all Thomson scattering data obtained in He jet at atmospheric pressure.

Using Stark broadening, a n_e,max = 5 × 10¹³ cm⁻³ has been observed by Hofmann et al (2011) [100] for a He RF powered jet with a grounded metal plate (with a hole at the axis of the plasma jet) in front of the jet. This particular electrode configuration is very unique and therefore difficult to compare with other plasma jet systems.

Two different He pulsed systems have been examined through Thomson scattering by Jiang et al (2019) [95] and Wu et al (2021) [98], both revealing high electron densities of 1 × 10¹⁴ and 1–4 × 10¹⁴ cm⁻³, respectively. Although these high values are remarkable for helium plasma jets, they are plausible when considering the power delivery to the plasma. Jiang et al (2019) [95] have operated their freely-expanding jet with a low 10 Hz frequency and pulses of less than 200 ns delivering up to 2 mJ per pulse, which is much higher than for instance the 10–40 μJ per pulse reported in [106] for the jet used in Hofmans et al (2020) [96]. Wu et al (2021) [98] have operated their jet at a comparable 8 kHz but their reported current of 500 mA is much higher than in the other works.

Lastly, high electron densities have been observed inside the capillary tube in two He AC powered jets: 9 × 10¹³ cm⁻³ in Tschang et al (2020) [116] and 1–8 × 10¹⁴ cm⁻³ in Jõgi et al (2014) [110] inside the capillary tube. The first case is unique since it is the only jet examined (through N₂ line ratios) in the presence of a strong external magnetic field (2 Tesla) impacting on a small liquid target [116]. Jõgi et al (2014) [110] have reported the highest values but comparison with the other papers is difficult since the densities are examined inside the capillary tube of a micro-plasma jet with a small inner diameter of 80 μm. Section 3.2.1 will show that this has a significant effect on the electron densities.

For Ar operated jets, figure 2 shows a wide range of electron densities as well. Similarly as for the helium results, these densities are not only the maximum obtained in a measurement series but also the spatially and temporally resolved results. To compare the various results, the maximum reported electron density of each paper will now be discussed. The lowest n_e,max for an Ar jet has been reported by Hübner et al (2013) [90] with 7 × 10¹³ cm⁻³. The streamer investigated in that work has been generated by an applied pulsed voltage (5 kHz, 500 ns) to a 1 mm needle housed in a glass tube with 1.8 mm inner diameter and 3 mm outer diameter. The distance from the tip of the needle to the end of the glass tube is not mentioned in the text, but from a photograph an estimation of 3 mm can be made. As a result of this short distance, there is little charging of the glass tube before the ionization wave propagates away in the argon–air mixing plume region.

Slightly higher n_e,max values of 1 × 10¹⁴ cm⁻³ have been reported in [100, 109, 111]. These works include plasma jets of various designs but all are operated using voltages with a high frequency, i.e. radio-frequency or microwave. Another radio-frequency operated argon plasma jet has been reported by van Gessel et al (2013) [89] with values of electron density ranging between 1–3 × 10¹⁴ cm⁻³. This RF-jet is modulated at 20 kHz with a duty cycle of 20%, thus with a repetitive plasma on- and off-times of 10 and 40 μs. The aforementioned Ar papers all examined a free-expanding jet without the presence of a target.

Qian et al (2010) [108] have reported maximum electron density values of 2 × 10¹⁴ cm⁻³, for a 38 kHz AC powered jet interacting with a glass target which is electrically grounded on the backside. A different AC powered jet (23 kHz) has been examined by Slikboer and Walsh (2021) [97] interacting with a liquid target (tap water), which yields maximum electron densities in the centre of the plume of 6 × 10¹⁴ cm⁻³ when the liquid is at floating potential and 1 × 10¹⁵ cm⁻³ when grounded. The same jet has been examined for a pulsed applied voltage (16 kHz, 5 μs on-time) in [99], measuring n_e,max values of 1 × 10¹⁵ cm⁻³ with a floating potential liquid and 3 × 10¹⁵ cm⁻³ when electrically grounded.

It is worth mentioning that Thomson scattering measurement performed in atmospheric pressure glow discharge in He [93] or in Ar [88] report respectively electron density of 2.5 × 10¹⁵ cm⁻³ and 8 × 10¹⁴ cm⁻³, corresponding to a similar range of electron densities as observed in RF or kHz plasma jets despite a very different type of discharge. In [82], electron densities of 5 × 10¹⁴ cm⁻³ are measured in an Ar microwave surfatron jet source with both microwave interferometry and H_β line broadening showing a good agreement between values obtained by these two techniques.

Whereas all the aforementioned papers have either used Stark broadening or Thomson scattering to determine the electron densities, the measured current has been used by Sup Lim et al (2020) [113] to estimate an electron density of 2–10 × 10¹⁵ cm⁻³ for an AC operated jet (50 kHz) impacting a small liquid reservoir in a cuvette. Besides the different diagnostic technique used, this jet is different from the others since it has a long quartz tube with a significant gap of 60 mm between the powered inner electrode and the edge of the tube.

Even though a wide range of electron densities has been reported for Ar plasma jets of different design and operating parameters, patterns can be found. Freely expanding RF-jets generate lower electron densities than AC or pulsed jets interacting with a surface. Results from Slikboer and Walsh (2021) [97, 99] using the same plasma jet and liquid target suggest that using a grounded target yields higher densities than a target at floating potential. They also suggest that AC powered jets have lower electron density than pulsed jets, similarly to what is observed for helium plasma jets. However, the works reported by Hübner et al (2013) [90] and by Sup Lim et al (2020) [113] seem to contradict this, which could be related to the significantly shorter and longer distances between the inner powered electrode and the edge of the capillary tube, respectively. The influence of this distance has not been investigated further experimentally. In fact, the only results reported inside the capillary tube (in between the electrodes) instead of the plasma plume are those of Schäfer et al (2010) [112], reporting values of 2–4 × 10¹⁴ cm⁻³ for an RF-operated Ar jet.

There is a large variety of atmospheric pressure plasma jet designs as seen in section 2.1, including electrode configurations that have pin and ring electrodes either outside or inside of the dielectric tube forming the jet, or some combination of electrodes located inside and outside [6]. Glass tubes are usually used, with variable length and diameter. In each design variation, an ionization wave propagates through and out of the tube and into the ambient air. A rare gas is usually flowed through the tube, either pure or with a small (<1%–5%) admixture of a molecular gas such as oxygen, nitrogen or water vapour.

There is also a large variety of targets used in experiments. In this work, we focus on solid targets. In models, the description of solid targets is based only on its dimensions and electrical properties. For plasma jets impacting on liquids, the problem is much more complex with an induced chemistry in the liquid and interdependent flow dynamics [8, 140, 141].

We should notice that for discharge simulations it is very important to define accurately the boundary conditions. In particular, it is crucial to define carefully the location of grounded surfaces and report it in publications, as noticed by Lietz and Kushner (2018) [142]. Inclusively, for comparisons with experiments, these may be adapted to ensure that the boundary conditions of Poisson's equation are well defined and are similar in experiments and simulations, as in Xiong et al (2012) [133].

A specific feature of simulations of plasma jets is the need to take into account the spatial variations of gas composition as the buffer gas exits the tube and mixes with air, as also noticed in the recent review by Babaeva and Naidis (2021) [143]. The simplest models can assume no helium–air mixing at the tube exit, as in [18, 19, 22, 144]. In this case, the discharge propagates only in a uniform He-based mixture, surrounded by air. To obtain a more accurate spatial distribution of gases in the plume region, flow calculations have to be coupled with the discharge model. Different approaches can be used to couple flow and plasma dynamics.

• Static flow models:

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As a first approximation, the effect of the plasma on the flow can be neglected. As the gas flow velocity is typically of the order of a few μm/μs (four orders of magnitude lower than the discharge velocity), a static gas hypothesis can be used. As such, during the μs timescale of discharge dynamics, the gas flow can be considered to be frozen with no diffusion of neutral species.

The spatial distribution of gases without plasma can be pre-calculated and provided to discharge models. This approach of coupling a static-solution gas flow with a transient plasma model has been used in most He jet modelling works, such as [123, 135, 146–148]. The flow calculations can be carried out as described in [149–151], by solving the stationary Navier–Stokes equations for laminar and turbulent flows of inhomogeneous gas mixtures self-consistently with the diffusion equation for the same mixing. A buoyant force term is added to the momentum Navier–Stokes equation. Usually, in plasma jets, the flow is laminar (Reynolds number is below 200) and thus the model can be solved for laminar flow conditions.

Furthermore, in kHz pulsed conditions with He buffer gas, the gas heating is usually negligible [13, 14] and is not expected to affect the discharge dynamics. Then, the gas is usually considered to stay at room-temperature (T_g = 300 K), and there is no need to solve the heat transfer equation. Finally, the spatial distribution of the gas composition in the plume region is determined by the input flow, the tube diameter and the tube-target distance.

In most He plasma jet experiments, the buffer He gas flows into ambient air. However, as the discharge usually propagates in a region with air molar fraction below 5%, models can consider that the He buffer gas flows into an atmosphere of only N₂ or O₂, as a first approximation to air. In particular, the discharge dynamics in the plasma plume has been shown to present similarities when using O₂ or air as surrounding gas for a He jet, due to the important role of electronegativity. N₂ and O₂ can also be present in the buffer He gas as impurities or as admixtures to promote reactivity. Figure 3 shows an example of calculated N₂ molar fraction distribution in a jet where He (with 10 ppm N₂) flows into a N₂ atmosphere.

• Full coupling:

The most complete approach considers the interdependence between plasma and flow by solving plasma and flow equations simultaneously using the same mesh and time-step. This approach can take into account possible effects of the plasma on the flow, through either gas heating or the electrohydrodynamic (EHD) force [152–157]. As an example, in Lietz et al (2017) [155] the plasma dynamics has been solved together with modified Navier–Stokes equations and it has been found that localized gas heating takes place in a He jet at the powered electrode during a voltage pulse of 100 ns and produces an acoustic wave that propagates at the gas flow velocity. However, usually the temporal and spatial scales to solve plasma and fluid equations are significantly different (ps and μm for plasma and ms and mm for fluid). In that case, it is more efficient to calculate these equations separately, using different meshes and time-steps. Furthermore, in kHz pulsed conditions with He buffer gas, the gas heating is usually negligible [13, 14] and is not expected to affect the discharge dynamics. We should notice that the use of long applied voltages or of gases other than helium, such as argon, is more likely to require the calculation of gas heating and the study of its impact on the discharge dynamics and structure than in the case of ns or μs pulses.

• Drift–diffusion model based on the local electric field approximation (LFA) and its limits:The simplest fluid models solve the continuity and momentum conservation equations, considering the drift–diffusion approximation, and the LFA. For the sake of simplicity, we henceforth call these models LFA models. The He jet models developed in [18, 19, 21, 22] used this approximation. The following conservation equations are solved in LFA models for electrons, positive ions and negative ions, coupled with Poisson's equation, in cylindrical coordinates (z, r):
  $$\begin{equation}\frac{\partial {n}_{\mathrm{k}}}{\partial t}+\nabla \cdot {\mathbf{j}}_{\mathrm{k}}={S}_{\mathrm{k}},\end{equation} \tag{ 1 }$$
  $$\begin{equation}{\mathbf{j}}_{\mathrm{k}}=({q}_{\mathrm{k}}/\vert e\vert ){n}_{\mathrm{k}}{\mu }_{\text{k}}\mathbf{E}-{D}_{\mathrm{k}}\nabla {n}_{\mathrm{k}},\end{equation} \tag{ 2 }$$
  $$\begin{equation}{{\epsilon}}_{0}\nabla \cdot ({{\epsilon}}_{\mathrm{r}}\nabla V)=-\rho -\sigma {\delta }_{\mathrm{s}},\end{equation} \tag{ 3 }$$
  $$\begin{equation}\mathbf{E}=-\nabla V;\hspace{25.0pt}\rho =\sum {q}_{\mathrm{k}}{n}_{\mathrm{k}},\end{equation} \tag{ 4 }$$
  where the subscript k refers to electrons and ions and n_k, q_k, j_k, μ_k and D_k are the number density, charge, flux, mobility and diffusion coefficient of charged species k, respectively. S_k is the total rate of production and destruction of species k by kinetic processes and by photoionization. V is the electric potential, E the electric field, e the electron charge, ₀ the vacuum permittivity, _r the relative permittivity and δ_s the Kronecker delta (equal to 1 on dielectric/gas interfaces). The numerical solution of Poisson's equation includes not only the geometrical Laplacian potential distribution, defined by boundary conditions, but also the contributions of volume net charge density ρ and surface charge density σ on dielectric surfaces (tube and target, if these have a dielectric character).Using the LFA, a local relation between the electron kinetics coefficients and the reduced electric field (E/N) is considered. The LFA assumes that the energy gain by electrons from the electric field is locally balanced by the losses due to collisions with neutrals, which in He at atmospheric pressure take place with a frequency of the order of 10¹² s⁻¹. The LFA has been shown to be well justified for the description of electrons in air streamers propagating in volume by comparisons with particle in cell/Monte Carlo (PIC/MCC) simulations in [158]. However, the LFA is not valid when the temporal variations of E/N are of the order of the electron–neutral collisional timescales. Although E/N in the discharge front is much lower in He plasma jets than in air streamers, so is the electron–neutral collision frequency, and the LFA may not be valid when the discharge interacts with surfaces. This is the case, in particular, when a positive ionization front in a He plasma jet impacts a target of very high dielectric permittivity, and when the discharge interacts with the inner surface of the dielectric tube at the location where a grounded ring is wrapped around the tube. In these situations, a positive sheath is formed due to the limitation on electron emission from the surface to the plasma [124]. It should be taken into account that, in that sheath, electrons transported through drift move in the sense of the E acceleration and are thus heated, while the electrons transported through diffusion in the opposite sense of acceleration are cooled. The LFA, with the approximation of mean electron energy (_m) only dependent on the local E/N, does not take these physical aspects into account, and thus overestimates or underestimates electron-impact rate coefficients [159, 160]. In the case of the interaction between positive discharges and surfaces, the LFA overestimates these coefficients. As a result, in that sheath, as the electric field and the electron-impact ionization rate reinforce each other and the sheath becomes thinner, the collapse of the sheath and unphysical results can be obtained.
• Drift–diffusion model based on the local mean electron energy approximation (LMEA):One possible solution to the problems raised by the LFA is to introduce corrections to the ionization rate proportional to the gradients of electron density or electric field [159, 161–163]. However, a more consistent approach to account for non-locality is to evaluate the mean electron energy (_m) through the inclusion of the electron energy conservation equation in the model, and then use the LMEA instead of the LFA. The energy equation is implemented in several fluid models for atmospheric pressure plasma jets [122–131, 164] and is recommended with respect to the use of the LFA in several works [165–168]. Adopting the same approximations as in [169, 170], the electron energy conservation equation is written:
  $$\begin{equation}\frac{\partial }{\partial t}({n}_{\mathrm{e}}{{\epsilon}}_{\mathrm{m}})+\nabla \cdot {\mathbf{j}}_{{\epsilon}}=-\vert {q}_{\mathrm{e}}\vert \mathbf{E}\cdot {\mathbf{j}}_{\mathrm{e}}-{{\Theta}}_{\mathrm{e}},\end{equation} \tag{ 5 }$$
  $$\begin{equation}{\mathbf{j}}_{{\epsilon}}=-{n}_{\mathrm{e}}{{\epsilon}}_{\mathrm{m}}{\mu }_{{\epsilon}}\mathbf{E}-{D}_{{\epsilon}}\nabla ({n}_{\mathrm{e}}{{\epsilon}}_{\mathrm{m}}).\end{equation} \tag{ 6 }$$
  The terms on the right side of equation (5) represent the source/loss terms due to acceleration/cooling by the electric field and the loss term Θ_e due to the power lost in collisions. Thus, equation (5) takes the same form of the continuity equation (1) with drift–diffusion approximation for the electron energy density that we redefine as n = n_{e m}. The electron-impact rate coefficients, the electron and electron energy transport parameters and Θ_e can be obtained from the EEDF as f(_m). A more complete description of the development of He LFA and LMEA models can be found for instance in [145, 171].
• Comparison of the discharge dynamics in a dielectric tube with the two fluid models:To illustrate the differences between the results of the two fluid models, we present a comparison on a simplified test-case in which both models can be used: discharge propagation in a 99% He–1% N₂ mixture inside a dielectric tube (with inner radius r_in = 1.25 mm, outer radius r_out = 2 mm and relative permittivity _r = 4), between an inner ring electrode (located between z = 1.3 cm and z = 1.8 cm and powered by an applied voltage of 50 ns rise-time and 4 kV plateau) and a grounded target located at z = 0. Figure 4 shows the electric field magnitude (E_t) during discharge propagation inside a dielectric tube at the same simulation time. With both approaches, the discharge is annular during its propagation in the tube. It can be observed that the LMEA leads to a slightly slower and thinner discharge than the LFA, with lower peak E_t, essentially due to the electron energy loss in the sheath between the plasma and the dielectric surface.
• Transport parameters and rate coefficients of electron impact reactions:For plasma jet simulations, it is crucial to take into account the dependence of electron swarm parameters and of rate coefficients of electron impact excitation and ionization reactions on both the particular gas mixture and the local reduced electric field E/N or mean electron energy _m. These parameters are often obtained from electron Boltzmann equation (EBE) solvers, such as the two-term solver BOLSIG+ [169], using the databases of cross sections in the LXCat platform [172]. Due to the buffer flow and its mixing with ambient air, in the discharge simulation in the plume it is necessary to take into account that the gas mixture composition is different in each cell. Usually, the electron kinetics parameters are computed and stored in lookup tables for different mixtures as function of E/N or _m. Although the discharge usually propagates in the plume in the region with air mixture below 5%, the parameters are sensitive to small changes in composition. The choice of mixtures for which the EBE is solved should take this into account and be resolved with precision below 1% air admixture in the relevant range of mixtures. Then, the electron kinetics parameters are found in each cell and at each timestep for the local values of composition and E/N or _m through interpolation or fitting of the tabulated values [135, 148, 173].As a first approximation, drift and diffusion of positive and negative ions is not essential to describe transient discharges in the ns time scale, due to the two-to-three order of magnitude slower motion of ions when compared to electrons. However, they have a role in the μs discharge dynamics in He jets. Positive and negative ion mobilities in different gases are commonly retrieved from LXCat databases [172] as function of E/N for E/N ⩽ 100 Td and can be extended to higher electric fields, for instance using the formulas from [174]. Diffusion coefficients for ions are often evaluated using the Einstein relation, considering ions and neutrals to be at the same temperature.
• Numerical constraints:Finally, we point out the temporal and spatial resolution needed to numerically describe discharge dynamics in these plasmas. The layers of strong charge separation at the discharge front or in plasma–surface interactions have dimensions typically of some 10s of μm. This corresponds to a few times the Debye length, i.e., the scale above which we can consider that charged particles act collectively in a plasma and that a macroscopic fluid formulation is accurate to describe their dynamics. This quantity is proportional to the square root of electron temperature T_e (T_e = (2/3)_m) and inversely proportional to $\sqrt{{n}_{\mathrm{e}}}$, and in these discharges is of the order of 10 μm. We should notice that the mean free path for electrons in He at atmospheric pressure is of the order of 0.3 μm [175]. Since it is much smaller than the usual thickness of charge separation regions, the use of the continuum/fluid approximation is justified. In order to follow the fast discharge dynamics with the spatial resolution of the Debye length and ensure the stability and accuracy of the numerical solution, the temporal resolution of the numerical simulations is often limited by the Courant–Friedrichs–Lewy conditions [176], which for these plasmas usually determine a time-step of the order of units or 10s of ps.

In numerical and experimental studies on helium plasma jets, usually the helium gas flow expands in open air or in a controlled atmosphere [107, 177, 178] (usually of He with additions or pure N₂ or O₂, or of air with H₂O addition) at the tube exit. In some studies, a low amount of gas admixtures (usually air-related species) is added to the helium gas flow to study the generation of reactive species and the optimization of plasma jet application, while preserving the easy breakdown conditions of He [179, 180].

Electron-impact ionization in helium and in helium–air mixtures is key to understanding the ignition and propagation of discharges in regions with low air fractions and the guiding of discharges in the plasma plume [22, 181]. Figure 5 shows a comparison of electron-impact ionization coefficients versus reduced electric field in helium–air mixtures with different molar fraction of air X_air [181]. These values take into account the dependence of the EEDF on the gas composition. It is interesting to notice that for E/N < 100 Td, the ionization rate coefficient is nearly independent of X_air for X_air < 10⁻² and decreases with increasing air content at larger X_air. Indeed, atomic noble gases, and He in particular, present much more energetic EEDF than molecular gases. The excited and ionized states of He have very high energy thresholds (19.8 eV for the lowest-energy excitation and 24.6 eV for ionization), in contrast with the low-energy electronically and vibrationally excited states of N₂ and O₂, leading to a depletion of the EEDF tail when these gases are admixed in He. Moreover, the relative proximity in energy between the first metastable state and ionization favours stepwise ionization.

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${\mathrm{H}\mathrm{e}}_{2}^{+}$

${\mathrm{N}}_{2}^{+}$

${\mathrm{O}}_{2}^{+}$

The description of plasma kinetics is gas-dependent and pressure-dependent. Different reaction schemes for He mixtures with air gases, at high pressures, have been proposed [78, 123, 126, 129, 134, 135, 139, 179, 181, 184–202]. It is interesting to note that in helium–air mixtures, Penning reactions leading to the ionization of air molecules through quenching of highly energetic metastable excited states He(2³ S, 2¹ S) can take place. The simulations in [18, 20, 22, 123, 125, 181, 200] have shown that Penning ionization may be the dominant electron-production mechanism in the plasma channel, but not in the discharge front, where electron-impact ionization prevails, as underlined very recently in [143]. These conclusions are supported by the simulation results in figure 6, where the spatial distribution of the total ionization source term is represented for different conditions at an instant when the discharge is propagating in the plume between the tube and the target. The first condition is that of a He flow with 10 ppm of N₂ not mixing with surrounding air gases at the tube exit, as in Naidis (2010) [18] and Boeuf et al (2013) [22]. In this case, the highest values of the ionization source term in the discharge front are off-axis and the discharge has a ring shape, which is a result of the ignition and propagation of the ionization front in the dielectric tube (figure 4). In the second condition, there are no air gases at the tube exit and therefore the gas composition (He with 10 ppm N₂) is spatially uniform, as in Du et al (2020) [203]. In that case there is no radial confinement of the discharge in the plume region, as verified in Breden et al (2012) [123]. In the third condition, there is He–N₂ gas mixing at the tube exit and the plasma and the flow are coupled using the static flow approximation. This is shown to be essential for discharge confinement in the plume, leading to its propagation within the region with N₂ molar fraction below 5%. The radial confinement is attributed to the influence of species molar fractions directly in the chemistry source terms and also in the calculation of the EEDF and electron-impact rate coefficients, as shown in figure 5. Finally, the last condition in figure 6 shows that the discharge structure is not altered by the absence of Penning reactions in the model, as reported by Naidis (2011) [181].

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In plasma jet discharges with long (of the order of a few microseconds) rise times of the applied voltage and long tubes (tens of cm), the discharge dynamics can be affected not only by fast electron-impact reactions, but also by long life-time species and active plasma chemistry in the plasma channel. In fact, experiments [204] and simulations [179] have reported significant effects of admixtures of N₂ up to 1.5% in the He buffer gas on discharge velocity, structure and luminous intensity, attributed to the role of plasma chemistry: electron-impact ground-state and stepwise (through collisions with He metastables) ionization coefficient, photoionization, Penning ionization and ion transfers between the primary atomic ion He⁺, the molecular ion ${\mathrm{H}\mathrm{e}}_{2}^{+}$ and the air ions, that have significantly higher recombination rates than the He ions. These reactions can affect the conductivity of the plasma column, whose importance for discharge dynamics has been put forward in [22, 132, 179, 205].

Most fluid models are used to simulate only a single discharge even if experiments are carried out in repetitive conditions. In pulsed conditions, charged and excited species are created during the pulse and then tend to disappear in the time between pulses. Thus, the initial densities to take into account in models are dependent on the reaction scheme, gas-mixture and pulse repetition. They can be essential for the production of seed electrons that is required for discharge propagation. In regions with helium fractions above 99% and molecular gas fraction above 10 ppm, ${\mathrm{N}}_{2}^{+}$ or ${\mathrm{O}}_{2}^{+}$ are expected to be the dominant positive ions between pulses in plasma jets flowing in dry air. If charged particle losses between pulses are determined by electron–ion recombination with rate coefficient k_rec ∼ 10⁻⁷ cm³ s⁻¹ [183, 184], the electron density n_e between pulses can be estimated as function of time after the pulse as n_e(t) ∼ 1/(k_rec t). We should notice that this is not the case when there is a high density of electronegative gases (typically above 1%), in which case the electron density is decreased faster through attachment. With period between pulses of the order of 1 ms (1 kHz pulser frequency), background preionization of electrons and positive ions is thus expected to be of the order of 10¹⁰ cm⁻³ [181]. Moreover, it has been suggested from the experimental studies in [7, 206, 207] that a uniform initial preionization density of 10⁹ cm⁻³ is necessary to allow the discharge to propagate in a repeatable mode. As such, high values of background preionization, above the natural values of 10⁰–10⁴ cm⁻³, are commonly used to model pulsed He discharges. The influence of the choice of preionization values on discharge dynamics in He jets has been reported in simulation works such as [22, 179].

We should notice that the preionization condition for repeatability sets the pulse repetition frequency in experiments in the kHz order of magnitude. Indeed, it has been observed experimentally in pulsed jets that a decrease of pulse frequency [206] and a decrease of applied voltage magnitude [208] can lead to stochasticity in discharge ignition and propagation. Conversely, an increase in frequency can lead to a repetitive accumulation of charges and excited species and thus to heating and arcing. Using sinusoidal applied voltages, as there is not such a clear relaxation period between discharges as in pulsed conditions, the background preionization is more difficult to estimate. However, the same evolution of stochasticity and heating with frequency and applied voltage is present [209]. Finally, we should notice that in most plasma jet systems it is unknown how much charge remains on surfaces in between pulses. Therefore, most models consider that charges on surfaces are neutralised before the ignition of each new discharge.

For a discharge ignited by an applied voltage with positive polarity, the propagation of the ionization front is highly sensitive to photoionization. In helium discharges, photons emitted from excited helium species have enough energy to ionize nitrogen, oxygen and water molecules, present as impurities in the helium flow or at the tube exit [18, 107, 126, 135, 142, 144, 179]. At atmospheric pressure, the ionizing radiation is emitted mainly by the helium excimer ${\mathrm{H}\mathrm{e}}_{2}^{\ast }$ formed by three-body conversion reactions (He* + He + M → ${\mathrm{H}\mathrm{e}}_{2}^{\ast }$ + M) involving excited helium atoms He*. The excimer emission does not experience resonant reabsorption, and can therefore produce non-local ionization, whereas the resonance radiation of excited helium atoms is trapped due to reabsorption by abundant ground state He. However, the resonance radiation of excited He atoms may become the main contribution of the ionizing radiation in plasma jets at lower pressures, as shown by Lietz et al (2020) [107] at a nominal pressure of 200 Torr. Finally, in some works on atmospheric pressure plasma jets, the contribution of photoionization of O₂ by emission from ${\mathrm{N}}_{2}^{\ast }$ has also been included [20, 107, 123, 210]. In the work by Lietz et al (2019) [135] it has been shown that this photoionization process can be essential for the ionization front propagation in regions of high air concentration where ${\mathrm{H}\mathrm{e}}_{2}^{\ast }$ is not generated or is rapidly quenched by reactions with N₂ and O₂. However, in classical plasma jet setups at atmospheric pressure, it has been verified by Lietz and Kushner (2018) [142] that the main photoionization process for the ionization front propagation in the tube and in the plume is due to the radiation emitted by the helium excimer. Currently, photoionization models of helium excimer radiation ionizing nitrogen, oxygen and water molecules used for plasma jets are rather simple and do not take all the complexity of the wavelength dependence of absorption and photoionization processes [18, 126, 179] (see discussion in section 5.5). A key quantity for current photoionization models is the photoionization cross section σ_ph for excimer radiation, which is about 3 × 10⁻¹⁷ cm² for nitrogen, oxygen and water molecules [18, 210]. In the work by Naidis (2010) [18], it has been assumed that the number of ionizing photons emitted per unit time in a given discharge volume is proportional to the number of excited atoms produced, per unit time, in this volume by electron impact. The proportionality factor ξ is varied between 0.1–1 in [18] and up to 10 in [179]. As photoionization is a non-local process, a classical way to calculate the photoionization source term is to use an integral formulation as in [18, 210], which is very costly computationally. In the work by Bourdon et al (2016) [179], it has been proposed to adapt the differential method SP3 [211] derived for the calculation of the photoionization source term in air to the calculation of the photoionization source term in a He–N₂ mixture based on the photoionization model proposed by [18], taking into account only one absorption length.

With repetitive discharges, preionization from previous discharges occurs and may decrease the role of photoionization in helium jet discharge propagation [212]. To illustrate the role of preionization and photoionization in simulation results, figure 7 presents the spatial distribution of the ionization source terms by electron-impact (S_e) and by photoionization (S_p) in the discharge front while propagating in a He–N₂ plasma plume (no attachment present). These source terms are dependent on gas mixing and represent the production of seed electrons, therefore defining the path of propagation of the discharge in the plume. Two cases of preionization (n_init) of electrons and ${\mathrm{N}}_{2}^{+}$ are compared: 10⁹ cm⁻³ and 10⁶ cm⁻³. Figure 7 shows that S_e is higher on the discharge front with n_init = 10⁶ cm⁻³ due to a higher peak electric field. However, 1 mm ahead of the front, it is significantly lower than with n_init = 10⁹ cm⁻³. In the case with n_init = 10⁹ cm⁻³ it is visible that S_e has higher values and extends further ahead of the discharge front than S_p. This shows that S_e is a more important source of seed electrons and has a major role structuring the shape of the discharge in the plasma plume, in agreement with the simulation results by Breden et al (2011) [20], according to which photoionization is not required to allow for discharge propagation in He jets. The extension of S_e is only possible due to the very low electric field threshold for ionization of He with small air admixtures (see section 2.2.4). However, with n_einit = 10⁶ cm⁻³, the values of S_e and S_p ahead of the discharge front are comparable and both mechanisms help to define the path of discharge propagation. We can conclude that the identification of the main propagation mechanisms is dependent on gas-mixture, reaction scheme and the choice of preionization values in models.

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The implementation of a surface in a discharge model includes the description of its interaction with charged particles. It is often considered that positive ions impacting a surface through electric drift and thermal motion are recombined, thus removing an electron from the surface, while negative ions are detached, providing an electron to the surface. As such, the flux of each ion j_i across the surface can be expressed as function of the normal vector pointing outwards from the surface (n) and the thermal velocity (v_{th, i}) [189, 213] as:

$$\begin{equation}{\mathbf{j}}_{\mathrm{i}}\cdot \mathbf{n}=\mathrm{min}(0,({q}_{\mathrm{i}}/\vert e\vert ){n}_{\mathrm{i}}{\mu }_{\mathrm{i}}\mathbf{E}\cdot \mathbf{n})-\frac{1}{4}{v}_{\text{th},\text{i}}{n}_{\mathrm{i}}.\end{equation} \tag{ 7 }$$

The interaction of electrons with different materials usually considers perfect absorption when electrons impact the surface. However, the perfect absorber assumption is questioned [214–216] and quantum-mechanical methods have been proposed for calculating the sticking and backscattering probabilities of electrons hitting a solid wall [217, 218]. Electron emission can also take place at surfaces. In the case of metals, it is often considered that electrons are emitted by effect of the electric field when it is directed inwards the surface, and thus a Neumann boundary condition for the electron flux between the plasma and the surface is usually considered. Another approach frequently taken, both at the surface of metallic and dielectric materials, consists in considering that secondary emission of electrons takes place by ion bombardment (Auger neutralization and de-excitation [219, 220]) and by photoemission or photodesorption [126, 221]. The electron flux on the surface boundary, considering perfect absorption, can thus be expressed as function of the flux of each positive ion (${\mathbf{j}}_{{\mathrm{i}}^{+}}$, inwards the surface or zero), the photon flux ( φ , inwards the surface), the secondary emission coefficient (γ_se) and the photoemission coefficient (γ_pe) as:

$$\begin{align}\hfill {\mathbf{j}}_{\mathrm{e}}\hspace{-1pt}\cdot \hspace{-1pt}\mathbf{n}& =\hspace{-1pt}\mathrm{min}(0,-{n}_{\mathrm{e}}\;{\mu }_{\mathrm{e}}\;\mathbf{E}\hspace{-1pt}\cdot \hspace{-1pt}\mathbf{n})\hspace{-1pt}-\hspace{-1pt}\frac{1}{4}{v}_{\text{th},\text{e}}{n}_{\mathrm{k}}\hspace{-1pt}-\hspace{-1pt}{\gamma }_{\mathrm{s}\mathrm{e}}\\ \hfill & \hfill \quad \times \sum\limits _{{\mathrm{i}}^{+}}{\mathbf{j}}_{\mathrm{i}+}\hspace{-1pt}\cdot \hspace{-1pt}\mathbf{n}\hspace{-1pt}-\hspace{-1pt}{\gamma }_{\text{pe}}\;\boldsymbol{\varphi }\hspace{-1pt}\cdot \hspace{-1pt}\mathbf{n}.\end{align} \tag{ 8 }$$

A correction to the usual expressions for ion and electron fluxes at the surfaces (equations (7) and (8)) has been proposed by Hagelaar et al (2000) [213], taking into account diffusive motion and reflectivity.

The coefficient for photoelectron emission γ_pe is estimated of the order of 0.01 [135, 222, 223]. Relatively high values of secondary emission coefficient, such as γ_se = 0.1, are often considered, as a way to roughly take into account other secondary emission processes such as field emission and secondary emission of electrons by impact of metastable species. In fact, the effective secondary electron emission coefficients for different dielectric materials have been reported to have values between 0.02 and 0.4 in the experimental study of Tschiersch et al (2017) [224]. Values of that order have been calculated theoretically through solid-state considerations for dielectric surfaces in Zhang et al (2021) [220] and through quantum-kinetic methods for γ_se for metallic surfaces in Pamperin et al (2018) [225] and for secondary emission through metastable impact in dielectric surfaces in Marbach et al (2012) [226]. An alternative approach for generation of secondary electrons by approaching ions is to treat energetic emitted electrons by kinetic Monte Carlo methods [107, 223, 227, 228]. Other methods consider thermionic electron emission [229] and field emission from metallic surfaces [229–231]. Monte Carlo techniques have also been used in plasma–surface interactions in plasma jet models to simulate ion trajectories [232].

As a result of the fluxes of all charged particles (each with subscript k), a net surface charge density σ is obtained on dielectric surfaces, calculated by time integrating charged particle fluxes through electric drift to the surface:

$$\begin{equation}\sigma ={\sigma }_{0}-{\int }_{t}\sum\limits _{\mathrm{k}}\frac{{q}_{\mathrm{k}}}{\vert {q}_{\mathrm{e}}\vert }{\mathbf{j}}_{\mathrm{k}}\cdot \mathbf{n}\,\mathrm{d}t,\end{equation} \tag{ 9 }$$

where σ₀ is the initial local surface charge density. Usually, no initial surface charges are considered on the dielectric surfaces. This has been shown in Slikboer et al (2019) [164] to be a reasonable assumption for targets, due to the almost inexistent surface charge densities on a dielectric BSO target, measured and calculated after the fall of an applied voltage pulse. Moreover, it has been claimed from simulations in [233] that no surface charge density remains in a dielectric tube after the pulse. On dielectric surfaces, as the electric conductivity of charges is very low, charges are considered immobile on the surface.

Other surface reactions, such as de-excitation of excited species and association of atomic radicals can be included in models, as in [192, 195, 234]. Another feature of dielectric surfaces is the change of permittivity that affects the electric field distribution.As a corollary of Poisson's equation (equation (3)), it affects the electric fields at the interface (in the plasma E_n and in the dielectric E_−n) between a plasma of permittivity ₀ and a dielectric of permittivity _{0 r} through Gauss's law:

$$\begin{equation}{{\epsilon}}_{0}{\mathbf{E}}_{\mathrm{n}}\cdot \mathbf{n}-{{\epsilon}}_{0}{{\epsilon}}_{\mathrm{r}}{\mathbf{E}}_{-\mathrm{n}}\cdot \mathbf{n}=\sigma .\end{equation} \tag{ 10 }$$

According to this law, the higher relative permittivity of the dielectric induces an increase of the magnitude of the electric field in the plasma.

In plasma jets, the discharge is generated inside a thin dielectric tube, which channels the gas flow. Inside the tube the gas-mixture is homogeneous and the discharge interacts radially with the dielectric surface while propagating along the direction of tube length. Two parameters are of key importance for the discharge interaction with the dielectric tube: the tube radius and its dielectric permittivity. The tube thickness is not considered one of the main parameters because it is similar in most works, of the order of 1 mm. Usually in plasma jets, typical tube radii are in the range of a few mm and the dielectric tube permittivity is of about 4. In the two following sections, we study the influence of these parameters on the discharge dynamics and structure, based on the following dedicated works in He [22, 41, 110, 123, 125, 256, 257] and in other gases [132, 222, 258–260]. One factor not discussed in this section is the influence of leftover surface charges on the tube walls as a source of electrons for repetitive discharge propagation. The leftover surface charge density in the tube is expected to be very small [233] and not a primary effect for discharge dynamics inside the tube.

3.2.1.1. Study of the discharge dynamics inside a tube with a relative permittivity of _r = 1.

To assess how the tube constrains spatially the discharge, we firstly use simulations to study a discharge in a dielectric tube with the same relative permittivity as the surrounding gas (_r = 1). In this section, we consider the same point-to-plane electrode geometry with a 1.5 cm gap, 99% He–1% N₂ mixture and applied voltages (V_P = +4 kV and V_P = −4 kV) as in section 3.1, and we add a 2 cm long dielectric tube. The inner radius of the tube is of 2 mm, therefore lower than the radius of the discharges studied without tube ($\sim 8$ mm). Figures 10 and 11 present the spatial distributions of the magnitude of electric field E_t at different times for V_P = +4 kV and V_P = −4 kV, respectively. For each case, E_t is represented at different instants with respect to propagation, as was the case in figure 8.

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We notice in figures 10 and 11 that the discharge propagation is significantly different than in the cases without tube. After the discharge ignition, the discharge front expands both radially and axially. For both polarities, as the discharge front encounters the tube surface at r = 2 mm, then it can only propagate axially, with maximum E_t located close to the tube surface. The maximum of E_t is significantly higher for positive than negative polarity at t = 100 ns and when the front hits the target. In general, the values are higher than the corresponding cases without a tube present. Unlike in the cases without tube, the structure of the discharge propagating in the tube is dependent on the polarity. For V_P = +4 kV, E_t has a clear maximum at the discharge front close to the tube, while for V_P = −4 kV it is almost radially uniform at the discharge front, as in the simulations by Xiong and Kushner (2012) [132]. When analysing the quasi-neutral channel left behind the discharge propagation, where the electric field is very low, we can notice that it fills the whole tube radius for V_P = −4 kV, while for V_P = +4 kV its radius is lower than the tube inner radius. The difference induced by polarity in the structure of discharges propagating in tubes (i.e. more homogeneous for negative discharge and more filamentary for positive discharge) has been reported in experiments in He [76, 261] and in simulations in Ne–Xe [132]. Finally, it is shown in figures 10 and 11 that the arrival of the discharge front on the target takes place slightly earlier with the tube than without the tube, the difference being greater for positive polarity (40 ns) than for negative polarity (15 ns).

The significant change in discharge structure with polarity is closely related to the distribution of net charge density ρ. In fact, with positive polarity there is a positive radial sheath of about 500 μm width between the plasma bulk and the tube wall, as observed in the simulations by Liu et al (2014) [125]. The sheath is created by the streamer mechanisms pulling outer electrons towards the plasma. As only a few electrons can be emitted from the dielectric surface, they cannot fill and neutralise the sheath region [124, 125]. In [145], by artificially adding a Neumann boundary condition to the simulations for the electron flux between the plasma and the tube surface, it has been verified that the limitation of electron flux is the main reason for the differences in discharge structure due to the polarity. Indeed, for negative polarity, the electrons are pushed from the plasma to outer regions. The surface constitutes a barrier to this movement, but not a limitation to the plasma filling the tube. The positive sheath in the positive discharge induces a higher peak E_t and higher ρ than for negative polarity. The absolute value of ρ at the discharge front reaches 100 nC cm⁻³ for V_P = +4 kV but only 20 nC cm⁻³ for V_P = −4 kV.

The different polarities also have an effect on net surface charge density σ deposited on the tube inner walls. In the positive polarity case, the tube is charged very slowly through positive ion recombination and secondary electron emission and σ remains at very low values during the propagation, below 0.040 nC cm⁻². As such, σ does not play an important role in the discharge dynamics in this case. Conversely, in the negative polarity case, surface charge deposition through electric drift flux of electrons is faster, due to the different mechanism of propagation and the higher mobilities of electrons with respect to positive ions (μ_e ∼ 100 × μ_i). In the current negative case, |σ| rises up to $\sim 0.8-1.0$ nC cm⁻² in proximity to the front of the propagating discharge and behind it.

For both polarities, the charge separation (in volume and at the surface) induced by the radial confinement imposed by the tube helps to transfer the electric potential from the powered electrode and leads to higher values of electric field than in the discharges without tube. Indeed, ρ with tube and V_P = +4 kV at the discharge front is about ten times higher than in the case without tube. As ρ is higher in the case with tube than without, so are |E_z | and E_r, with maxima around 12 kV cm⁻¹ each, rather than 7 and 4 kV cm⁻¹, respectively. The same tendency has been observed in numerical studies of air discharges propagating in thin dielectric tubes [222, 258]. The high electric field in the sheath region explains the off-axis peak ionization observed in figure 10, as reported by Liu et al (2014) [125]. In the case with tube and V_P = −4 kV, E_z has a maximum close to 9 kV cm⁻¹, higher than in the case without tube (∼6 kV cm⁻¹). This value is lower than for positive polarity, as in the simulations of Ne–Xe discharges in capillaries by Xiong and Kushner (2012) [132].

As a result of the higher fields, charged particle density is generally higher in the cases with tube than without tube. Figure 12 gives insight into the radial distribution of electron and positive ion densities (n_e and n_i) in the different cases of polarity and presence or absence of tube. The radial profiles are represented at a fixed axial position at 5 mm from the target (z = 0.5 cm), at a chosen instant when the discharge front has already crossed this axial position but has not yet hit the target at z = 0. The number density of the net charge can be inferred from the figure as: n_ρ = n_i − n_e.

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As can be noticed in figure 12, the radial profiles of charged species densities extend much more broadly radially without tube than in the cases with the tube inner wall at r_in = 2 mm. The densities n_e, n_i and n_ρ are significantly higher in the cases with tube, up to a factor 3 at r = 0, and to one order of magnitude in the radial edges. The same tendency has been observed in a numerical study of positive air discharges in thin dielectric tubes [222, 258]. In fact, in the current results, |n_ρ | for both polarities without tube is always lower than 2 × 10¹⁰ cm⁻³, while with tube for V_P = +4 kV it reaches 5 × 10¹¹ cm⁻³ in the sheath between the plasma and the tube. The tube also induces a change of structure in the charged particle profiles. Instead of being centred as in the cases without tube, for V_P = +4 kV, the densities of charged species increase radially until around r = 1.5 mm and then decrease radially in the sheath until the tube wall at r = 2 mm. A similar radial structure in the positive sheath of an air discharge in a capillary tube has been reported by Jánský and Bourdon (2011) [259]. For V_P = −4 kV, the densities of charged species also increase radially but have their maxima closer to the tube at r = 2 mm. Moreover, these densities are generally lower than for V_P = +4 kV, as in the simulations of Ne–Xe ionization wave propagation in capillaries by Xiong and Kushner (2012) [132]. The maxima n_e shown in figure 12 agree with those measured through Thomson scattering in He jets for positive polarity at the exit of capillary tubes [91, 94], close to 10¹² cm⁻³, while n_e at r = 0 is in the range of values estimated from electric field measurements inside the tube [44], between 10¹¹ and 10¹² cm⁻³ (see figure 2).

In the cases under study, the higher peak electric fields induced by the tube lead to a higher velocity of discharge propagation when including the dielectric tube, as can be noticed through the chosen time instants in figure 12. The difference in velocity provoked by the tube confinement is significantly higher for positive polarity than for negative polarity (up to 5 cm μs⁻¹ higher velocity for positive polarity and less than 3 cm μs⁻¹ higher velocity in the negative case). In long (above 10 cm length) tubes, the propagation has been found to be faster for negative than for positive polarities in an experimental study of Ne discharges [260] and in a numerical study of Ne–Xe discharges [132]. Conversely, in the experiments in [262] it has been shown that the velocity of propagation of He discharges inside a long dielectric tube is higher for positive polarity. The dependence of velocity on polarity appears to be dependent on particular conditions of pulse shape, tube length and tube radius.

3.2.1.2. Influence of the tube permittivity and radius.

Discharge dynamics in the tube, preceding its propagation in the plume and its interaction with targets, is characterised not only by the tube, but also by the electrode geometry, the shape and polarity of the applied voltage and the choice of buffer gas. Different setup configurations and their potential applications have been reviewed [4, 6, 8, 9]. Moreover, the effect of different setup parameters on discharge dynamics has been described in the review by Lu et al (2014) [5]. The influence of electrodes on discharge propagation distance and on production of reactive species in the tube and in the plume has been highlighted in experimental works [265–268] and in the simulation work by Lietz and Kushner (2018) [142] for negative discharges.

In particular, negative discharge simulations in [142] have found that the following factors increase the discharge intensity and the production of reactive species: placing the powered electrode closer to the outlet of the tube; placing the powered electrode inside the tube rather than around it; adding a grounded ring around the tube; placing a coaxial grounded cylinder within 5 cm from the axis. The same modelling work has found that a more intense ionization wave can also be obtained by shortening the pulse rise-time and, in case the powered electrode is placed around the tube, for an optimal tube relative permittivity of _r = 4. Other modelling studies [126, 137, 269] have evaluated the influence of setup parameters such as the pulse repetition frequency, gas flow rate, gas composition and power on plasma parameters.

In this section, we provide additional insights from simulations on some of the main changes in positive discharge dynamics induced by variations of setup parameters which are often varied in plasma jet studies: the electrode geometry and the shape of the applied voltage.

3.2.2.1. Influence of the electrode geometry.

In this section, we have chosen different typical electrode configurations (represented in figure 13) used in plasma jets to study the influence of the electrode geometry on the discharge propagation inside the dielectric tube. The tube geometry is the same as in figures 10 and 11, with 2 mm inner radius and _r = 1. The pulse of positive applied voltage is also the same, with 50 ns rise-time and V_P = +4 kV. In all the cases, there is a grounded metallic plane at z = 0 and a grounded cylinder at r = 10 cm. Simulation details are given in [145]. The six cases of different electrode geometries assessed are listed below:

• Case 1 (C1): powered point with 1 mm radius and spheric tip at z = 3.5 cm (not represented in figure 13). In this case the length of the domain and the inter-electrode distance are 2 cm higher than in the other 5 cases.
• Case 2 (C2): powered point with 1 mm radius and spheric tip at z = 1.5 cm, as in figures 10 and 11.
• Case 3 (C3): same as in case 2, with an additional cylindrical powered plane, serving as point holder, placed at z = 2.0 cm, with 3 mm radius.
• Case 4 (C4): powered ring inside the tube at z = 1.5 cm, with 0.4 mm inner radius and 2 mm outer radius, i.e. filling the tube.
• Case 5 (C5): powered ring around the tube at z = 1.5 cm, with 3 mm inner radius and 3.5 mm outer radius.
• Case 6 (C6): same powered ring inside the tube as in case 4, with an additional grounded ring around the tube, with 3 mm inner radius and 3.5 mm outer radius, between z = 0.7 cm and z = 1.0 cm.

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In the first three cases, with point electrode, the Laplacian distribution of electric field at the tip of the powered electrode is very similar and so is the ignition time at about t = 40 ns, as shown for case 2 in figure 10. However, with the powered plane behind the point (case 3), the radial component of electric field has a lower magnitude in the region next to the point. With the powered ring inside the tube (case 4), the Laplacian distribution of electric field is significantly different than with the point electrode, even though the time of ignition is similar. The maximum of electric field is located on the edge of the ring, at r = 2 mm, and therefore the ignition is not centred. If the powered ring is outside the plasma (case 5), wrapped around the tube, the electric field in the plasma is much lower and the ignition is much slower. Only at around t = 150 ns we can consider that a discharge is starting to propagate. Finally, with powered ring inside the tube and grounded ring wrapped around it (case 6), which is the case in many experiments, the ignition is similar to case 4, but it takes place slightly ($\sim 10$ ns) earlier and with higher electric fields due to the proximity to the ground.

Figure 14 shows the effect of the electrode configurations on the distribution of electron density, at an instant when the discharge is close to reaching the grounded plane, along with the corresponding time. For the first three cases, with point electrode, the tubular discharge structure is the same. Nevertheless, the longer interelectrode distance in case 1 induces a lower average velocity of discharge propagation (∼8 cm μs⁻¹ in case 1 and ∼15 cm μs⁻¹ in cases 2 and 3). The powered plane in case 3 only induces a change in the region next to the point, along with a slightly lower surface charge deposition than in case 2. Then, if we consider the propagation with powered inner ring (case 4), despite the different ignition with respect to cases 1 to 3, the tubular structure of n_e ends up being similar during most of the propagation. However, with solely a powered ring wrapped around the tube (case 5), the electric fields in the plasma are always lower than in the cases with an inner powered electrode and thus charge production, discharge radius and velocity of propagation significantly decrease, in agreement with the simulations by Lietz and Kushner (2018) [142]. Finally, by adding the grounded ring around the tube (case 6), the discharge front is firstly directed towards the grounded ring. The tube is significantly charged and then the discharge propagation continues towards the grounded target. Overall, the introduction of the grounded ring around the tube leads to a faster discharge propagation, higher surface charge deposition close to the grounded ring and higher charge production, but the tubular structure of discharge propagation between the grounded ring and the target does not significantly change with respect to cases 1 to 4. We can conclude that for positive polarity, the discharge structure and dynamics at the end of the tube and then at the start of the propagation in the plume is very similar, as long as the powered electrode is placed inside the dielectric tube and the applied voltage is the same. However, if simulations want to quantitatively characterize a particular jet used in experiments, it is recommended to take into account the particular electrode geometry. In particular, we should take into account that, if both metallic electrodes (powered electrode and grounded target) are exposed to the plasma, a conductive channel may form after discharge propagation through continuous electron emission from the cathode.

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3.2.2.2. Influence of the shape of the applied voltage.

In this section, we study the influence of the shape of the applied voltage on the discharge characteristics. The same geometry is used as in section 3.2.1 (2 cm domain, powered point electrode without point holder, r_in = 2 mm and r_out = 3 mm), but with _r = 4. First we compare a DC applied voltage of +4 kV (shape 1) with shape 2: a 50 ns rise-time and then a constant applied voltage plateau at V_P = +4 kV, which is the shape used in the previous sections. We show in figure 15 the temporal profiles of the discharge velocities of propagation inside the tube, obtained by following the maximum of |E_z | every 5 ns, for both shapes. We see that the ignition takes place at around t = 5 ns for the DC voltage (shape 1). For shape 2 with a rise time of 50 ns, the ignition happens at around t = 35 ns, when the applied voltage is close to 2.5 kV. However, after the voltage shape 2 reaches the plateau, the discharge structure is approximately the same for both voltage shapes, as is the velocity of propagation in the tube, around 10 cm μs⁻¹, until approximately 0.5 cm from the target. Then, the proximity of the discharge front to the ground leads to an increase of the velocity. It is thus shown that the velocity of propagation is proportional to the value of applied voltage at each instant in time and that, for short rise-times (of the order of a few tens of nanoseconds), the discharge dynamics is not significantly affected by the different ignition times. Then, we study two voltage pulses with a 2.2 μs rise-time, as in [75, 76, 179, 270]. We should notice that this rise-time is of the same order of magnitude as those of sinusoidal voltages with kHz–MHz frequencies, e.g. 25 μs rise-time for f = 10 kHz. Shapes 3 and 4 have maxima, respectively, of +4 kV and +10 kV.

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Figure 15 shows that for shapes 3 and 4 the discharge starts propagating when the applied voltage is around 1.5 kV and then the discharge velocity of propagation increases proportionally to the applied voltage until the discharge is close to the grounded target. Although not presented here, the discharge front radius is also proportional to the applied voltage. The dependence of discharge ignition on the applied voltage and the proportionality of discharge radius and velocity with the applied voltage have been reported in many works at atmospheric pressure, starting with air streamer simulations [239, 241], He jet experiments [271] and He jet plume simulations [147]. However, it should be noticed that both in air streamers and helium plasma jets, the values of electric field in the ionization front and the electron density in the plasma are approximately independent of the applied voltage, as these depend mostly on local charge separation in the discharge front [96, 239, 241].

Firstly, we should notice that the presence of a target influences the plasma conditions already during discharge propagation and before discharge impact. On the one hand, due to a constriction of the free gas flow, the gas mixing (between the buffer gas and the environment gases) in the plasma plume is altered by the target. In section 2.2.2 the importance of gas mixing for discharge dynamics has been explained. On the other hand, the presence of a target with a given dielectric permittivity and eventually leftover surface charges (in repetitive pulsed conditions) or a potential bias, affects the spatial distribution of electric potential. This has an influence on the electric field directing discharge dynamics, and thus on the speed and structure of the discharge. For these reasons, the position of the target with respect to the tube, its inclination and its permittivity affect the discharge. The spatial range where the target influences significantly the discharge dynamics depends on particular conditions.

The effect of two different targets on discharge propagation is observed in figure 18, from both experimental and numerical results from Viegas et al (2020). The axial position of the discharge front is represented as a function of time, for the cases of a free jet (no target), of a jet with metallic target at floating potential and with a grounded metallic target. We should notice that the floating metallic target is described in these simulations as a solid of extremely high dielectric permittivity (to guarantee an isopotential volume) and approximately infinite conductivity [48], similarly to the work by Babaeva et al (2019) [238]. Moreover, both targets are considered in simulations to emit as many electrons as required by the discharge, through field emission, according to a Neumann continuity condition.

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It is visible from both measurements and simulations in figure 18 that the influence of the target on the propagation depends on its electrical character. With a grounded metallic target, the spatial distribution of electric potential is severely affected and the discharge propagation is accelerated since the tube, resulting in a difference with the free jet case of around 100 ns at the target position. This faster propagation towards grounded targets has been shown in many experimental and numerical works [48, 78, 131, 238, 286, 292]. Conversely, when the metallic target is at a floating potential, its presence accelerates the discharge propagation only in the plasma plume and by only around 30 ns, with respect to the free jet case. This acceleration is due to the high target permittivity, the change in gas mixing resulting in an increased helium content in the plume and the increased electron emission from the target. This conclusion is reinforced by the simulation results in figure 19, where the discharge dynamics are represented for cases with dielectric targets at floating potential of different relative permittivity _r: low _r = 4, intermediate _r = 20 and high _r = 80. In that figure, discharge propagation is the same for every case until the discharge front approaches the target by only 3 mm. Then, when the discharge is very close to the target, its velocity is proportional to _r, as shown in other studies [288].

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Figure 19 also shows that the electron density during discharge propagation is affected by _r mostly close to the target. This is confirmed by electron density measurements in Klarenaar et al (2018) [94] close to the different targets listed in figure 17. In the works by Kovačević et al (2018) [46] and Sobota et al (2019) [47], by measuring the peak electric field in the discharge front and using different targets (free jet, distilled water, saline water, glass at floating potential, glass attached to a grounded plate and metallic grounded) at different positions, it has also been observed that the peak electric field is affected by the target only in the last 2 mm of propagation. These trends have been confirmed by different simulation works [78, 130, 200, 238, 286, 288]. In particular, the work by Viegas and Bourdon (2020) [78] has shown that the peak electric field of the discharge impacting the target, directed perpendicularly to the target, is proportional to _r and increases when grounding the target.

Another feature observable in the experimental results in figure 17, for the case of a glass target, and in the simulation results in figure 19, is that the discharge spreads radially on top of dielectric target surfaces. The charging of the surface during this radial spreading has been confirmed by measurements of electric field inside dielectric targets, based on the Pockels effect [66–69, 71, 164]. The charging and spreading are present with both perpendicular and inclined targets, although the spreading is not symmetric in the latter case [68, 69, 71, 73, 296]. Moreover, in the work by Slikboer et al (2019) [164], with perpendicular jet incidence on a BSO target (_r = 56) and positive polarity of applied voltage, it has been shown that in some conditions a star-shaped pattern can be formed by the discharge on top of the target, rather than a radially symmetric spreading.

The spreading on floating potential targets in figure 17 is less apparent with the high-permittivity water and copper targets than with the glass one. It is also visible in figure 19 that the spreading is inversely proportional to _r. This distinction has been described firstly for plasma jets and dielectric targets attached to a grounded plate by the simulations by Norberg et al (2015) [288] and has been confirmed in other jet simulations [78, 129, 200, 289] and experiments [94, 290, 292]. Targets with lower values of _r have lower capacitance and a shorter time (RC time constant in [288]) is required to locally charge the surface and locally raise its electric potential. As a consequence, for targets with lower values of _r, this leads to a quicker decrease of the axial component of electric field E_z (perpendicular to the target) in gas phase, determined by the balance between charge separation in volume and charge deposition on the surface [148], and the rise of the radial component E_r (parallel to the target), that sustains the radial propagation of the discharge on top of the surface. Following Gauss's law (equation (10)), the electric field from the plasma penetrates more into targets of lower _r. Conversely, the electric field E_z mediating charge transport between the plasma and the surface increases with the target's _r. This has been shown to lead to higher surface charging with higher _r, both globally during the pulse and locally at the point of discharge impact [78, 200, 288]. Indeed, in the simulations by Viegas and Bourdon (2020) [78], raising _r of floating potential targets from 1 to 80 has been shown to raise the values of surface charge density by more than two orders of magnitude and the values of total surface charge by about a factor 5. Grounding the dielectric target has also been shown to lead to a significant increase of E_z and surface charging, which is supported by new experimental measurements using Mueller polarimetry [297]. However, it should be noticed that E_r on top of the target surface is also higher for a grounded dielectric target than for a floating potential target of the same permittivity [78].

The experimental results in figure 17 show that, with floating potential high-permittivity water target, emission from the plasma plume persists during the pulse after the discharge impact. This is even more evident with the floating copper target in the same figure, in which case the visible discharge appears to propagate from the target towards the tube. This phenomenon is related to the return stroke, a second discharge front propagating in opposite direction to the first front, starting from the target. The axial propagation of the return stroke is visible from simulation results in figure 18 for the cases of both floating and grounded metallic surfaces. In fact, the return stroke has been identified in several recent experimental and numerical works as being an important feature of discharge dynamics in jets interacting with conductive targets [35, 48, 76, 94, 97, 99, 145, 238, 288–290, 294, 295, 298–300]. In particular, in the work by Darny et al (2017) [76], it has been measured for both polarities of applied voltage that metastable production in the plasma and E_z outside the tube have a second peak corresponding to the return stroke propagation, at the same time that E_r outside the tube drops. These features have been qualitatively reproduced by simulations [295]. However, the return stroke is known from earlier works on air streamers and leaders [121, 301, 302].

Indeed, the return stroke takes place due to the slow or inexistent local charging of high _r or high conductivity targets after the discharge impact. This is evidenced by the recent works on Ar by Slikboer and Walsh (2021) [97, 99], where a return stroke is only observed when a liquid target (tap water) under exposure of an argon jet is grounded via low resistance. When there is a high impedance path between the target and the ground, the target is charged before the discharge impact and a breakdown takes place without the visible presence of a return stroke. When the local charging of the target is slow or inexistent, a high voltage drop remains in the plasma channel after the first discharge impact, along with an axial component of electric field E_z (as discussed for air streamers in [302] and studied in plasma jets in [145, 288]). The E_z remaining between the target and the plasma can be sufficiently high to generate enough ionization to launch an opposite polarity ionization wave (as discussed for air streamers in [301, 302] and for plasma jets in [48, 295]). This ionization wave has a different character than the first one, as it propagates in an already ionized channel. Figure 18 shows that it propagates faster than the first discharge front, in agreement with experiments [76, 99]. The propagation of this second wave redistributes the electric potential in the system and partially neutralizes the surface charge in the tube and the remaining charge separation (net charge density) in the plasma channel, at the same time that it produces additional ionization in the channel (as discussed for air streamers in [121] and for plasma jets in [145]). As long as the ionized plasma channel persists, along with a gradient of electric potential within it (and particularly between the target and the powered electrode), successive but damped reflections can continue to propagate (as discussed for air streamers in [302] and studied in plasma jets in [76, 238, 294, 298]). A complementary explanation for the return stroke rests on the concept of impedance mismatching [76]. As the plasma channel connecting the powered electrode and the target forms a transmission line, the mismatch between the ionized channel impedance and the very low impedance of the conductive target leads to a reflection phenomenon when the high voltage applied on the powered electrode is being partially transferred through the plasma channel up to the target, with a reflection coefficient calculated from the impedance values as studied for air streamers in [303].

The quantification of discharge parameters associated to the return stroke or induced by the return stroke will be addressed in section 4.3.

Finally, one last aspect of plasma–target interaction is the light emission observed after the end of the rectangular applied voltage pulse, as can be seen in an example shown in figure 20. This has been shown to be dependent on the target, being more intense for the case of a floating potential copper target than a grounded one [48]. Indeed, as the pulse of applied voltage falls and the electrode becomes grounded, an electric field is generated between the newly-grounded electrode and the charged plasma. It has opposite sign to the one of the first discharge propagation. This electric field distribution propagates as a rather diffuse charge relaxation event that neutralizes the plasma channel to a great extent. This event is sometimes called a second breakdown or a second discharge, due to the light emission and current it produces. While the return stroke propagates during the pulse between a low-potential target and a powered electrode, this event propagates after the pulse between a newly-grounded electrode and the plasma, potentially assisted by a highly charged target. The charge relaxation event has been reported in many works, not only on plasma jets without target and with different targets, but more generally on DBDs, e.g. in plane-plane or coplanar geometry [45, 48, 66, 68, 78, 94, 97, 99, 148, 164, 233, 288–290, 292, 298, 304–307]. Recently, it has been described in more detail for pulsed jets in [48]. Using AC voltages, this event takes place at the reversion of the applied voltage polarity [66, 68, 97, 305, 306], while in pulsed jets it takes place at the fall of the pulse.

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The quantification of discharge parameters associated to the charge relaxation event or induced by this phenomenon will be addressed in sections 4.3 and 4.4, but an example is briefly discussed here based on the results in figure 20. The light emission and electric field induced inside a dielectric target are shown for a 1 μs helium-in-air pulsed plasma jet operated at an inclined impact angle of 45 degrees with a tube-target gap of 10 mm [70]. At a time delay of 0.4 μs relative to the rise of the pulse, the ionization wave coming from the plasma jet on the left-hand side is seen to impact the surface of the target (at a distance of 1.2 mm in figure 20). Charge deposition takes place and an induced electric field of 6 kV cm⁻¹ is observed. Due the inclined angle and surface charge accumulation, there is a continuation of the ionization wave along the surface of the target in the direction of the helium flow (towards the right). This continues until the end of the high voltage pulse at t_d = 0.9–1.0 μs. While at that point the light emission is seen only at the head of the ionization wave, the induced electric field is observed over its entire path creating a surface structure of 7 mm length. The electric field induced by the discharge propagation on the surface is lower by approximately 2 kV cm⁻¹ than the field at the impact point. When the applied high voltage pulse ends (after t_d = 1.0 μs), the second light emission event (charge relaxation event) is seen to affect the impact point. Then, in the next 500 ns, a very weak ionization wave travels the same path of the first discharge along the targeted surface. This results on the neutralization of surface charges and on the reduction of the induced electric field to zero.The charge relaxation event is particularly visible on the surface of a target at an inclined impact angle since the ionization wave is able to travel further.