Introduction

Over the last two decades, a quite large number of investigations have been devoted to the study of electrical discharges in (or in contact with) liquids, primarily due to their wide range of technological applications [1,2,3,4,5]. These applications include but are not limited to, agriculture, disinfection, water decontamination, and environmental preservation [6,7,8]. These kinds of discharges can be classified into three main groups: (i) discharges in the liquid phase, (ii) discharges in the gaseous phase with one or both liquid electrodes, and (iii) discharges in multiple phases, such as bubbles or liquids in the form of spray or droplets [5].

The glow-type discharge with a liquid-cathode belongs to the second group. This specific discharge type shares similarities with low-pressure glow discharge between metallic electrodes [9, 10]. Numerous experimental investigations have been developed for atmospheric pressure glow discharges with liquid electrodes. Optical Emission Spectroscopy (OES) is commonly employed as an experimental diagnostic tool to estimate important discharge parameters such as electric field strength, electron density, and characteristic temperatures (see e.g., [1, 11,12,13]). However, the number of numerical models that have been developed to acquire information on plasma features, like neutral and ionic composition, water content, electron energy distribution function (EEDF), and heating and charge production mechanisms, are scarce [14,15,16,17].

The model developed in [14] focused on the plasma composition of a diffuse discharge in atmospheric pressure air ignited between two liquids (tap water) electrodes. The particle composition was calculated assuming that the plasma is in chemical equilibrium with electrons and heavy particles following two separated Maxwellian distributions, and with the population among different excited states following Boltzmann distributions; the corresponding temperatures being the parameters of the model. By comparison with experimental data, the ratios between these characteristic plasma temperatures, as well as the upper limit of the water vapor percentage in plasma were determined. The results show that the electrical neutrality is mainly due to NO+ and electrons for a water vapor percentage lower than 20% and to H3O+ for higher water vapor percentages. In [15], an experimental and theoretical study of a DC glow discharge in atmospheric pressure air with distilled water as cathode was presented. The EEDF and related electron features were obtained by means of the numerical solution of the Boltzmann equation for electrons. For these purposes, a water molar fraction up to 20% was considered. The electron density was estimated from the plasma conductivity for a discharge current density range of ~ 3.3–4.8 A/cm2. The results indicated that the EEDF essentially exhibits a non-Maxwellian behavior, especially in the high-energy electron range (referred to as the “tail” of EEDF), primarily due to the substantial impact of water vapor on this electron population, which considerably decreases the energy of electrons, due to the large cross section at low energies of the water molecules in comparison to other major components of humid air plasmas as nitrogen and oxygen molecules. On the other hand, the vibrational–translational relaxation was identified as the primary mechanism for gas heating. The measured gas temperature was almost unaffected by the discharge current in the range 20–50 mA. In [16], a zero-dimensional model of the plasma composition for the conditions considered in [15] was reported. The model solves the Boltzmann equation for electrons and considered the vibrational kinetics for ground states of N2, O2, H2O and NO. The electric field strength, averaged gas temperatures, and emission radii of the plasma column are the parameters of the model. The electron concentration is not calculated since the ionization mechanisms are unknown. It is estimated from plasma conductivity measurements. Water molar fraction up to 5% was considered. The numerical results indicate a high-vibrational temperature of N2 ground state, and that the main neutral species in the plasma are O2(a1Δ), O2(b1Σ), O(3P), NO, NO2, HNO3, H2O2, and OH. Numerous reactions involving charged particles (such as charge exchange, ionization, and recombination phenomena) were not considered. Because obtaining the ionic composition of the plasma was not one of the aims of the model. In [17], a numerical model was developed to determine the plasma composition over the surface of the electrolyte cathode. The model considers a glow discharge in humid air operating at a discharge current value of 25 mA. The gas temperature is fixed at 1000 K. Again, the electron density was estimated based on plasma conductivity and was not calculated. Its value was kept constant during the calculations. The authors reported that H2O+ is the main positive ion, independently of the water content. It is important to mention that the model did not consider the NO+ ion in the kinetic scheme. This revision shows that there is currently no consensus on the ionic composition nor the dominant ionization mechanisms in the positive column of humid air glow-type discharges with water cathode. This work is an attempt to address this issue.

In this work, a zero-dimensional numerical model describing the elementary processes occurring within the positive column of a millisecond DC-pulsed humid air glow-type discharge with a water-cathode is reported. The model self-consistently calculate the ionic composition as well as the relevant mechanisms responsible for the charged particle balance in the discharge. The EEDF is obtained by solving the electron Boltzmann equation. A water molar fraction up to 20% is examined. Additionally, to validate the simulation results, measurements of the electric field strength, emission discharge radius, and gas temperature in the positive column of the discharge were carried out.

Experimental

Water-Cathode Glow-Type Discharge

The discharge is ignited in a pin-to-water electrode configuration with an inter-electrode distance L = 12 mm. A cone-shaped tungsten-thoriated bar is the anode of the discharge while the distilled water contained in a 1–L grounded reservoir made of AISI 304 acts as the cathode (Fig. 1). To ignite the discharge, a variable autotransformer was used to adjust the discharge current I. This autotransformer was connected to the primary circuit of a high-voltage AC power transformer (25 kV, 50 Hz) with high dispersion reactance (95.3 ± 0.5 kΩ). This high impedance creates negative feedback between the discharge current and voltage, eliminating the necessity for external ballasts. Furthermore, a full-wave semiconductor rectifier bridge was used in the secondary circuit to fix the electrode polarity. The measurement of the discharge voltage V was conducted using a high-voltage probe (Tektronix P6015A, 1000X, 3 pF, 100 MΩ) connected to an oscilloscope (Tektronix TDS 2004C) with a sampling rate of 1 GS/s and an analog bandwidth of 70 MHz. Simultaneously, the discharge current I was inferred by employing a (low inductance) 100 Ω shunt resistor.

Fig. 1
figure 1

Schematic of the experimental setup

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The voltage and current discharge waveforms are shown in Fig. 2. The current waveform shows an almost sinusoidal form with a peak value of about 155 mA (100 mA RMS value) as it is controlled by the high dispersion reactance of the transformer. Voltage signals for three different times measured from the discharge ignition (1, 30, and 300 s) are presented. At the beginning of each pulse, voltage spikes of about 4 kV are found, corresponding to the discharge ignition by a streamer-to-spark high-voltage transition [5]. Then, after the breakdown, the voltage drops to about 2 kV due to the high-impedance of the transformer and the discharge stabilizes to a glow-type discharge. The measured voltage includes not only the voltage drop in the gas gap but also the voltage drop in the electrode sheaths (mainly in the cathode) as well as in the resistive cathode (i.e., the distilled water). According to Fig. 2, the V–I characteristic curve has a positive slope immediately after the discharge is ignited (reaching a maximum value of about 4 kV for the peak current). This is due to the high voltage drop in the cathode due to the initial low electrical conductivity of the distilled water (5 µS/cm). However, as the time increases, the electrical conductivity of the water soon increases (to a value of about 50 µS/cm after 30 s, and about 180 µS/cm after 300 s), and the voltage drop in the liquid cathode decreases; thus, leading to the typically negative slope in the V–I characteristic of such discharge between metallic electrodes (e.g., [18]). The increase in the electrical conductivity of the water in contact with air discharges is always observed due to the ions formed in the liquid volume [19].

Fig. 2
figure 2

Current and voltage waveforms of the discharge corresponding to a current pulse of 155 mA peak value. The voltage waveforms corresponding to three times after the discharge ignition are shown: 1 s (red), 30 s (blue), and 300 s (green) (Color figure online)

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Measurement of the OH (A → X) Band Emission in the Positive Column

OH (A2Σ+–X2Π, Δv = 0) band emission from the discharge was measured with a monochromator (Spectral Products DK480, f = 480 mm, f/7.8, 2400 l/mm grating) coupled to a CCD linear detector (Alphalas CCD-S3600-D-(UV), 3648 active pixels (16 bit-ADC), 200–1200 nm) with an exposure time of 10 ms. Moderately high-resolution spectra of the OH band were obtained for discharge current pulses of 70, 106, and 155 mA peak values (50, 75, and 100 mA RMS values, respectively) with a resolution of 0.055 nm FWHM. A UV high-quality-convex lens (focal length, f = 100 mm) was mounted facing the discharge, collecting the light emitted only from the positive column, and redirected (through a UV–Vis optical fiber) toward the entrance slit, of 25 µm width, of the spectrometer (Fig. 3). The time-averaged rotational temperature of the OH (A → X) transition in the wavelength range λ = 306–311 nm is obtained by the Boltzmann plot method using the MassiveOES software [20,21,22,23]. Ten emission spectra were averaged.

Fig. 3
figure 3

Optical arrangements employed in the characterization of the of the positive column of the discharge. Top: optical emission spectroscopy. Bottom: photograph of the discharge

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Measurement of the Emission Discharge Radius and the Electric Field Strength in the Positive Column

To measure the emission radius of the discharge at the vicinities of the middle of the positive column, a large number (~ 100) of visible photographs were taken for several discharge current pulses with different peak values. To achieve this, a UV-convex lens (f = 100 mm) was used for focusing each captured image onto the sensor of a CCD camera (Lumenera Lt225m, resolution 2048 × 1088 pixels), with an exposure time of 0.1 ms (Fig. 3). These images were saved in BMP format and digitized using an 8-bit gray-level frame grabber. Only the brightest images (corresponding to the passage of the discharge current by its peak value and having a similar pixel intensity distribution) were selected and analyzed. To eliminate any interference from background noise, a background image (with the discharge off) was subtracted to each image of the discharge. The discharge emission radius was then defined as the radial distance at which the pixel intensity dropped to 50% of the maximum (axial) intensity [24, 25]. In addition, as the discharge emission radius in atmospheric pressure air glow-type discharges is close twice smaller than the current-carrying radius [25], the current density at the positive column j corresponding to each peak current value is then calculated by assuming that the current-carrying radius is equal to the diameter of the luminous region. A similar approach was employed in [25, 26]. The results are presented in Fig. 4.

Fig. 4
figure 4

Experimentally inferred current density in the positive column versus the discharge current

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To determine the electric field strength (E) in the positive column, the discharge voltage was measured as a function of the interelectrode distance L for several current pulses having different peak values. To do this, the anode was moved using a micrometer to change L between 1 and 12 mm with an uncertainty of about ± 500 µm. The value of E is then determined from the slope of the fitting line. To ensure that the resistive voltage drop in the water is negligible small with respect to that in the gas gap, such measurements were carried out with an acidified cathode (i.e., for t > 300 s).

The electron number density for each single discharge current value is then inferred from the electrical conductivity of the plasma considering the experimentally measured average gas temperature. As the mobility of electrons depends largely on the amount of water vapor in the plasma (which is initially unknown), the electron number density is calculated for a humid air plasma containing a mole fraction of water up to 20%.

Plasma model

The model considers a fairly complete kinetic block consisting of 216 reactions in a mixture of water vapor with dry air ([N2]:[O2] = 4:1) with the participation of molecular species, N2(X1∑g+,v), N2(A3∑u+), N2(B3Πg), N2(a’1∑u−), N2(C3Πu), O2, O2(a1Δg), O2(b1∑g+), NO, H2O, OH, H2, N2O, NO2, NO3, O3, HO2, HNO2, HNO3 and HNO; atomic neutral species, N(4S), N(2D), N(2P), O(3P), O(1D), O(1S) and H(1S); positive ions, NO+, N2+, O2+, H2O+, H3O+, and O+; negative ions, O−, O2−, O3−, H− and OH−; and electrons (e). For the simulations, two different water molar fractions χ are considered, 2 and 20%. The reactions are presented in the Table 1 of the Appendix 1. The proposed kinetic model does not incorporate reactions involving species such as N4+, O4+, HN2+, and hydrated ions. At gas temperatures above 900 K, the N4+ and O4+ cluster ions are efficiently destroyed owing to their low dissociation energy [27]. Consequently, the absence of this cluster of ions interrupts the chain of hydrated ions production as O2+(H2O) and H3O+(H2O)n (the subscript n indicates the coordination number) [28].

The local balance of neutral, ions, and electrons, are obtained through the time-dependent continuity equations,

$$\frac{{\partial \left[ {n_{k} } \right]}}{\partial t} = S_{k} - \frac{{D_{k} \left[ {n_{k} } \right]}}{{\Lambda^{2} }} - \frac{{\left[ {n_{k} } \right]}}{\tau }$$
(1)

The subscript k denotes the k-th species, with [nk] representing its respective number density. D refers to the diffusion coefficient, τ represents the time scale of the convective flow, and Λ denotes the characteristic diffusion length, estimated from the discharge radius R inferred from optical experiments. The diffusion coefficients of the neutral species are corrected according to the gas temperature [29], considering the data reported in [30] for Tg = 300 K. The rate at which species k is produced or destroyed (negative in this case), because of reaction i, is denoted by the term S. The number density of the dominant species N2 is determined by the conservation of the pressure (p = 105 Pa). Simultaneously, the densities of O2 and H2O are calculated through the atoms conservation laws of N, O, and H, considering the initial gas composition. The dominant positive ion NO+ is determined by applying the plasma quasi-neutral condition.

The balance equations that govern the average vibrational energy εV of the N2 molecule and the average translational energy of the neutral species (gas) are described as follows,

$$\frac{{\partial \left( {\left[ {N_{2} } \right] \varepsilon_{V} } \right)}}{\partial t} = \eta_{V} \sigma E^{2} - \left[ {N_{2} } \right]\frac{{\varepsilon_{V} - \varepsilon_{V} \left( {T_{g} } \right)}}{{\tau_{VT} }} - \frac{{\left[ {N_{2} } \right] \varepsilon_{V} - \left[ {N_{2} } \right]_{0} \varepsilon_{V} \left( {T_{0} } \right)}}{\tau },$$
(2)
$$\begin{aligned} \frac{\partial }{{\partial t}}\left( {T_{g} \mathop \sum \limits_{k}^{{}} \left[ {n_{k} } \right] c_{pk} } \right) & = \eta_{T} \sigma E^{2} + \left[ {N_{2} } \right]\frac{{\varepsilon_{V} - \varepsilon_{V} \left( {T_{g} } \right)}}{{\tau_{VT} }} - Q_{CD} + Q_{R} \\ & \quad \quad - \frac{{T_{g} \mathop \sum \nolimits_{k}^{{}} \left[ {n_{k} } \right] c_{pk} - T_{0} \mathop \sum \nolimits_{k}^{{}} \left[ {n_{k} } \right]_{0} c_{pk} }}{\tau } - \frac{{\dot{m}_{{{\text{H}}_{{2}} {\text{O}}}} H_{v} }}{{V_{p} }} \\ \end{aligned}$$
(3)

where εV is related to the vibrational temperature Tv by \({\varepsilon }_{V}=q \hslash \omega /\left[\text{exp} \left(q \hslash \omega /\left({k}_{B} {T}_{V}\right)\right) -1\right]\). Here, q represents the elementary charge, kB is the Boltzmann constant and \(\hslash\) ω denotes the vibrational quantum (= 0.29 eV) of the N2 molecule. The expression \({\varepsilon }_{V}({T}_{g})\) corresponds to the equilibrium vibrational energy value (i.e., for Tv = Tg) [29, 31]. The vibration–translation relaxation time (V–T), τVT, takes into account relevant V–T relaxation phenomena in water–nitrogen–oxygen mixture [29, 32, 33]. The rate of V–T due to N2–O collisions is taken from [34]. Water vapor strongly contributes to the acceleration of V–T relaxation of N2(X1∑g+, v) molecules: through N2–H2O vibration–vibration (V–V) collisions followed by the extremely fast V–T relaxation process of H2O–H2O [32, 33]; as well as the fast V–T relaxation process by N2–H2O collisions. The rate of V–T relaxation of N2(X1∑g+, v) molecules by collisions with H2O molecules is taken from [29]. σ is the electrical conductivity (= q [ne] µe). The coefficients \({\eta }_{V}\) and \({\eta }_{T}\) depict the quantity of electronic energy that is transferred into the vibrational and translational modes, respectively. The term QR denotes the fast-gas heating in chemical reactions [34,35,36] described with allowance for electron-impact predissociation, electron-ion recombination, quenching of excited species, and H2O dissociation. cpk is the specific heat of the heavy species k (= 5/2 kB, 7/2 kB, and 8/2 kB for atoms, diatomic molecules, and triatomic molecules, respectively). QCD (= 8 κ (Tg−T0)/Rt2) is the radial heat transport, assuming that the spatial variation of the gas temperature follows a radial parabolic profile [37]. κ is the translational component of the heavy particles thermal conductivity of humid air as a function of the molar fraction of water vapor [38, 39], Rt is the thermal radius(fixed to a value of 1 mm), and T0 (= 300 K) the ambient gas temperature. Finally, Hv is the water latent heat of vaporization, \({\dot{m}}_{{\text{H}}_{2}\text{O}}\) is the rate of water evaporation, and Vp is the volume of the plasma column.

The last term on the right-hand side of Eqs. (1) and (2), as well as the fifth term in Eq. (3), account for the axial (vertical) transport resulting from the weak longitudinal flow effects by natural convection [9, 40]. The convective transport time \(\tau\) was estimated using a simple approximation of the convective flow velocity according to Bernoulli’s equation [25, 40, 41].

Experimental studies on liquid-cathode glow discharge at atmospheric pressure have shown that the liquid is effectively transferred to the plasma column through the formation of droplets and fine plumes, mainly by ion sputtering [42,43,44]. In the plasma column, the water droplets quickly vaporize resulting in plasma cooling. This cooling mechanism due to the increase in water evaporation with increasing discharge current is considered through the last term on the right-hand side in Eq. (3). The time-averaged evaporation rate was measured from the mass loss of liquid over several discharge cycles as,

$$\langle m_{{{\text{H}}_{{2}} {\text{O}}}}\rangle = \frac{1}{t}\mathop \smallint \limits_{0}^{t} \dot{m}_{{{\text{H}}_{{2}} {\text{O}}}} {\text{d}}t$$
(4)

Under the considered conditions, the time-averaged evaporation rate was 3.4 ± 0.2 mg/s.

Then, by assuming that the instantaneous evaporation rate has a quadratic dependence on the discharge current [43, 45], the instantaneous and average evaporation rates can be correlated as,

$$\frac{{\dot{m}_{{{\text{H}}_{{2}} {\text{O}}}} \left( I \right)}}{{\langle \dot{m}_{{{\text{H}}_{{2}} {\text{O}}}} \rangle }} = 2 \left( {\frac{I}{{ \hat{I} }}} \right)^{2} ,$$
(5)

where \(\widehat{I}\) is the peak value of the discharge current pulse. The Eq. (5) is an approximation based on experiments conducted under DC conditions. However, it is expected that this relationship will be maintained under the conditions considered because the current variation timescale (10 ms) is slow enough as compared to the characteristic timescales of the plasma processes. This point will be discussed in the next section.

The local-field approximation was used to calculate the transport and reaction coefficients of electrons (i.e., as a function of reduced electric field E/N, where N is the neutral numerical density), employing a Boltzmann solver [46]. This approach is typically satisfied in molecular plasmas at atmospheric pressure (e.g., [16, 47, 48]). The transport and reaction coefficients of electrons are calculated for a humid air plasma containing a mole fraction of water up to 20%. The cross-section data are taken from the Morgan database [49]. The ‘effective’ electron temperature, Te, was found from the mean electron energy (= 3/2 kB Te). The parameters ηT and ηV were calculated as a function of the E/N values with the help of the Boltzmann solver [46]. In Eq. (3), the parameter ηT takes into account the energy transferred from electrons to the gas heating through elastic collisions, rotational excitation of the N2 and O2 molecules, the vibrational excitation of the H2O(010) state followed by the fast V–T relaxation in H2O(010)–H2O collisions, and the vibrational excitation of O2 molecules. The model also considers the impact of super-elastic collisions with vibrationally excited molecules on the EEDF, which has a strong influence on high-threshold electron-impact reactions, such as electronic excitation, dissociation, and ionization. To achieve this, the Boltzmann equation for electrons was solved setting the excitation temperature to the vibrational temperature and the transition energy to the first vibrational threshold (0.29 eV) of the N2 molecule.

The increases in the thermal dissociation rate of N2 under non-equilibrium conditions were considered through the Losev-β model [50, 51]. The electric field strength in the discharge column is calculated from Ohm’s law,

$$j = \sigma E$$
(6)

as a function of the experimentally inferred j values (Fig. 4). Discharge current density values for I < 50 mA were extrapolated by fitting the experimental data.

Balance Eqs. (1), (2), and (3) are solved numerically by a finite-difference explicit second-order Runge–Kutta predictor–corrector method [52]. To provide an accurate numerical integration, a short time-step of 1.0 × 10−10 s is used due to the stiffness of the problem, related to the wide range of timescales associated with various plasma processes. Appropriate initial conditions were used. The used initial conditions did not have a significant impact on the calculations results. Achieving a dynamic equilibrium required integrating the equations over several discharge periods, so that the densities of the species converge within an error of about 10−3 to their equilibrium values.

Results and Discussion

The presented results correspond to the millisecond DC-pulsed (rectified AC) discharge conditions according to the experimental arrangement shown in Fig. 1. Glow-type discharges in humid air with water cathode have been studied in several experiments (e.g., [3, 12, 13, 15, 16, 25, 53,54,55,56,57,58]). In most of these experiments the discharges were operated with DC currents, except for the experiment [57] operated at 60 Hz AC current. However, a direct comparison between the calculations and the literature data is still possible because the discharge current timescale (10 ms) is much larger than the characteristic timescales of the heavy particles processes in the discharge column. Consider for instance the timescale for diffusive losses of gas particles from a discharge column of radius R: tD ≡ Λ2/D, where D is the diffusion coefficient of the neutral particles (which is proportional to Tg3/2/p) and Λ ≡ R/2.4 is the characteristic diffusion length for a cylindrical volume [9]. D ~ 7 × 10−4 m2/s for air molecules at Tg ~ 3000 K and R is about 1 mm. Under the conditions considered it follows that tD is less than or equal to 0.3 ms, much shorter than the discharge current timescale (10 ms). To validate this statement, simulations were carried under DC conditions for several current values. The results indicate that the variations in gas and vibrational temperatures were about 1%; while for electron temperature and plasma number density the differences were around 4%; when compared to the millisecond DC-pulsed simulations. Note also that these results add consistency to the approximation (5), obtained by comparison with experiments conducted under DC conditions.

The time-averaged experimental spectrum of the OH(A–X) transition for a discharge current pulse having a peak value of 155 mA is presented in Fig. 5A. The corresponding Boltzmann plot up to the rotational quantum number J = 16.5 is shown in Fig. 5B. Note that the Boltzmann plot has a unique slope, indicating that the OH(A) rotational population follows a Boltzmann distribution under the conditions considered [1, 11]. Assuming that the rotational temperature of OH is in equilibrium with the translational (gas) temperature [11], a time-averaged gas temperature ⟨Tg⟩ = 3440 ± 460 K is obtained. Spectral fitting is not used to determine the gas temperature since this method does not exclude high rotational quantum numbers that may be overpopulated, leading to an overestimation of the gas temperature [1, 11]. For the other discharge current pulses having peak values of 70 and 106 mA, the Boltzmann plot shows 3530 ± 720 K and 3550 ± 840 K, respectively. These results are in agreement with previous investigations on DC glow-type discharge with water-cathode [12, 13] showing that the gas temperature measured by OES does not depend on the current discharge in the range 10–100 mA.

Fig. 5
figure 5

A Time-averaged experimental spectrum for OH(A–X) transition emission in the positive column of the discharge for a current discharge pulse with a peak value of 155 mA (100 mA RMS). CCD exposure time = 10 ms. B Corresponding Boltzmann plot up to the rotational quantum number J = 16.5

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Figure 6 shows the simulated gas temperature Tg, the vibrational temperature Tv of the N2(X1∑g+,v) molecules, and the electron temperature Te versus the discharge current for water molar fractions of χ = 2 and 20%. It can be observed that the plasma is in thermal non-equilibrium, with a significant departure between Te and Tg. The primary process of gas heating is V–T energy relaxation, which occurs mostly due to collisions between N2 and H2O molecules. Accordingly, the departure between Tv and Tg decreases markedly with the increase in water content in the discharge. Despite this increase in the V–T energy relaxation with increasing plasma water content, the gas temperature does not increase appreciably (rather decreases slightly) due to the rise in the thermal conductivity of the plasma caused by the increase in the water content. The cooling of the gas by water droplets evaporation is not important, reaching < 10% of the total heat losses, being higher than the losses caused by the H2O dissociation via reaction (R64) with an energy consumption of 5.11 eV per dissociation event. Due to the values of the mean electron energy (of about 1 eV), the electron power is mainly transferred to the vibrational modes (ηV ~ 0.9) of N2 molecules. It is remarkably that the simulated values of both Tg and Tv are almost independent of the discharge current for χ = 20%. It should also be noted that the gas temperature is lower than those typically obtained in similar discharges but operating with metal electrodes in dry air (see e.g., [18, 31]). The experimental values of Tg measured by OES in this work, as well as those for Tg and Tv reported in the literature for this kind of discharge [12, 13, 15, 53,54,55] are also shown in Fig. 6. Experimental vibrational temperature data was derived from the radiative intensity distribution of the N2(C3Πu → B3Πg). According to numerical estimations the vibrational temperature of the (C3 Πu,v) exited state are similar to that of the lower levels of the (X1 Σg+,v) state in this type of discharges [15]. It is observed that the best agreement between the simulations and the experimental is achieved for χ = 20%.

Fig. 6
figure 6

Simulated distributions of the gas temperature, vibrational temperature of the N2(X1∑g+,v) molecule, and electron temperature, versus the discharge current. Relevant experimental data are also presented

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The electron temperature is almost unaffected by the discharge current due to minimal variation in the reduced electric field, showing a decrease with increasing plasma water content. Experimental data of the electron excitation temperature of hydrogen atoms Texel (with values around 0.5–0.6 eV) are also shown in Fig. 6 [13, 25]. It is observed that the experimental values of Texel and simulated Te values show some discrepancy. The reported Texel measurements were obtained from the intensity ratio of the Hα over Hβ lines (with excitation energies of about of 12–13 eV), assuming a Maxwellian EEDF and a Boltzmann distribution for the population of excited states of hydrogen. However, the method may be doubtful because the EEDF is essentially non-Maxwellian under the conditions considered. The water vapor content decreases the number of electrons in the high energy ‘tail’ of the distribution [15, 25]. Consequently, high-level populations of hydrogen atoms may differ from the Boltzmann distribution. Furthermore, it is not possible to generate Boltzmann plots for the determination of Texel and, consequently, verify the assumption of the Boltzmann distribution due to the small intensity of the hydrogen emission and the superposition with other nitrogen emission lines. As a result, the evaluation of the excitation temperature of these hydrogen levels will be underestimated. This may be the reason explaining the discrepancies between the simulated electron temperature and the reported values of the hydrogen excitation temperature. It is worth to noting that the calculated Te values ensure that electron losses (essentially by dissociative recombination) are compensated by electron-impact ionization collisions with neutral particles with ionization energies close to or greater than 10 eV in the positive column of the discharge; thus, ~ 1 eV seems a realistic value for Te in this kind of discharges in contact with liquids (see the review [4] and references therein, as well as [59]).

Figure 7 shows the number density of several neutral species versus the discharge current for χ = 2% and 20%. The main mechanism of the production and destruction of NO molecules for χ = 2% can be described by the following reactions:

$${\text{OH }} + {\text{ O}}\left( {^{{3}} {\text{P}}} \right) \to {\text{O}}_{{2}} + {\text{ H}}\left( {^{{1}} {\text{S}}} \right), \quad \left( {{\text{R1}}00} \right)$$
$${\text{O}}_{{2}} + {\text{ H}}\left( {^{{1}} {\text{S}}} \right) \to {\text{OH }} + {\text{ O}}\left( {^{{3}} {\text{P}}} \right), \quad \left( {{\text{R81}}} \right)$$
$${\text{HNO }} + {\text{ O}}\left( {^{{3}} {\text{P}}} \right) \to {\text{OH }} + {\text{ NO}},\quad \left( {{\text{R1}}0{7}} \right)$$
$${\text{NO }} + {\text{ H}}\left( {^{{1}} {\text{S}}} \right) \to {\text{HNO}}.\quad \left( {{\text{R85}}} \right)$$
Fig. 7
figure 7

Simulated distributions of the number density of several neutral species in the positive column versus the discharge current. Relevant experimental data are also presented

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The thermal dissociation of nitroxyl (HNO) is the main mechanism of the production of NO molecules, through (R107), while the destruction process is governed by the formation of HNO by (R85). The production of H(1S) atoms is mainly governed by the reaction (R100) (which in turn is the dominant destruction mechanism of O(3P) atoms and OH molecules); while its destruction is mainly due to reaction (R81) for I < 100 mA, being replaced by reaction (R85) for higher discharge currents. In addition, the reaction (R81) is the dominant mechanism of production of O(3P) atoms and OH molecules. The production of OH by thermal dissociation of HNO in collision with O(3P) atoms (R107) becomes also comparable to the reaction (R81) for I > 100 mA. With the increment in the water content, the neutral kinetics are strongly affected, producing a reduction in the population of NO molecules and an increase in the hydrogen-containing species. The main reactions scheme for χ = 20% can be described by:

$${\text{OH }} + {\text{ OH}} \to {\text{H}}_{{2}} {\text{O }} + {\text{ O}}\left( {^{{3}} {\text{P}}} \right),\quad \left( {{\text{R93}}} \right)$$
$${\text{H}}_{{2}} {\text{O }} + {\text{ H}}\left( {^{{1}} {\text{S}}} \right) \to {\text{H}}_{{2}} + {\text{ OH}},\quad \left( {{\text{R92}}} \right)$$
$${\text{H}}_{{2}} + {\text{ O}}\left( {^{{3}} {\text{P}}} \right) \to {\text{OH }} + {\text{ H}}\left( {^{{1}} {\text{S}}} \right),\quad \left( {{\text{R1}}0{6}} \right)$$
$${\text{OH }} + {\text{ O}}\left( {^{{3}} {\text{P}}} \right) \to {\text{O}}_{{2}} + {\text{ H}}\left( {^{{1}} {\text{S}}} \right),\quad \left( {{\text{R1}}00} \right)$$
$${\text{OH }} + {\text{ HNO}} \to {\text{H}}_{{2}} {\text{O }} + {\text{ NO}},\quad \left( {{\text{R98}}} \right)$$
$${\text{NO }} + {\text{ H}}\left( {^{{1}} {\text{S}}} \right) \to {\text{HNO}}.\quad \left( {{\text{R85}}} \right)$$

The concentration of OH molecules increases about one order of magnitude due to the thermal dissociation of H2O molecules in collisions with H(1S) atoms via (R92), which also serves as the primary destruction pathway for H(1S) atoms. The increasing water content also enhances the production of H2 molecules, which increases two orders of magnitude, through (R92). The production of H(1S) is mainly governed via (R100) for I < 70 mA, and via (106) for higher current values. The increase in the concentration of H(1S) atoms decrease in turn the NO molecules concentration via thermal recombination (R85), which exceeds the production of NO molecules through (R98). The reduction in the concentration of NO (with a low ionization energy of 9.27 eV) in turn induces a change in the ionization kinetics with increasing water content.

The loss and production of O(3P) atoms are also influenced by the water content. The impact of production reactions (R81) and (R93), as well as destruction reactions (R100) and (R106), becomes comparable, resulting in a slight reduction in the concentration of these atoms. The concentrations of OH and O(3P) radicals are in the range 1021–1023 m−3, according to reported experimental data [4, 56]. In particular, the best agreement between the simulations and the OH experimental data [56] is again achieved for χ = 20%. As indicated in [16], the formation of the OH is not the electron-impact dissociation of the H2O molecules via (R10), but the thermal dissociation of O2 via (R81) for the lower water content, and the thermal dissociation of H2O via (R92) when the molar fraction of water rises. The concentration of other oxygen–hydrogen particles like HO2 radical and H2O2 are not important under the relatively hot gas conditions considered. As expected, the numerical density of ozone is very low because it is thermally destroyed by (R90) under hot gas conditions as indicated in [16]. The main excited particle (not shown) is the O2(a1Δg) metastable. Independently of the water molar fraction, the formation is governed by the electron-impact excitation (R16), which is favored due to its low excitation energy (= 0.98 eV), and the destruction occurs mainly through quenching reactions with NO molecules via reaction (R114). The impact on the plasma dynamics of oxygen-excited atoms with the increment in the water molar fraction is weak because of the efficient quenching of the O(1D) and O(1S) states by water molecules via (R159) and (R165), respectively.

The mechanisms involved in electron production and loss versus the discharge current for χ = 2% and 20%, are depicted in Fig. 8. It is observed that the dominant process for electron production in the positive column is the electron-impact ionization of the NO molecule (R1), regardless of the plasma water content (up to 20%). On the other hand, the electron-ion recombination of NO+ (R31) is the dominant mechanism contributing to electron loss. However, as the plasma water content increases the concentration of the NO molecules in the plasma column decreases (see Fig. 7), mainly due to the formation of HNO by the reaction (R85). The consequent deficit in the NO+/electron pair production for χ = 20% is compensated by the electron-impact ionization of O2 via (R3),

$${\text{e }} + {\text{ O}}_{{2}} \to {\text{ e }} + {\text{ e }} + {\text{ O}}_{{2}}^{ + } ,\quad \left( {{\text{R3}}} \right)$$
$${\text{O}}_{{2}}^{ + } + {\text{ NO }} \to {\text{ NO}}^{ + } + {\text{ O}}_{{2}},\quad \left( {{\text{R2}}0{6}} \right)$$

followed by the rapid charge-exchange through (R206). Besides, as the level of water rises, the attachment cross-section for H2O becomes greater than the O2. Consequently, the water dissociative attachment process with a high energy threshold of 4.36 eV (R171) speeds up compared to the oxygen dissociative attachment reaction (R169), partially compensated by the detachment reaction (R192). Under the relatively cold gas conditions simulated, the associative ionization reaction in collisions between N(4S) and O(3P) atoms via (R20), which typically rules the charge production in glow-type discharges operating in atmospheric pressure dry at currents levels of the order of 100 mA (Tg > 4000–4500 K [18, 31]), does not play any significant role in the charge production.

Fig. 8
figure 8

Simulated mechanisms of electron production and loss in the positive column versus the discharge current

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The simulated the electric field strength as well as the reduced electric field in the positive column as a function of discharge current for χ = 2% and 20% are depicted in Fig. 9. Experimental data obtained in this work as well as available in the literature [3, 15] are also presented. The measured electric field strength in the positive column ~ 70–100 V/mm is usually higher than that for metallic cathodes (e.g., [60]). As quoted before, this is because H2O molecules have cross-sections considerable larger than those of N2 or O2 at low electron energies. A good agreement between the simulated and experimental results is observed. It is also observed that the simulated reduced electric field increases from ~ 35 to 40 Td when χ increases from 2 to 20% (1 Td ≡ 10−21 Vm2). This is because the ionization energy of O2 molecules (12.1 eV) is larger than that of NO molecules, whose concentration decreases with increasing water content in the plasma, as discussed above.

Fig. 9
figure 9

Simulated distributions of electric field strength and the reduced electric field in the positive column versus the discharge current. Relevant experimental data are also presented

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Figure 10 shows the number density of the main ions and electrons as a function of discharge current for χ = 2 and 20%. It is observed that the electron number density grows as the discharge current increases, reaching approximately 1019 m−3 disregarding the water molar fraction. The plasma electrical neutrality is mainly due to NO+ and electrons. This agrees with [14], which shows that electrical neutrality is mainly due to NO+ and electrons when the water molar fraction is lower than 35%. However, the number densities of OH− and H3O+ increase close to one order of magnitude as the water molar fraction increases. The number density of OH− is relatively high because they are produced by ion-exchange reactions (R196 for χ = 2%, and R198 for χ = 20%), but it is still small as compared to the number density of electrons. The experimental electron density data obtained in this work as well as relevant available experimental data are also included for comparison. Most of the available experimental data on electron density have been obtained from plasma conductivity (e.g., [15, 16, 57, 58]) or measuring the Stark broadening of the Hβ Balmer line emission (e.g., [13]). Although some dispersion between the values of the electron number density obtained from the plasma conductivity can be observed (which may be mainly attributed to the fact that the emission and conduction radii of the discharge may be not equal, depending on the experimental conditions) the simulation results show the correct order. On the other hand, the number density value reported in [13] is more than one order of magnitude larger than the simulated values. However, this density corresponds to the electron density in the cathode region and not in the positive column (since the Hβ Balmer emission is practically only observable in the cathode region [13]). Note that a higher electron density can be expected near the cathode of the discharge due to its filamentary structure [13, 54].

Fig. 10
figure 10

Simulated distributions of the number density of main ions and electrons in the positive column versus the discharge current. Relevant experimental data are also presented

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Conclusions

A numerical investigation of a millisecond DC-pulsed glow-type discharge in atmospheric pressure air with a water cathode was reported. A zero-dimensional model with a complete block of chemical reactions that self-consistently describes the gas heating mechanisms and ionic composition in the plasma was considered. A water molar fraction up to 20% was examined. Moreover, to validate the model, the electric field strength, emission discharge radius, as well as the OH (A → X) band emission in the positive column were measured for discharge currents up to 155 mA. A comparison between the model results and the experimental data suggests that the fraction of water in the plasma is around 20% for the conditions considered. For such water content in the plasma, the simulation shows that:

  • An electron temperature ~ 1 eV seems a realistic value in this kind of discharge in contact with liquid. The NO+ is the dominant ion, mainly formed by electron-impact ionization of NO molecules. The electron-impact ionization of O2 molecules, followed by a quick conversion to NO+, also plays a role. The associative ionization in atomic collisions does not play any significant role due to the rather low temperature of the gas. Consequently, the electric field strength in the plasma is quite large. The electron number density is close to 1019 m−3, showing good agreement with those experimentally inferred from plasma conductivity.

  • The main neutral species in the positive column of the discharge are NO molecules as well as O(3P) and OH radicals. The concentration of OH radicals is of the order of 1022 m−3.

  • The humid air plasma is in thermal non-equilibrium. The gas temperature remains remarkably constant with a value around 3500 K, somewhat lower than those typically obtained in similar discharges but operating with metal electrodes in dry air. The simulated results agree with the obtained experimental data.