Extended reaction kinetics model for non-thermal argon plasmas and its test against experimental data

The modelling studies were performed by means of a time-dependent, spatially one-dimensional fluid model based on a hydrodynamic description of the plasma, where the spatial variation takes place along the x-axis. The particle number densities of the species included in the RKM are obtained by solving their balance equation

$$\begin{equation}\frac{\partial }{\partial t}{n}_{j}\,(x,t)+\frac{\partial }{\partial x}{{\Gamma }}_{j}\,(x,t)={S}_{j}\,(x,t)\,.\end{equation} \tag{ 2 }$$

Here, n_j is the particle number density of the species j, Γ_j is the particle flux and S_j denotes the source term, which describes the gain and loss of particles in the plasma due to collisional and radiative processes. The spatiotemporal evolution of the mean electron energy u_e is described by the energy balance equation

$$\begin{equation}\frac{\partial }{\partial t}{w}_{\mathrm{e}}(x,t)+\frac{\partial }{\partial x}{Q}_{\mathrm{e}}(x,t)=-{e}_{0}{{\Gamma }}_{\mathrm{e}}(x,t)E(x,t)+{P}_{\mathrm{e}}(x,t),\end{equation} \tag{ 3 }$$

where w_e = n_e u_e and Q_e denote the energy density and energy flux of electrons, respectively. The first term on the right-hand side of equation (3) represents the power input from the electric field and the second term describes the gain and loss of electron energy due to the different particle collision processes listed in tables 2 and 3.

In addition, the applied fluid model takes into account Poisson's equation

$$\begin{equation}-\frac{\partial }{\partial x}\left({\varepsilon }_{\mathrm{r}}{\varepsilon }_{0}\frac{\partial {\Phi }(x,t)}{\partial x}\right)=\sum\limits _{j}{q}_{j}{\,n}_{j}\,(x,t)\,\end{equation} \tag{ 4 }$$

for the determination of the electric potential, Φ(x, t), and electric field, E(x, t) = −∂Φ(x, t)/∂x. Here, ɛ_r and ɛ₀ are the relative and vacuum permittivity and q_j is the particle charge.

For the determination of the fluxes of heavy particles, the common drift–diffusion approximation

$$\begin{align}\hfill {{\Gamma }}_{j}\,(x,t)& =\mathrm{sgn}({q}_{j}){n}_{j}\,(x,t){b}_{j}\,(x,t)E(x,t)\hfill \\ \hfill & \quad -\frac{\partial }{\partial x}\left({D}_{j}\,(x,t){n}_{j}\,(x,t)\right)\,\hfill \end{align} \tag{ 5 }$$

is applied. Here, b_j and D_j represent the mobility and diffusion coefficient of species j, respectively, and the function sgn(q_j ) determines the sign of q_j . The electron particle and energy flux are expressed by the relations

$$\begin{align}\hfill {{\Gamma }}_{\mathrm{e}}(x,t)& =-\frac{1}{{m}_{\mathrm{e}}{\nu }_{\mathrm{e}}}\frac{\partial }{\partial x}\left[({\xi }_{0}+{\xi }_{2}){n}_{\mathrm{e}}(x,t)\right]\hfill \\ \hfill & \quad -\frac{{e}_{0}}{{m}_{\mathrm{e}}{\nu }_{\mathrm{e}}}E(x,t){n}_{\mathrm{e}}(x,t),\hfill \end{align} \tag{ 6 }$$

$$\begin{align}\hfill {Q}_{\mathrm{e}}(x,t)& =-\frac{1}{{m}_{\mathrm{e}}{\tilde{\nu }}_{\mathrm{e}}}\frac{\partial }{\partial x}\left[({\tilde{\xi }}_{0}+{\tilde{\xi }}_{2}){w}_{\mathrm{e}}(x,t)\right]\hfill \\ \hfill & \quad -\frac{{e}_{0}}{{m}_{\mathrm{e}}{\tilde{\nu }}_{\mathrm{e}}}\left(\frac{5}{3}+\frac{2}{3}\frac{{\xi }_{2}}{{\xi }_{0}}\right)E(x,t){w}_{\mathrm{e}}(x,t).\hfill \end{align} \tag{ 7 }$$

These relations were obtained by an expansion of the electron velocity distribution function (EVDF) in Legendre polynomials and the derivation of the first four moment equations from the Boltzmann equation of the electrons [70, 71]. The dissipation frequencies of the momentum (ν_e) and energy $({\tilde{\nu }}_{\mathrm{e}})$ flux as well as the transport coefficients ξ₀, ξ₂, ${\tilde{\xi }}_{0}$, and ${\tilde{\xi }}_{2}$ are given as integrals of the isotropic part f₀ and the first two contributions f₁ and f₂ to the anisotropy of the EVDF over the kinetic energy of the electrons as detailed in [72].

The transport coefficients for electrons were determined in the same way as the rate coefficients by solving their steady-state Boltzmann equation in multi-term approximation. The values of rate and transport coefficients of the electrons were used in the fluid-Poisson model as a function of the mean electron energy [73] (local-mean-energy approximation).

Notice that the present modelling approach does not include the impact of superelastic collisions of electrons on the EVDF. The influence of these processes can be taken into account similar to e.g. [74]. Furthermore, it should be mentioned that the present model does not couple self-consistently the time-dependent, spatially one-dimensional fluid-Poisson equations with the solution of the time- and space-dependent electron Boltzmann equation because of the large complexity of that approach and the expected enormous computational expenditure. Corresponding self-consistent approaches were reported e.g. in [75–79].

The mobilities of atomic and molecular argon ions were applied as functions of the reduced electric field E/N considering the data of [80, 81] and their diffusion coefficients were obtained using Einstein's relation [57]. The diffusion coefficients for the metastable argon atoms, Ar[1s₅] and Ar[1s₃], were determined according to ND_m = 1.7 × 10¹⁸ cm⁻¹ s⁻¹ [80], where N is the background gas density.

In order to solve the set of equations (2)–(7), appropriate conditions were applied at the plasma-facing surfaces. Flux boundary conditions for electrons, heavy particles and the electron energy density as well as the emission of secondary electrons caused by ions impinging onto the walls were considered in accordance with [37, 82, 83]. Furthermore, surface charge accumulation at the dielectric surfaces was considered for the calculation of the electric field and potential in the test cases 1 and 2 dealing with DBDs (cf figure 1(a)).

The considered DBD cell consists of two copper electrodes, both covered by 1 mm thick quartz dielectrics. The distance between the dielectric surfaces is d = 3 mm. The electrodes have rectangular shape with a length of 6.6 cm and width of 1.1 cm and provide the discharge area A = 7.62 cm². The DBD is enclosed by a frame made of poly(methyl methacrylate) with quartz side panels, which make the discharge accessible for optical measurements. At the inlet of the plasma source, argon gas with a purity of 99.999% (Ar 5.0) [86] enters through a variable throttle valve (SS-1RS6MM, Swagelok, USA). The gas flow is driven through the system by a Venturi pump (VP00-060H, Vaccon Company Inc., USA) placed at the outlet of the plasma source. The measurements of the applied voltage and electrical current were performed by means of a 75 MHZ high-voltage probe (P6015A, Tektronix Inc., USA) and a current probe (TCP0030, Tektronix Inc., USA), respectively. The measured voltage signals at the powered electrode during the discharge were used as input for the modelling studies.

The modelling of this DBD was performed with the help of COMSOL Multiphysics^® software [87] using the time-dependent solver. The system of partial differential equations (2)–(7) was solved by the finite element method using linear and quadratic Lagrange elements for the fluid equations (2) and (3) and Poisson's equation (4), respectively. The computational domain consists of a plasma region and two dielectric parts (cf figure 1(a)). In the plasma region, a non-uniform mesh with 1500 elements was employed, whose size decreases from the middle of the domain towards the dielectric walls. In both dielectric regions, a uniform mesh with 50 elements was used in accordance with [88]. For all calculations, the time step was adaptively determined considering the maximum relative error tolerance of 10⁻⁴. The model calculations were performed until a stable periodic state of the discharge was reached. The initial number density of electrons was 2 × 10¹² m⁻³ and the number densities of atomic and molecular ions were 1 × 10¹² m⁻³, assuring quasi-neutrality at the beginning of the calculations. An initial mean electron energy of 3 eV and zero potential in the gap were assumed.

The calculated spatiotemporal evolution of the electron number density and the temporal course of the discharge current

$$\begin{equation}I(t)=\frac{A}{d}{\int }_{{\Delta }}^{{\Delta }+d}\left[\sum\limits _{i}{q}_{i}{{\Gamma }}_{i}\,(x,t)+{\varepsilon }_{0}\frac{\partial }{\partial t}E(x,t)\right]\mathrm{d}x\end{equation} \tag{ 8 }$$

in the first seven voltage periods obtained by use of the 23-species RKM is shown in figure 2. It can be seen that the electron density exhibits a symmetric behaviour with respect to both half-periods of the applied voltage after few periods. Approximately equal intensities of positive and negative current peaks indicate that the calculated discharge current follows the same behaviour.

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The corresponding comparison of the measured and calculated current in the stable periodic state of the discharge represented in figure 3 shows that the measured results are very well described by the model calculation using the 23-species RKM. In addition to the good prediction of the time of the breakdown occurrence, the symmetry of the calculated current is also confirmed by the experimental results. It is important to note that the 15-species RKM could not be applied under the given conditions to analyse the discharge behaviour because the applied voltage was too low to ignite the discharge.

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In order to reveal the reasons for different results obtained by means of the 15-species and 23-species RKM, their temporal evolutions of the spatially averaged mean electron energy ${\bar{u}}_{\mathrm{e}}(t)$ and electron number density ${\bar{n}}_{\mathrm{e}}(t)$ during the first voltage period are displayed in figure 4. Here and in the following, the spatially averaged value $\bar{g}(t)$ of a property g(x, t) is determined according to

$$\begin{equation}\bar{g}(t)=\frac{1}{d}{\int }_{{\Delta }}^{{\Delta }+d}g(x,t)\mathrm{d}x.\end{equation} \tag{ 9 }$$

Figure 4(a) shows that the mean electron energy predicted by both RKMs increases synchronously with increasing energy input during the first 0.7 μs. Afterwards, ${\bar{u}}_{\mathrm{e}}(t)$ obtained by use of the 23-species RKM shows a faster increase with increasing time. A quite similar behaviour is noticed from the comparison of ${\bar{n}}_{\mathrm{e}}(t)$ in figure 4(b), except for that the number densities obtained by both RKMs begin to differ after t = 1.4 μs. This means that at t = 1.4 μs the electron number densities are still approximately equal, while the mean electron energies are already different. In the further temporal course, larger differences between the results of the 15-species and 23-species RKM are found for ${\bar{u}}_{\mathrm{e}}(t)$ and ${\bar{n}}_{\mathrm{e}}(t)$. In particular, the results obtained by the 15-species RKM make clear that the electron number density remains too low to lead to ignition of the DBD.

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Figure 5 shows the corresponding spatial variation of the number density and mean energy of the electrons in the gap at the time t = 1.4 μs. The electron number densities determined by both models are almost equal. They exhibit a maximum close to the momentary anode at x = 1.0 mm followed by a continuous decrease towards the momentary cathode at x = 4.0 mm. The mean electron energy is almost constant along the gap. The one obtained by the 23-species RKM is larger by about 0.2 eV than that obtained by the 15-species RKM. Electron-impact excitation processes of ground state argon atoms were found to be responsible for different electron energy losses at the present conditions resulting in different values of the mean electron energy.

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In order to analyse the electron energy loss due to electron-impact excitation of Ar[1p₀] in more detail, the corresponding frequencies of electron energy loss ${\tilde{\nu }}_{\mathrm{e},i}^{\mathrm{L}}={\varepsilon }_{i}{k}_{i}{n}_{\text{Ar}[1\text{p}0]}$ are displayed as a function of the mean electron energy u_e in figure 6(a). Here, i denotes the index of the process in table 2 and ɛ_i and k_i are the electron energy loss and the rate coefficient of the process i, respectively. For a better presentation and to allow a direct comparison between the data of the 15-species and 23-species RKM, the frequencies of specific processes are summed according to the formula ∑_i ɛ_i k_i n_Ar[1p0]. In particular, the electron energy loss frequencies due to electron-impact excitation processes of Ar[1p₀] to the 1s levels (i = 2, 4, 6, 8), the 2p levels from 2p₁₀ to 2p₅ (i = 10, 12, 14, 16, 18, 20) and the 2p' levels from 2p₄ to 2p₁ (i = 22, 24, 26, 28) are shown and denoted by ${\tilde{\nu }}_{\mathrm{e},\mathrm{1}\mathrm{s}}^{\mathrm{L}}$, ${\tilde{\nu }}_{\mathrm{e},\mathrm{2}\mathrm{p}}^{\mathrm{L}}$ and ${\tilde{\nu }}_{\mathrm{e},\mathrm{2}{\mathrm{p}}^{\prime }}^{\mathrm{L}}$, respectively. It can be noticed that all frequencies used in the 23-species RKM are smaller than those of the 15-species RKM above u_e = 5 eV. This implies that the 23-species RKM predicts less electron energy loss due to excitation processes with Ar[1p₀] atoms. It is also a direct consequence of the use of more recent data on electron-impact collision cross sections in the 23-species RKM when compared to the 15-species RKM [37].

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Although the difference between the mean electron energies presented in figure 5 is only around 0.2 eV in main parts of the discharge region, the change of the electron particle gain frequency due to ionisation of ground state argon atoms, ${\nu }_{\mathrm{e},\mathrm{1}{\mathrm{p}}_{\mathrm{0}}}^{\mathrm{G}}={k}_{242}{n}_{\text{Ar}[1\text{p}0]}$, can become quite large. Figure 6(b) shows the corresponding frequencies of the 15-species and 23-species RKM as function of u_e in the range from 5 eV to 6 eV. Generally, the presented frequencies show a similar monotonously increasing behaviour with increasing u_e. In particular, the rapid increase of the mean electron energy between 5.3 and 5.5 eV by only 0.1 eV is accompanied by an increase of ${\nu }_{\mathrm{e},\mathrm{1}{\mathrm{p}}_{\mathrm{0}}}^{\mathrm{G}}$ by one order of magnitude. Therefore, the production of electrons in the case of the 23-species RKM is higher, taking into account that the generation of electrons is largely controlled by the aforementioned process.

Furthermore, a comparison between the calculated and measured number densities of the metastable Ar[1s₅] atom is performed. For that purpose, a laser atom absorption spectroscopy (LAAS) system (EasyLAAS, neoplas control GmbH, Germany) with a 200 MHz photodetector (HCA-S, Femto, Germany) was used for the time-dependent absorption measurements at the wavelength of 811.53 nm, which corresponds to the $\mathrm{A}\mathrm{r}\left(\mathrm{1}{\mathrm{s}}_{\mathrm{5}}\right.$–$\left.\mathrm{2}{\mathrm{p}}_{\mathrm{9}}\right)$ transition. Details regarding the experimental setup are given in [89]. The measurements were performed in the middle of the gap, i.e. at x = 2.5 mm. Figure 7 shows the results of the Ar[1s₅] number densities, corresponding calculated rates of main gain and loss processes of Ar[1s₅] as well as the mean electron energy u_e(x = 2.5 mm, t) during one period of the applied voltage U_a(t) in the periodic state of the discharge.

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Figure 7(a) demonstrates clearly that the measured temporal evolution of the Ar[1s₅] number density is very well described by the fluid-Poisson model using the 23-species RKM. The good agreement confirms an adequate choice of the processes defining the gain and loss of Ar[1s₅] atoms as well as appropriate values of their rate coefficients in the modelling approach.

As it can be seen from figure 7(b), the mean electron energy reaches its largest values shortly before breakdown. Here, the direct electron-impact excitation process Ar[1p₀] + e → Ar[1s₅] + e (i = 2) is mostly responsible for the production of Ar[1s₅] during the ignition phase. With breakdown, the mean electron energy decreases rapidly leading to a much smaller production of Ar[1s₅] atoms. Note that there is a continuous contribution to the production of Ar[1s₅] atoms due to the quenching (i = 293 and 352) and de-excitation (i = 33) of higher levels. This is compensated by the loss in two- and three-body quenching processes (i = 354–356).

Data of PROES measurements for the setup under consideration provide the possibility for an additional validation of the proposed 23-species RKM. The PROES setup used in the experimental studies includes a CCD camera (PicoStar HR12, LaVision GmbH, Germany) with a resolution of 520 × 520 pixels connected to an imaging spectrograph (Shamrock 750, Andor, United Kingdom) with a width of the entrance slit of 200 μm and a grating constant of 600 mm⁻¹. Further details of the PROES measurements can be found in [89].

The measurements reveal the spatiotemporal dynamics of Ar[2p₁] atoms in the gap based on the emission at the wavelength of 750.39 nm, which corresponds to the transition from Ar[2p₁] to Ar[1s₂]. The experimental results are presented in figures 8(a) and (b). They correspond to the emission during the breakdown events in the first and second half of the applied voltage period in the stable state of the discharge, respectively. Zero values of the time axis represent the moment when the magnitude of the discharge current reaches its maximum value. Since the regions with the highest intensity of light emission correspond to the largest production of Ar[2p₁] atoms, the rate of production of Ar[2p₁] due to all electron-impact excitation processes is presented in figures 8(c) and (d). In addition, the spatiotemporal behaviour of the mean electron energy together with contour lines of the electron number density are exhibited in figures 8(e) and (f). The corresponding electric field variation is shown in figures 8(g) and (h). Notice that the momentary cathode is at x = 1 mm during the first half-period (figures 8(a), (c), (e) and (g)) and at x = 4 mm during the second half-period (figures 8(b), (d), (f) and (h)).

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From the PROES measurement results, regions with the highest light emission can be observed at the moment of the gas breakdown in front of the momentary cathode (figures 8(a) and (b)). These regions are very well reproduced by the calculated excitation rate (figures 8(c) and (d)). This concerns their similar occurrences with respect to time and to the spatial position of the emission maxima. Note that electron-impact excitation of ground state argon atoms to the Ar[2p₁] state has the largest contribution to the total production of Ar[2p₁] during breakdown. Since the rate of this collision process is determined by the formula k₂₈ n_e n_Ar[1p0], it can be concluded that the good agreement of the measured emission and calculated excitation rate is a further proof of proper spatiotemporal profiles of n_e, u_e, and E presented in figures 8(e)–(h).

Thus, we can summaries that the proposed 23-species RKM reproduces the plasma behaviour very well at the investigated conditions and that it can be used for different analyses of DBDs in argon plasmas.

The measured data used for this further evaluation of the 23-species RKM was obtained for an atmospheric-pressure DBD, which has previously proven as suitable for the analysis of the thin-film deposition in mixture of argon with small amounts of HMDSO or TMS [40–42]. The plane-parallel discharge configuration with rectangular steel mesh electrodes is 10 mm long (in gas flow direction) and 80 mm wide. Thus, the discharge area is A = 8 cm². Dielectric layers made of borofloat glass with a thickness Δ of 2 mm are glued onto the electrodes. The gas gap d is 1 mm. The purity of argon gas used in the experiment was 99.9999% (Ar 6.0) [40]. The discharges were driven by a sinusoidal voltage provided by a high-voltage generator (G2000, Redline Technologies, Baesweiler, Germany) and the gas flow rate was 6 slm. A sketch of the plasma source is given in figure 1(a) and the discharge parameters are listed in table 4.

Two independent approaches were used to measure the discharge current as described in [42]. In the first case, the grounded electrode was connected in series to a 1 Ω precision resistor R and the voltage U_R measured across the resistor was used to determine the discharge current I_{exp, R}(t) = U_R(t)/R. In the second case, a capacitor C with a capacitance of 32.5 nF was placed instead of the resistor. The measured transferred charge Q(t) was used for the determination of the current I_{exp, C}(t) = dQ(t)/dt. All measured signals were monitored by a digital oscilloscope (MDO3052, Tektronix, Beaverton, OR, USA, 500 MHz bandwidth).

In order to analyse the described plasma source by means of the fluid-Poisson model, the same finite-difference method as used in [37, 72] was applied to solve the set of equations (2)–(7). In case of the balance equations of charged particles and mean electron energy, the exponential scheme based on Scharfetter and Gummel [90] was used for the spatial discretisation. The central-difference method [91] was applied for the solution of Poisson's equation and the balance equations of metastable argon atoms. The calculation domain was divided into 500 non-equidistant spatial intervals, logarithmically refined towards the boundaries. The time step size was automatically adapted to keep the relative error of the mean electron energy below the tolerance of 10⁻⁴. The same initial conditions as in section 4.1 were used.

Figure 9 shows the comparison of the calculated discharge currents using the 15-species and 23-species RKM and measured data during one voltage period in the stable periodic state of the discharge. The measured current signals I_{exp, R}(t) and I_{exp, C}(t) agree very well, which proves the reliability of the measurements. The current peaks representing the breakdown of the discharge are visible in the results of both models. However, the instants of the breakdown events predicted by the two models are different. In case of the 23-species RKM, the moment of the occurrence of the current peaks is in good agreement with the experimental data. The intensity of the calculated current peak around t = 3 μs also agrees well with measured currents and the agreement around t = 9 μs is satisfactory.

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The results obtained by use of the 15-species RKM show a delayed breakdown compared to the experimental and the calculated 23-species RKM results. It is found again that the electron energy loss due to electron-impact excitation processes of ground state argon atoms resulting from the 15-species RKM is larger than in the case of the 23-species RKM. Thus, the mean electron energy obtained using the 15-species RKM is smaller, which leads to a lower electron production due to electron-impact ionisation processes. This implies that more time is needed to meet the breakdown condition when using the 15-species RKM and explains the delayed breakdown compared to the 23-species RKM.

The corresponding temporal evolution of the spatially averaged electron number density ${\bar{n}}_{\mathrm{e}}(t)$ as well as the discharge current I obtained by model calculations is represented in figure 10. It is found that ${\bar{n}}_{\mathrm{e}}(t)$ obtained by use of the 15-species RKM is mostly lower than that predicted by the 23-species RKM. The larger electron number density of the 23-species RKM at the moment of gas breakdown leads to higher absolute values of the current peaks. Furthermore, the results of the 23-species RKM demonstrate that the current peaks in the two half-periods have different absolute values. This behaviour of the discharge current agrees with the measured data (figure 9). It is affected by a different temporal evolution of ${\bar{n}}_{\mathrm{e}}(t)$ in the two half-periods (figure 10(a)). The results obtained using the 15-species RKM do not show this feature of I (figure 10(b)). Here, both current peaks have almost equal intensities.

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The asymmetric behaviour of I(t) is strongly connected with an asymmetric spatiotemporal evolution of the electron number density n_e(x, t) as shown in figure 11. Here, every breakdown event is highly influenced by the previous one, which indicates that memory effects are responsible for the observed asymmetric discharge behaviour.

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The reproduction of the temporal evolution of a measured current is quite difficult, regardless of the type of the model. Based on the analysis for test case 2 it can be said that good agreement between calculated and measured discharge currents substantiates that the 23-species RKM can also be used well for investigations of DBDs at atmospheric pressure.

The third test case of the proposed 23-species RKM is performed for a megahertz-driven μAPPJ. This plasma source consists of two parallel stainless steel electrodes covered with quartz glass windows that allow access for optical measurements. The electrodes are 20 mm long and 1 mm wide resulting in a discharge area A of 20 mm². The gap d is 1.3 mm so that the embedded discharge channel has the volume V = 26 mm³. Argon gas with the same purity level as in the first test case (99.999%) was used here [86]. A function generator (AFG3252, Tektronix) was used to power the discharge with a sinusoidal voltage. A schematic representation of the plasma source is shown in figure 1(b) and the discharge parameters are listed in table 4.

PROES measurements of the spatiotemporal evolution of the plasma emission at 811.53 nm are conducted similarly as in the first test case (section 4.1) with the aid of a high-repetition rate-gated iCCD camera (PicoStar HR12, LaVision GmgH, Germany). Further details about the experimental setup and measurement procedures can be found in [38].

The discharge generated in the plasma jet under the investigated conditions represents an atmospheric-pressure RF glow discharge. This discharge can be operated in two different modes. In the α-mode, the electron production is mainly determined by bulk ionisation, while secondary electrons generated at the cathode surface are the main channel of electron production in the γ-mode [38]. For the numerical modelling of discharges in these two modes and in the α–γ transition regime, the measured dissipated power was used as input. In each period the applied voltage amplitude U₀ was automatically adapted by the relation

$$\begin{equation}{U}_{0}^{\text{new}}={U}_{0}^{\text{old}}{\left(\frac{{P}_{\mathrm{exp}}}{{P}_{\text{calc}}}\right)}^{c},\quad 0< c\leqslant 1\,\end{equation} \tag{ 10 }$$

until reaching periodic state. Here, P_exp denotes the measured power and P_calc is the calculated current averaged power. The constant c has been taken to be 0.25 in accordance with [38]. The use of this procedure ensures that P_exp and P_calc have the same values when the periodic state of the discharge is reached. A similar numerical procedure as in the second test case (section 4.2) was used to solve the fluid-Poisson model. Details about the used initial and boundary conditions are given in [38].

The experimental studies of the μAPPJ reported in [38] were supplemented by time-dependent, spatially one-dimensional fluid model calculations using the 15-species RKM. The results of PROES measurements for the emission line at 750.39 nm (Ar[2p₁] → Ar[1s₂] transition) at the dissipated power of 0.6 and 1.3 W, respectively, were compared with the calculated ground state excitation rate of the lumped Ar[2p'] state comprising the states 2p₄–2p₁. At the powers used in the experimental studies, the discharges operate in the α-mode and in the transition regime from α-mode to γ-mode, respectively. To reproduce these discharge modes, the applied power in the model calculations was chosen to be 1.4 and 3.0 W, respectively. Since the argon emission at 750.39 nm is mainly caused by direct electron-impact excitation of Ar[1p₀], the comparison became possible and showed very good agreement.

A corresponding analysis was also performed for the emission at 811.53 nm determined by the radiative transition Ar[2p₉] → Ar[1s₅]. The measured data were compared with the calculated total excitation rate of the lumped Ar[2p] state comprising the states 2p₁₀–2p₅. This excitation rate includes ground state and stepwise excitation due to electron collisions. Opposite to the comparison for the emission line at 750.39 nm, certain differences between measured and calculated results were found.

The proposed 23-species RKM includes all individual 2p states, i.e. a direct comparison with the PROES results for the emissions at 750.39 nm and 811.53 nm becomes possible. The results of model calculations using the 23-species RKM show that the main contribution to the emission at 750.39 nm comes from direct electron-impact excitation of Ar[1p₀]. Generally, the modelling results show good agreement with the measured emission at 750.39 nm, similar to the results obtained by use of the 15-species RKM shown in [38]. Therefore the focus of the analysis is on the emission at 811.53 nm.

Figure 12 shows corresponding results related to the emission at 811.53 nm, where the applied power of 0.6 and 1.3 W was used for the modelling studies using the 23-species RKM, in accordance with the measurements. The figure includes the periodic behaviour of the calculated discharge voltage and current as well as the spatiotemporal behaviour of the measured argon emission, the calculated production rate of the Ar[2p₉] state due to all electron-impact excitation processes and the calculated mean electron energy u_e(x, t). Notice that the measured emission and calculated excitation rates are normalised with respect to their maximum values at higher power for direct comparison.

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As expected, the results for the discharge voltage and current (figures 12(a) and (b)) show larger values at higher applied power. The electrical properties can be related to the emission intensity as well as patterns measured by PROES. When analysing the results of the PROES measurements at 811.53 nm (figures 12(c) and (d)), two maxima labelled as I and II can be clearly noticed. Although the emission at lower power (0.6 W) is much weaker than at higher power (1.3 W), the same emission pattern can be identified. When comparing these results with the calculated total electron-impact excitation rates of Ar[2p₉] (figures 12(e) and (f)), very good agreement is found for both applied powers. The same emission maxima are reproduced by the model calculations. Especially, their occurrence in time corresponding to the maximum current magnitude as well as their positions in space are reproduced convincingly. Their existence can be explained by the spatiotemporal evolution of the mean electron energy. It can be seen that the maximum values of the calculated production rate (figures 12(e) and (f)) correspond to the maximum electron energy of bulk electrons (figures 12(g) and (h)) indicating that the bulk electrons are responsible for the excitation. The analysis of the individual electron-impact excitation rates of Ar[2p₉] shows that ground state excitation is the predominant process followed by the excitation of Ar[1s₅]. Note that for the investigated conditions, secondary electrons accelerated by the strong electric field in front of the momentary cathode have the highest mean energy in the gap, but a small contribution to the total Ar[2p₉] production due to the low number density of electrons in this region.

It turns out that the influence of secondary electrons on the plasma behaviour cannot be studied from the emission at 811.53 nm, as it was the case for the emission at 750.39 nm [38]. Based on this analysis, it can be concluded that the proposed 23-species RKM provides the possibility to understand the plasma behaviour not only in DBDs, but also in other types of gas discharges, such as the μAPPJ configuration with plane electrodes.