Non-equilibrium vibrational and electron energy distribution functions in mtorr, high-electron-density nitrogen discharges and afterglows

We consider a reduced electrical field E/N = 100 Td applied for ${\tau }_{\mathrm{pulse}}$ = 40 ms to a molecular nitrogen gas with p = 5 mtorr and T = 500 K, constant during the simulation, while the post-discharge regime is followed up to 1 s. The initial distributions (t = 0) are Boltzmann vdf and Maxwell eedf at 500 K, respectively, while a very low initial degree of ionization α = 10⁻¹⁰ is considered. The initial distributions start following the application of electrical field until the achievement of a first stationary eedf in the so-called cold gas approximation, i.e. in an environment poor of vibrational and electronic excited states. Perturbations to this stationary state are observed when a well-developed vibrational distribution is formed which in turn represents a new heating for electrons through superelastic vibrational collisions with cold electrons. A new stationary state can be formed when the concentration of atomic species becomes important. In this case the atoms affect the eedf because they present inelastic losses lower than the corresponding ones involving molecules. The post-discharge regime is dominated by superelastic electronic and vibrational collisions. On the other hand we can imagine that the electron density will strongly increase during the pulse following the trend of eedf and vdf, while a decrease is expected when the electrical field is turned off. To understand the relevant results we can define different relaxation times which will be recovered in the relevant results. It should be noted that the relaxation times involving free electrons depend on the electron density. In this case we select the electron density at t ∼ 30 ms. Typical relaxation times can be estimated as

$$\begin{eqnarray}{\tau }_{{\rm{V}}-{\rm{V}}} & = & {({N}_{{{\rm{N}}}_{2}}{k}_{\mathrm{0,1}}^{\mathrm{1,0}})}^{-1}={({10}^{14}\times {10}^{-14})}^{-1}=1{\rm{s}}\\ {\tau }_{{\rm{V}}-{\rm{T}}} & = & {({N}_{{{\rm{N}}}_{2}}{k}^{\mathrm{1,0}})}^{-1}={({10}^{14}\times {10}^{-17})}^{-1}={10}^{3}{\rm{s}}\\ {\tau }_{{\rm{e}}-{\rm{V}}} & = & {({n}_{e}{k}_{{\rm{e}}-{\rm{V}}}^{\mathrm{1,0}})}^{-1}={(2{10}^{10}\times {10}^{-9})}^{-1}=0.5\ {10}^{-1}{\rm{s}}\\ {\tau }_{\mathrm{eedf}} & = & {({N}_{{{\rm{N}}}_{2}}{k}_{{\rm{e}}-{\rm{V}}}^{\mathrm{1,0}})}^{-1}={({10}^{14}\times {10}^{-9})}^{-1}={10}^{-5}{\rm{s}}\end{eqnarray} \tag{ 2 }$$

where ${\tau }_{{\rm{e}}-{\rm{V}}}$ describes the relaxation time for pumping vibrational energy in the molecules and ${\tau }_{\mathrm{eedf}}$ the relaxation time for getting a stationary condition for eedf in the cold gas approximation. ${\tau }_{{\rm{V}}-{\rm{V}}}$ and ${\tau }_{{\rm{V}}-{\rm{T}}}$ represent vibration–vibration and vibration–translation energy transfer relaxation times involving molecular nitrogen. Of course the reported relaxation times must be taken on a qualitative basis since at longer times large concentrations of atoms and vibrationally and electronically excited states can appear, characterized by completely different relaxation times. The longer pulse considered in this case study is such to generate important concentrations of excited states and of electrons as well as of atomic nitrogen which will in turn affect, as anticipated, the eedf. These qualitative considerations can be recovered by the results reported in figure 1.

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Let us first examine the time evolution of the electron temperature, as 2/3 of the average electron energy, $\langle \varepsilon \rangle$ calculated from the eedf, as well as the N₂ vibrational temperature, estimated as the average vibrational energy from vdf. Figure 1(a) shows the evolution of temperatures from the beginning of the discharge up to 40 ms as well as in the post-discharge. In particular we can note that in the post-discharge ${T}_{e}\approx {T}_{v}$, are in agreement with the results in [41]. Both these quantities start from their initial small values presenting different trends especially in the early evolution. In fact T_e reaches a quasistationary situation in times of the order of 10⁻⁵ s keeping this value up to times of about 10⁻² s. From 10⁻² s to 3 10⁻² s the e–V processes start pumping vibrational energy in the system and superelastic vibrational collisions increase the electron average energy. On the other hand the ${T}_{{{\rm{N}}}_{2}}$ keeps a constant low value up to 10⁻² s, starting to increase in the time interval for $t\gt 2$ 10⁻² s. In the post-discharge we can note a sudden decrease of T_e due to the turn-off of the electrical field, while ${T}_{{{\rm{N}}}_{2}}$ relaxes from the value of 2 eV at the end of pulse to 1.4 eV in the post-discharge.

This behavior reflects the action of the applied electrical field on the achievement of a quasistationary situation for eedf in qualitative agreement with the relaxation time ${\tau }_{\mathrm{eedf}}$. On the other hand in the time interval $t\lt {\tau }_{{\rm{e}}-{\rm{V}}}$ the vibrational and electronic states of N₂ are not populated so that the corresponding average energy of molecular nitrogen does not increase. The situation completely changes from $t\gt {\tau }_{{\rm{e}}-{\rm{V}}}$ when vibrational and electronic energies of nitrogen start being activated. The coupling between  the vibrational and electronic energy of N₂ molecules with the eedf through superelastic collisions is such to increase the electron temperature as reported in figure 1(a). This coupling lasts up to 4 10⁻² s when both temperatures present a maximum which then evolves toward another stationary situation. On the other hand the fraction of atomic nitrogen at the end of the pulse (figure 1(b)) reaches a value of 0.46 sufficient to affect the eedf partially eliminating the vibrational inelastic and superelastic vibrational collisions of electrons. Moreover a high molar fraction of electrons, around 6.7 10⁻³, is formed at the end of pulse, with this fraction slightly increasing in the afterglow.

Figure 2 reports the time-dependent evolution of vdf. At the end of the pulse a quasi Boltzmann distribution is present up to 8 eV which decays in the afterglow. At 1 s one can observe a small deactivation of vdf due to multiquantum V–T transitions involving atomic nitrogen.

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Figure 3 reports the time evolution of eedf starting from a very cold eedf (Maxwell, T = 500 K) under the discharge (a) and afterglow (b) conditions. We can note that the eedf's are very smooth either in the discharge or in post-discharge conditions. The structures due to superelastic electronic collisions are not present in this case study for the combined action of superelastic vibrational collisions, the thermalizing action of electron–electron collisions and the presence of atoms. On the other hand the eedf's reported in figure 3(a) show the passage from the molecular cold gas situation to a final situation (40 ms) where e–V (inelastic and superelastic) processes as well as electron–electron Coulomb collisions play an important role. The eedf in the post-discharge (figure 3(b)), after cooling due to the turn-off of the electric field, presents a regular evolution controlled by superelastic vibrational collisions in combination with electron–electron collisions.

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The action of atomic nitrogen also becomes important in the dissociation and ionization processes as one can appreciate by looking figures 4(a) and (b). In figure 4(a) the time-dependent rates for the following dissociation channels are reported:

$$\begin{eqnarray}&&e+{{\rm{N}}}_{2}(v=0)=e+2{\rm{N}}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&e+{{\rm{N}}}_{2}(v\gt 0)=e+2{\rm{N}}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{{\rm{N}}}_{2}+{{\rm{N}}}_{2}(v)={{\rm{N}}}_{2}+2{\rm{N}}\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{\rm{N}}+{{\rm{N}}}_{2}(v)=3{\rm{N}}\end{eqnarray} \tag{ 6 }$$

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We can see that the contribution of direct electron impact is the most important under the discharge conditions, while the dissociation involving atomic nitrogen becomes very important in the post-discharge due in particular to the multiquantum dissociation vibrational jumps [42–44]. Note that the dissociation involving nitrogen–nitrogen collisions becomes important only under the post-discharge conditions. In figure 4(a) the contributions of dissociations from v = 0 and $v\gt 0$ are reported. One can observe that the $v\gt 0$ contribution becomes important from 20 ms on exceeding up to more than two orders of magnitude the v = 0 contributions in the post-discharge.

Also interesting is the ionization rates due to the processes

$$\begin{eqnarray}&&e+{{\rm{N}}}_{2}(v=0)=2e+{{\rm{N}}}_{2}^{+}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&e+{{\rm{N}}}_{2}(v\gt 0)=2e+{{\rm{N}}}_{2}^{+}\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&e+{{\rm{N}}}_{2}({A}^{3}{{\rm{\Sigma }}}_{u}^{+})=2e+{{\rm{N}}}_{2}^{+}.\end{eqnarray} \tag{ 9 }$$

The relevant rates are reported in figure 4(b) showing the importance of channels (8) and (9) compared to the v = 0 contribution for $t\,\gt$ 30 ms on. In the post-discharge the sum of contributions are up to an order of magnitude higher than the v = 0 contribution.

In this case study the same conditions of case study 1 are considered, the only difference being ${\tau }_{\mathrm{pulse}}$ = 30 ms. Under the discharge conditions the results for this case study can be recovered from the previous figures up to t = 30 ms. In contrast in the post-discharge there are many differences because the shorter pulse creates lower concentrations of vibrationally and electronically excited state as well of free electrons. The temperatures and the molar fractions of the different species are reported in figures 5(a) and (b) only in the post-discharge regime. We can see that the N₂ vibrational temperature decays from 0.49 (end of pulse) to 0.41 eV in the post-discharge, while the electron density decays from the initial molar fraction of 2 10⁻⁴ to 10⁻⁵ at the end of the post-discharge, the ${A}^{3}{{\rm{\Sigma }}}_{u}^{+}$ state decaying from 10⁻² to 10⁻⁵ under the same conditions.

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Both vibrational energy and the ${A}^{3}{{\rm{\Sigma }}}_{u}^{+}$ state act in the post-discharge conditions especially on the vdf and eedf behavior reported in figures 6(a) and (b). In particular figures 6(a) and (b) report the behavior of the vdf and eedf in the post-discharge regime, respectively. The vdf presents a small decay showing however a well-pumped distribution sustained by forward and reverse e–V processes. The low concentration of atoms is unable to deactivate the distribution. More interesting is the evolution of eedf. In this case the relaxation occurs in the presence of a quasi-constant vibrational (small) temperature as well as with the concentration of molecular excited states, in particular the ${A}^{3}{{\rm{\Sigma }}}_{u}^{+}$ state, decaying with time. The ${A}^{3}{{\rm{\Sigma }}}_{u}^{+}$ state is responsible for the structures present in the eedf which tend to disappear following its decay. On the other hand the bulk of eedf is controlled by the vibrational temperature of the system so that the eedf converges at 1 s to a Maxwell distribution with ${T}_{e}\approx {T}_{v}=0.41\,\mathrm{eV}\approx 4800\,{\rm{K}}.$

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Figure 7 reports the time evolution of the different dissociation and ionization rates. The electron impact processes follow the evolution of eedf with a small peak at 30 ms due to the effect of superelastic vibrational collisions on eedf. The heavy particle dissociation rates follow on the contrary the time evolution of vdf. In the afterglow all the rates present a decay mitigated by the presence of vibrational states which promote the relevant reactions involving the high-lying vibrational states. Note that at the end of the pulse and in the post-discharge the $v\gt 0$ contribution plays an important role in the dissociation, while (figure 7(b)) $v\,=$ 0, $v\gt 0$ and ${A}^{3}{{\rm{\Sigma }}}_{u}^{+}$ state contributions have the same importance in the ionization rate.

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This case study deals with a reduced value of the pulse duration, ${\tau }_{\mathrm{pulse}}$ = 20 ms. We expect in this case lower molar fractions of vibrationally and electronically excited states, of free electrons as well as lower electron and vibrational temperatures. The situation can be understood by looking at the previous figures as well as at figures 8(a) and (b). From the relevant figures we can see that the vibrational temperature at the end of the pulse is $\approx 0$, while the electron temperature is about 1.5 eV. The molar fraction of the ${A}^{3}{{\rm{\Sigma }}}_{u}^{+}$ state and free electrons are about 10⁻⁴ s and 10⁻⁶ s, respectively, keeping approximately the same values in the post-discharge, while T_e abruptly decreases to very small values.

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Figures 9(a) and (b) show the corresponding behavior of vdf and eedf under the post-discharge conditions. We can see that the vdf's are less populated than in the previous cases showing however a peculiar behavior, i.e. a redistribution of quanta from low-lying vibrational states to high-lying ones. The behavior of eedf in the post-discharge regime presents well-structured forms due to the interplay of holes and humps, respectively, created by inelastic vibrational collisions and superelastic electronic collisions involving the metastable ${A}^{3}{{\rm{\Sigma }}}_{u}^{+}$ state. Dissociation and ionization rates are reported in figure 10.

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It is difficult to compare the present results with the numerous experimental results mentioned in the introduction. Concerning the vibrational distributions we must remind the reader that the present results refer to the vibrational distribution of the ground state of nitrogen while the experimental ones refer to the vibrational distribution of electronically excited states. Moreover the experimental distributions sample the low-lying vibrational levels of the excited state which in general are of Boltzmann type. From the Boltzmann plots of these distributions one obtains the vibrational temperature of the excited state. In general very high vibrational temperatures are observed in line with the theoretical results previously reported. However the theoretical results indicate the presence of a non-equilibrium portion of vdf involving the high-lying excited states. As an example in an inductive low pressure Ar(5%) N₂(95%) operating a p=1.4 mtorr, T_e=2.61 eV, T_g=400 K, n_e=10¹⁰ cm⁻³, i.e. a degree of ionization of ${\alpha }_{\exp }$=2.8 10⁻⁴ [9], a vibrational temperature of 9100 K was experimentally determined from the second positive system of nitrogen (${C}^{3}{{\rm{\Pi }}}_{u}\to {B}^{3}{{\rm{\Pi }}}_{g}$). Our results for case 1 report a value of the vibrational temperature at the end of the discharge of 2 eV = 23 200 K with a degree of ionization of α = 6.7 10⁻³, while for case 2 we obtain a vibrational temperature of 0.49 eV ≈ 5700 K and a degree of ionization of ${\alpha }_{2}$ = 2 10⁻⁴. Keeping in mind that to a first approximation the vibrational temperature depends on the degree of ionization we can assume that case 2 is close to the experimental conditions. Moreover we can correct the theoretical vibrational temperature of case 2 by the following equation:

$$\begin{eqnarray}&&{T}_{v}=5700({\alpha }_{\exp }/{\alpha }_{2})=7980\,{\rm{K}}\end{eqnarray} \tag{ 10 }$$

in satisfactory agreement with the experimental T_v value.

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Moreover the present results are in line with those reported by Kemaneci et al [13] in low pressure O₂ plasmas. In particular the strong non-equilibrium character of the time-dependent ${{\rm{O}}}_{2}(v)$ vdf under the discharge and post-discharge conditions [13] are to some extent in qualitative agreement with our vibrational distributions.

A similar agreement is found comparing our values of case 2 with the experimental value obtained in [12]. These authors present experimental vibrational temperatures in the pressure range 0.04–100 Pa together with the measurements of electron density in the same pressure range. The measured degree of ionization for 5 mtorr = 0.667 Pa amounts to about 1.55 10¹⁰ cm⁻³ corresponding to ${\alpha }_{\exp }$ = 1.6 10⁻⁴. Inserting this value in equation (10) we get a theoretical T_v value of 4560 K, not too far from the experimental value of 4800 K. Note that in this case the vibrational temperature refers to the ground state of nitrogen as a kinetic deconvolution of the second positive system of the molecule.

Concerning the eedf our results show a non-Maxwell behavior either in the molecular plasma situation or in the atomic one (case 1). We see that the degree of ionization $\alpha \,\gt \,$ 10⁻³ is able to maxwellize the distribution in both the discharge and post-discharge conditions. The hole in the eedf distribution due to e–V inelastic collisions persists only in the absence of superelastic vibrational collisions, i.e. in the post-discharge regime of case 3, as well as at low electron densities. To an extent this behavior is qualitatively in line with the experimental results reported by Lee et al [6]. These authors investigate the E–H transition in low pressure inductively N₂-Ar plasmas as a function of ICP power. In the E regime the electron density remains small presenting an abrupt change at a given pressure as a function of power. The measured eedf in the E region presents a deep hole due to the inelastic vibrational losses near 3 eV not contrasted by electron–electron thermalizing collisions, while in the H region the eedf is essentially Maxwellian.