We have started analysis of global model results with testing the sensitivity of global model results on the air content in helium. We made calculation with 5% and 1% of air in helium, in both case with 1% of the H_{2}O in air. All results presented in this section were obtained at 30ns after the pulse application. The results of calculation are presented in Figure 2 together with OH density interval evaluated from minimal and maximal value of spatial distribution in Figure 5a from [14]. It is obvious that the global model in the case with 5% of air (upper part of Figure 2) gives OH densities which are up to two orders of magnitude higher than the peak value in 2D simulation [14]. Only at the lowest values of *n*_{e} and *T*_{e}, our results lie in the interval of OH densities predicted by the fluid model in [14]. In the case with 1% of air and with the same amount of water, calculated curves intersect OH interval from [14] for all three values of *n*_{e}, but still are over targeted interval at the higher electron temperatures. In addition to above presented, this calculation additionally supports choice of 1% of air as input parameter.

Figure 2 (Colour online) OH density as a function of *T*_{e}
determined by global model with: 1% of air (full lines with symbols); and 5% of air (dashed line with symbol) compared on interval of [OH] from [14] (parallel dashed lines) for 1% of H_{2}O in air

Observed disagreement between results of global and fluid modelling is to some extent expected for the case of nanoseconds pulsed discharge, where streamer breakdown occurs. The growth of streamer is associated with rapidly development of ionization front, caused by the large gradient of electric field which produces hot electrons with *T*_{e} up to 9eV [14, 15]. Indeed, this rapid development imposes some restrictions even on the fluid models, as have been discussed in papers [21, 22]. On the other hand, the data point in Figure 2 was calculated with assumption of the constant electron density at given electron temperature, in whole modelled domain. So, our global simulation does not include temporal development of *n*_{e} and *T*_{e}, and comparison with results of fluid simulation [14] is noted straightforward and quantitative, but more qualitative.

As it is noted above, in fluid simulation [14] the water fractions in air were 1%, 0.1% and 0.01%, and the calculated peak OH densities, from the Figure 5d of paper [14], are 6.3·10^{12}cm^{−3}, 7.0·10^{11}cm^{−3} and 7.0·10^{10}cm^{−3}, respectively. As a test, we have run our code with these H_{2}O fractions, but only for *n*_{e} = 3.0·10^{13}cm^{−3}. The Figure 3 shows that in all three cases the global model predictions of OH density are over interval calculated by the fluid model [14], for the electron temperatures higher than 3 eV. Also, the results of our simulation, for given H_{2}O amounts gives similar dependence of [OH] on *T*_{e}, with values which decrease by an order or magnitude with decreasing H_{2}O content.

Figure 3 (Colour online) OH density as a function of *T*_{e} obtained by global model with 1% of air for different H_{2}O fractions (full lines with symbols) compared to results of fluid model[14] (parallel dashed lines). Vertical arrows connect compared data sets

Calculated results with *n*_{e} = 1·10^{13}cm^{−3} and *n*_{e} = 6·10^{13}cm^{−3} are symmetrically positioned below and above that for *n*_{e} = 3·10^{13}cm^{−3} in Figure 3, as is expected from dependencies in Figure 2, and they are not shown for clarity of presentation. So, one can conclude that overlapping of our results and results of fluid modelling [14], with 0.1% and 0.01% H_{2}O will be as in the case of 1% of H_{2}O shown in Figure 2, and we will proceed further analysis only for 1% of H_{2}O in air.

Another important task in modelling of atmospheric pressure plasmas generated in mixtures which contain water is revealing the main production and losses of OH radicals, as is done in [5, 8, 13, 14]. In order to reveal the main production and loss mechanisms, we have calculated the percentage contribution of all processes which determined OH radical’s kinetics, according the reactions list in [16, 17], with *n*_{e} = 6·10^{13}cm^{−3} and for the *T*_{e} interval 1.5eV to 9eV. The results are presented in , and in Figure 4 and Figure 5. The percentage contributions in the case of *n*_{e} = 1·10^{13}cm^{−3} are not presented, but they are discussed in text. The lists the main OH production and loss processes, which are labelled with R1-R5 and R6-R12, respectively.

Figure 4 Contribution of different processes from on OH production as a function of the electron temperature for *n*_{e} = 6·10^{13}cm^{−3}

Figure 5 Contribution of different processes from to OH destruction as a function of the electron temperature for *n*_{e} = 6·10^{13}cm^{−3}

Compared to results of fluid model [14], processes R2 and R4 remain dominant in OH production in case of the global model, unlike the electron-impact dissociation of H_{2}O with contribution that is now several orders of magnitude lower and is omitted from . Similar results are obtained in [4], where contribution of electron induced water dissociation to overall OH production is about 2%.

Electron impact dissociation of water is recognized as one of dominant mechanisms of OH production in plasmas with water content around several thousand ppm (particles per million) [5]. Furthermore, global model reveals additional processes of OH production (R1, R3 and R5) which were not included in the list of processes considered in fluid models [13, 14]. Dissociation of H_{2}O by O(^{1}S) is of particular interest, since it carries more than half of the OH production at low electron temperature, and around 7% at 9 eV, as it is shown in Figure 4.

On the other hand, the contribution of process R4 on OH production significantly increases at higher *T*_{e}, mostly caused by increased production of H_{2}O^{+} ions, since the rate coefficient for H_{2}O^{+} recombination with electrons () does not depend on the electron temperature [15]. According to our results, the production of H_{2}O^{+} ions dominantly goes through charge transfer reactions with nitrogen ions (N^{+},
$\begin{array}{}{\displaystyle {\mathrm{N}}_{2}^{+}}\end{array}$
and
$\begin{array}{}{\displaystyle {\mathrm{N}}_{4}^{+}}\end{array}$
), for the entire *T*_{e} interval from 1.5eV-9eV. Further analysis reveals that nitrogen ions production pathways are based on processes of Penning ionization with He metastable and on He^{+} ions conversion. So, as a final conclusion, an increase of contribution of process R4 in the OH production (Figure 4), and in the total OH density (Figures 2, 3) with *T*_{e}, is related to the increase of electron-impact rate coefficients, given as function of *T*_{e} [16, 17] (in Arrhenius parametric form). The calculations results with low electron density, *n*_{e} = 1·10^{13}cm^{−3} show that contributions of processes R4 and R5 are approximately reduced for factor 2.5 in whole *T*_{e} interval, and now processes R1 and R4 (booth including H_{2}O^{+} ions) carry around 67% of OH production at 9eV, while R2 and R3 dominate at low *T*_{e}.

Figure 5 shows an analysis of the percentage contribution of processes R6-R10 on destruction of OH radicals. Our simulation reveals that the electron dissociation of OH (R8) dominates at low *T*_{e}, mainly caused by the fact that the electron density is kept constant during 30 ns of simulation period, despite temporal growth during streamer development [14, 21, 22]. On the other hand, the processes involving positive ions He^{+} and N^{+} start to dominate in OH destruction at higher *T*_{e}, above 7eV [8]. This is a direct consequence of an increased production of positive ions through electron impact and Penning ionization. The quenching of OH by N and O atoms (R9 and R10) was stated in [5] as an important loss mechanism, and has around of 20% contribution according to our results.

The percentage contributions of destruction processes shown in Figure 5 are slightly affected by decreasing of electron density. The calculation with *n*_{e} = 1·10^{13}cm^{−3} reveals that relative contributions of R8 in this case are approximately equal at low *T*_{e} and differ within 2% at higher *T*_{e}, compared to values in Figure 5. The net contribution of processes R6 and R7 is now few percent lower at higher *T*_{e}, with decreasing of R6 contribution and increasing of R7 contribution within 5%. The three-body recombination reaction H + OH + M → M + H_{2}O was stated in [5, 13, 14] as the main OH loss mechanism. According to results of our simulation, its contribution is approximately two orders of magnitude lower then contribution of R6-R8, in the whole modelled range of electron temperatures. Except for differences in kinetic scheme and used rate coefficients for this process [14, 16], observed disagreement may be explained by higher density of He^{+} and N^{+} ions according to our simulation, since global model calculation, generally, could give higher ion densities.

The question that is imposing after presented analysis is: whether the results of our global model may be improved? Having in mind that low temperature atmospheric pressure plasmas are strongly non-equilibrium, it is obvious that the choice of electron energy distribution functions (EEDF) would affects the kinetics of species [25, 26, 27]. Particularly, the non-equilibrium EEDF for higher energies will be quite different from equilibrium Maxwell-Boltzmann (MB) EEDF, due to considerable inelastic losses in processes of electronic excitation and ionization of helium gas. The rate coefficients for electronic collision calculated with assumption of MB EEDF and with non-equilibrium one, by solving of Boltzmann equation (BE), may be different by several orders of magnitude and lead to quite different chemical composition of modelled plasmas [25, 26, 27]. Some of rate coefficients in our calculation, taken from [16, 17] were expressed as a parametric function of electron temperature *T*_{e} in [eV], with assumption of MB EEDF. The results presented in Figures 2-5, were calculated in this manner. On the other hand, with idea to test the influence of non-equilibrium EEDF on global model results, the rate coefficients for electron-molecules processes were obtained by solving Boltzmann equation with the two-term approximation solver BOLSIG+ [28]. In BOLSIG+ calculation, we have used cross section data from the MORGAN database [29] for He, O_{2}, N_{2} and H_{2}O as an input, for 1% of air in mixture and 1% of H_{2}O in air (78% of N_{2} and 21% of O_{2}). Figure 6 shows the test results, through the impact of non-equilibrium rate coefficients on OH densities. It is obvious that BE curves are significantly shifted to the lower OH densities, and they now approach to OH interval from [14] at higher *T*_{e}, especially for *n*_{e} = 6·10^{13}cm^{−3}. Decreasing of OH density in BE case is a direct consequence of a drop of H_{2}O^{+} density, since its production is now reduced by using of lower rate coefficient.

Figure 6 (Colour online) Influence of non-equilibrium EEDF on global model results. The OH densities drops are marked by a black arrow. Dotted curves are calculated with the non-equilibrium rate coefficients (BE EEDF)

Figure 7 The plasma composition calculated by means of global model with 1% of air content, with the water content is given as percentage of the air: 1% (black bars) and 0.01% (white bars)

Another fact that would be stressed is concerned on choice of electron-impact rate coefficients. The fundamental characteristic of a streamer discharge is formation of a streamer head, an area with increased electric field strength, in front of which electron temperature reaches very high values (around 9 eV) and ionization frequency is maximized, as can be seen in Figures 3b, c, d in ref. [14] and in papers [21, 22]. So, adopting of the rate coefficients from [16, 17], where they were used for *T*_{e} up to 3eV, in the modelling of streamer-like discharge with *T*_{e} up to 9eV, is questionable, as was stated in literature [5, 20]. And finally, in Eq. (1), which describes time evolution of particle density, diffusion loss and loss by the flux of particles directed to the wall are omitted. Taking into consideration very short pulse duration of 30ns and atmospheric pressure, diffusion can be negligible [18]. From Figure 1a in [14] characteristic diffusion length for cylindrical geometry is estimated to be 1/*λ*^{2} = 2.235·10^{6}m^{−2} (for *r* = 2.25mm and *z* = 3mm), and with OH diffusion coefficient in He, D_{OH}= 0.87·10^{−4} m^{2}/s [30] we obtain diffusion loss frequency ≈ 200 1/s. Compared to dominant destruction mechanisms R6-R10, OH loss due to diffusion is four orders of magnitude lower.

Why use a global model? Despite a large number of different species and comprehensive set of processes that determine kinetics of different species, numerical solving of the system of Eq. (1) is not time-consuming and the calculation time for a single data point in Figures 2-6 is shorter than 10 seconds. The second advantage of global modelling is reflected through fast assessment of the plasma’s chemical composition. In each run of code, we obtained densities of all 68 species included in , and we are able to investigate the changes in chemical composition induced by different input parameters. So, global model allows us to assess the densities of many reactive oxygen and nitrogen species, as shown in Figure 7. The comparison of plasma’s chemical compositions for two different amounts of water in air (1% vs 0.01%) is also presented in Figure 7. The main difference arises in density of water-based species, with concentrations that are several orders of magnitude lower in case with 0.01% of H_{2}O in air. The overall amount of oxygen and nitrogen reactive species is not affected by changes in the water content.

Table 1 List of plasma species included in global model

Table 2 The list of the main OH production/loss processes with the rate coeflcients taken from [16, 17] and references therein.

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