Microwave air plasmas in capillaries at low pressure I. Self-consistent modeling

Research in micro-plasma sources has known a growing interest over the last years due to their various potential applications in several fields, such as photonics with the development of powerful UV sources [1], materials with the synthesis of nanostructures [2, 3], micro-propulsion [4], chemical analysis and sensing [5, 6], environmental issues [7], and biomedical applications due to the production of chemically active species [8–11], especially if the plasma is produced in air and under controlled microfluidic flow conditions [12]. Usually, the generation of micro-plasmas (i.e. non-equilibrium plasmas confined to dimensions below 1 mm) is performed under atmospheric-pressure conditions, thus resulting in the production of chemically reactive species through a combination of heavy-species kinetic mechanisms and boundary-dominated phenomena [13, 14]. However, if technically possible, the generation of micro-plasmas at pressure conditions different than atmospheric can be also of practical interest. Indeed, the very-intense excitation predicted by the pd scaling in this case will favor both the emission of energetic radiation and the production of highly-reactive species, two main ingredients for biological applications [8, 15]. Moreover, in a context where standard working conditions are to be modified, the use of surface-wave discharges [16–18] is of particular interest since they ensure a flexibility of operation [19] in terms of wave frequency (from  ∼100s of kHz to 10s of GHz), gas pressure (∼10⁻⁴–10⁴ Torr), power (∼1–10³ W), dimensions (∼10⁻⁴–10 m) and geometry (e.g. planar and cylindrical).

This work studies microwave-sustained micro-plasmas in dry air ($80\%{{\text{N}}_{2}}:20\%{{\text{O}}_{2}}$), produced using a surface-wave (SW) excitation ($f=\omega /\left(2\pi \right)=2.45$ GHz frequency) within a small radius capillary (R  =  345 μm) at low pressure (p  =  300 Pa). The micro-plasmas are characterized using both self-consistent modeling and experimental diagnostics, and the main results of the study are presented in two cluster papers. The present paper is devoted to the self-consistent modeling of the system, discussing the theoretical formulation adopted, its numerical implementation, and the simulation predictions for the different model quantities, as a function of the electron density. The experimental investigation of the micro-plasmas is presented in a companion paper [20], which shows the experimental setup and the optical emission spectroscopy diagnostics adopted, assessing the validity of the model by comparing simulations and measurements under typical working conditions, as a function of the coupled power and the position along the plasma column.

In principle, the modelling of a low-pressure plasma excited by a high-frequency electric field can adopt a stationary description, since it is expected that ${{\nu}_{T}}\left({{m}_{e}}/M\right)/\omega \ll {{\nu}_{T}}/\omega \ll 1$ [21, 22], with ${{\nu}_{T}}$ the total electron–neutral collision frequency and m_e / M the electron-to-heavy-species mass ratio. These inequalities reveal that the times for the electron particle / energy relaxation and the electron transport evolution, respectively, are much longer than the oscillation period of the applied electric field, thus showing that the problem is of direct-current (DC) nature even for its most mobile species.

Note that the total electric field experienced by the plasma is ${{\vec{E}}_{\text{tot}}}\left(\vec{r},t\right)={{\vec{E}}_{s}}\left(\vec{r}\right)+E\left(\vec{r}\right){{\text{e}}^{j\omega t}}{{\vec{e}}_{z}}$, with ${{\vec{E}}_{s}}=-\vec{\nabla}V$ the space-charge electrostatic field (V being the corresponding potential) and E the high-frequency field. In the case of a plasma column axially-excited within a cylindrical tube, the electrostatic field is aligned mainly in the radial direction, except near the column end where a charge separation in the axial component can also develop. Therefore, the DC radial description of the charged-particle dynamics (balance and transport) involves only the electrostatic space-charge field (or potential), yet connected to the description of the electron energy balance where the DC component of collisional heating involves the high-frequency field [22]. In this work, the balance of energy for the electrons is set by solving the homogeneous two-term electron Boltzmann equation (EBE) for given values of the high-frequency electric field, the excitation frequency, the gas temperature and the ionization degree. The solution to the EBE, for the excitation conditions considered, yields the electron energy distribution function (eedf) that can be used to calculate the electron transport parameters and rate coefficients needed for the description of the charged-particle dynamics.

The unusual small-radius, low-pressure extreme conditions adopted here introduce new modeling challenges, for example as compared to studies developed for discharges with tube radii of the order of the cm. The major consequences are twofold: (i) the development of a very-high maintenance electric field, to compensate for the enhanced radial losses, which can pose some technical problems in the evaluation of the eedf at high kinetic energies; (ii) the need for an adequate description of the charged-particle radial transport, valid for the pR  <  0.08 Torr cm conditions adopted here, where the classical ambipolar theory fails.

A careful review of the existing transport theories for charged particles was made by Phelps [23], who clarified the transition from low pressure (collisionless plasma) to high pressure (collisional plasma), and from low electron-density (independent field-free motion) to high electron-density (ambipolar motion). Figure 1 presents a schematic that summarizes these transitions. Here, the transition from low to high pressure is characterized by the ratio $\Lambda/{{\lambda}_{+}}$, where $\Lambda=R/2.405$ is the characteristic diffusion length for infinite cylindrical geometry and ${{\lambda}_{+}}$ is the ion mean-free-path; whereas the transition from low to high electron-density is associated with the ratio $\Lambda/{{\lambda}_{\text{D}}}$, where ${{\lambda}_{\text{D}}}$ is the Debye length. The high-pressure case ($\Lambda/{{\lambda}_{+}}\gg 1$) was treated first by Schottky [24], who deduced the classical ambipolar diffusion coefficient D_a, by considering electrons in Boltzmann equilibrium with the electrostatic potential and a single type of positive ions with non-inertial behavior. In this high-pressure range, Allis and Rose [25, 26] derived an effective diffusion coefficient dependent on the ratio $\Lambda/{{\lambda}_{\text{D}}}$, to account for the transition between ambipolar and independent motion. The low-pressure ambipolar range ($\Lambda/{{\lambda}_{+}}\ll 1$ and $\Lambda/{{\lambda}_{\text{D}}}\gg 1$) was explored first by Tonks and Langmuir [27], who described the so-called free-fall regime by considering electrons with a Boltzmann distribution and a single type of positive ions with inertial motion. A unified transport theory for dense plasmas ($\Lambda/{{\lambda}_{\text{D}}}\gg 1$) was later proposed by Self and Ewald [28], who combined the transport equations of electrons and ions to describe a plasma column controlled by ambipolar motion at various gas densities, thus bridging the gap between the low and high pressure regimes in a continuous way. The results of Self and Ewald were later expressed by Ferreira and Ricard [29] in the convenient form of another effective diffusion coefficient D_se, introduced so as to extend the Schottky condition to all pressures as ${{D}_{se}}/{{\Lambda}^{2}}=\nu$ (with ν the electron net creation frequency). As expected, the value of D_se depends on the ratio $\Lambda/{{\lambda}_{+}}$, giving ${{D}_{se}}\to {{D}_{a}}$ at high pressure and ${{D}_{se}}<{{D}_{a}}$ as the pressure decreases.

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In principle, the results presented in [28, 29] for ${{D}_{se}}/{{D}_{a}}$ versus $\Lambda/{{\lambda}_{+}}$ could be used at intermediate/low pressures ($\Lambda/{{\lambda}_{+}}<1$) to describe the transport of charged particles in the present case. However, the values of D_se proposed by Ferreira and Ricard were obtained for a single type of positive ions and no negative ions, whereas the present study addresses N₂-O₂ mixtures in the presence of several positive ions and at least one negative ion, which demands a revision of these results in order to fit our working conditions.

The presence of negative ions introduces additional problems when revising the transport theory. Indeed, depending on their density and temperature, negative ions modify the electrostatic space-charge potential, remaining confined to the plasma core due to their low mobility and influencing the radial diffusion length of all other charged particles. In principle, the transport of charged particles can be described by solving numerically the full set of continuity and momentum equations for electrons and ions, as it was done by Edgley and Von Engel [30] for the general case of an arbitrary pressure. However, the approach is hard to implement and the results obtained are not easily translated into a simplified diffusion theory, which explains why other authors have chosen to adopt some extra assumptions, such as the quasi-neutrality condition [31–33] or the imposition of a particular negative ion profile [34]. The work of Ferreira [31] restricts the validity range of Edgley and Von Engel's model to intermediate/high pressures, by assuming quasi-neutrality and by neglecting the positive-ion inertia. A generalization of this formulation to the case of several positive ions is presented in [35]. Conversely, Franklin [32, 33] analyze the low-pressure asymptotic limit by neglecting all collisional processes, thus abandoning the possibility of generalizing the unified transport theory proposed by Self and Ewald. The assumption of a constant density ratio between negative ions and electrons was introduced by Rogoff [34]. The resulting transport description becomes much simpler to handle, but the assumption is valid only if the attachment / detachment rates are large when compared with the ionization rate.

In this work, we revise the unified transport theory proposed by Self and Ewald [28], keeping the assumptions of quasi-neutrality and flux conservation, but considering the presence of several positive ions and the O⁻ negative ion with low density (see table 1). The study is limited to the region of low/intermediate pressures ($p\leqslant {{10}^{3}}$ Pa), for which the negative ion has negligible drift velocity, remaining confined to the plasma core [32, 33]. Under these hypotheses, it is possible to retrieve a system of equations formally identical to the one of [28], whose numerical solution can be avoided using the results of [29] as an abacus. Even with this correction upon the effective diffusion coefficient D_se, the self-consistent maintenance electric field is expected to continue exhibiting considerably high values. Therefore, the electron cross sections and the corresponding kinetic calculations are extended up to 1000 eV, to describe the effect of the high-energy mechanisms impacting significantly on the tail of the eedf.

Table 1. Summary of the species considered in the model.

Species
e
N2(X,$\text{v}=0$–59)	O2(X)
N2(A)	O2(a)
N2(B)	O2(b)
N2(C)	O(3P)
N2(a)	O(1D)
N2(a')	O3
N2(w)	O${}_{3}^{\star}$
N(4S)
N(2D)
N(2P)
NO(X)	NO2(X)
NO(A)	NO2(A)
NO(B)
N+	O+
N${}_{2}^{+}$(X)	O${}_{2}^{+}$
N${}_{2}^{+}$(B)	NO+
N${}_{3}^{+}$	O−
$\text{N}_{4}^{+}$

The organization of this paper is the following. Section 2 describes the theoretical formulation implemented in the numerical code, detailing the transport theory and the kinetic model adopted. The presentation of the latter focuses on a revision upon the electron-scattering cross sections and the kinetic scheme for the ions, for which detailed tables are presented in appendix. Section 3 introduces the numerical code developed, presenting its work-flow and convergence criteria. Section 4 is dedicated to the presentation and discussion of simulation results, focusing on the analysis of the charged-particle transport at various radii and on calculations, as a function of the electron density, of the concentrations and the creation / destruction rates of the main plasma species. Section 5 concludes.

The present study focuses on microwave-sustained micro-plasmas in dry air ($80\%{{\text{N}}_{2}}:20\%{{\text{O}}_{2}}$) produced under the following typical working conditions: excitation frequency $f=\omega /\left(2\pi \right)=2.45$ GHz, tube radius R  =  345 μm, gas pressure p  =300 Pa, gas temperature ${{T}_{\text{g}}}\simeq 1000$ K, electron density ${{n}_{e}}\simeq 4\times {{10}^{12}}$ cm⁻³, and electron temperature (defined as 2/3 of the electron mean energy) ${{T}_{e}}\simeq 4$–7 eV. For generality sake, in a non-equilibrium situation the electron temperature can be replaced by the electron characteristic energy ${{u}_{k}}\equiv {{D}_{e}}/{{\mu}_{e}}$, with D_e and ${{\mu}_{e}}$ the corresponding electron free-diffusion coefficient and mobility, respectively.

For the purpose of this work, we have developed the LoKI (LisbOn KInetics) numerical code, an in-house simulation tool that keeps the same algorithmic structure and calculation blocks used in previous publications [35, 36]. In particular, LoKI couples two main calculation blocks that solve (i) the homogeneous two-term EBE (for the gas mixture considered, including first and second kind collisions as well as electron-electron collisions); (ii) the system of zero-dimensional (0D, volume average) rate balance equations for the most relevant charged and neutral species of the plasma.

For the typical working conditions considered, the plasma is characterized by the following parameters: $\Lambda/{{\lambda}_{\text{D}}}\simeq 15$, corresponding to a dense plasma; $\Lambda/{{\lambda}_{+}}\simeq 0.2$ corresponding to low pressure; ${{\nu}_{T}}/\omega \simeq 0.1$, allowing for a stationary description. In these estimations, ${{\lambda}_{+}}=\left({{M}_{+}}{{\mu}_{+}}/e\right){{\left(3e{{T}_{e}}/{{M}_{+}}\right)}^{1/2}}$ [37] is the mean-free-path of the most abundant O${}_{2}^{+}$ ion, where ${{\mu}_{+}}$ is the ion mobility (see table 2), M₊ is the ion mass and e is the electron charge; ${{\nu}_{T}}\left({{\text{s}}^{-1}}\right)\simeq {{10}^{-7}}N\left(\text{c}{{\text{m}}^{-3}}\right)$ is the total electron–neutral collision frequency, where $N=p/\left({{k}_{\text{B}}}{{T}_{\text{g}}}\right)$ is the gas density (k_B is the Boltzmann constant). Under these ambipolar steady-state conditions, the radially-averaged creation / destruction balance equations for the plasma species can be written as:

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{n}_{k}}\frac{{{D}_{k}}}{{{\Lambda}^{2}}}={{S}_{k}} \end{array}\end{eqnarray} \tag{ 1a }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{n}_{i}}\frac{{{D}_{si}}}{{{\Lambda}^{2}}}={{S}_{i}} \end{array}\end{eqnarray} \tag{ 1b }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{n}_{e}}\frac{{{D}_{se}}}{{{\Lambda}^{2}}}={{S}_{e}} \end{array}\end{eqnarray} \tag{ 1c }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{S}_{n}}\simeq 0 \end{array}\end{eqnarray} \tag{ 1d  }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{n}_{e}}+{{n}_{n}}=\underset{i=1}{\overset{p}{\sum}}{{n}_{i}}. \end{array}\end{eqnarray} \tag{ 1e }$$

Table 2. Transport parameters (mobility and free-diffusion coefficient) for the positive ion species considered in the model.

Ion species i	${{\mu}_{i}}$ (cm2 V−1 s−1)	Di (cm2 s−1)	Reference
N+ , N${}_{2}^{+}$, N${}_{3}^{+}$	$1.87\times (760/p)({{T}_{\text{g}}}/273)$	$[1.7\times {{10}^{18}}{{({{T}_{\text{g}}}/273)}^{0.5}}]/N$	[38]
$\text{N}_{4}^{+}$	$2.33\times (760/p)({{T}_{\text{g}}}/273)$	$[1.7\times {{10}^{18}}\,{{({{T}_{\text{g}}}/273)}^{0.5}}]/N$	[38]
O+	$2.11\times (760/p)({{T}_{\text{g}}}/293)$	$[1.5\times {{10}^{18}}\,{{({{T}_{\text{g}}}/273)}^{0.5}}]/N$	[39]
O${}_{2}^{+}$	$2.24\times (760/p)({{T}_{\text{g}}}/273)$	$[1.5\times {{10}^{18}}\,{{({{T}_{\text{g}}}/273)}^{0.5}}]/N$	[40, 41]
NO+	$1.91\times (760/p)({{T}_{\text{g}}}/273)$	$[1.5\times {{10}^{18}}\,{{({{T}_{\text{g}}}/273)}^{0.5}}]/N$	[42]

Note: In this table, p is in Torr, T_g is in K and N is in cm⁻³.

Equations (1a)–(1c) are for the neutral species k, the positive ions i  =  1, 2,...p and the electrons, respectively, with densities n_k, n_i and n_e, and net creation rates S_k, S_i and S_e, defined by the plasma chemistry (due to the ambipolarity, (1c) can be obtained from the sum of (1b) for all positive ions, see section 2.1.3); equation (1d) is for the negative ion, with density n_n and net creation rate S_n, for which the radial flux is assumed negligible small (see section 2.1.2); and equation (1e) is the charge-conservation condition that can be used to calculate the maintenance electric field as an eigenvalue solution to the problem (see sections 2.3 and 3). In (1a)–(1c), D_k, D_si and D_se are the diffusion coefficients of neutral species k, positive ion species i and electrons, respectively. The latter coefficient can be calculated for low/intermediate gas pressures from the transport theory proposed by Self and Ewald [28], to be revised by considering the presence of several positive ions and a single negative ion with low density.

In principle, it is possible to criticize the use of a fluid-type approach to describe the charged-particle radial transport, in situations where both the ion and the electron mean-free-paths are larger than the tube radius: ${{\lambda}_{+}}/\Lambda\simeq 5$; ${{\lambda}_{e}}/\Lambda\simeq 1/\left(N{{\sigma}_{t}}\Lambda\right)\simeq 35$, with ${{\sigma}_{t}}\simeq {{10}^{-16}}$ cm² a representative value of the total electron–neutral cross section. However, the 0D model adopted focuses on the average gain-loss of particles, thus not addressing explicitly the issues of locality versus non-locality in transport; moreover, the estimation of the charged-particle losses due to diffusion and wall recombination uses an effective diffusion coefficient D_se that modifies the transport loss-rate according to the pressure conditions. As mentioned, the estimation of D_se is based on a unified transport theory for dense plasmas [28] that embeds the classical ambipolar diffusion regime and the free-fall regime as limiting cases.

2.1. Transport theory for low-pressure electronegative dense plasma

In a dense plasma, the transport of charged particles can adopt an ambipolar description. At low pressure, the classical ambipolar diffusion theory does not apply and a free-fall description must be used instead, to consider also the ion inertia under the presence of the ambipolar electric field. The theory derived by Self and Ewald [28] provides a smooth transition between the two asymptotic theories, but only for an electropositive plasma composed of electrons and a single type of positive ions. The following revises this theory, by further considering several positive ions and a single negative ion with low density.

The continuity equations for electrons e, positive ions i  =  1, 2,... p and negative ion n, with densities n_j, drift velocities ${{\mathbf{v}}_{j}}$, fluxes ${{\mathbf{\Gamma}}_{j}}\equiv {{n}_{j}}{{\mathbf{v}}_{j}}$ and source terms S_j (j  =  e, i, n), write

$$\begin{eqnarray}\begin{array}{*{35}{l}} \boldsymbol{\nabla} \cdot \left({{n}_{e}}{{\mathbf{v}}_{e}}\right) & ={{n}_{e}}\left({{\nu}_{I}}-{{\nu}_{a}}+\alpha {{\nu}_{d}}\right)\equiv {{S}_{e}} \end{array}\end{eqnarray} \tag{ 2a }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} \boldsymbol{\nabla} \cdot \left({{n}_{i}}{{\mathbf{v}}_{i}}\right) & ={{n}_{i}}{{\nu}_{{{I}_{i}}}}\equiv {{S}_{i}} \end{array}\end{eqnarray} \tag{ 2b }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} \boldsymbol{\nabla} \cdot \left({{n}_{n}}{{\mathbf{v}}_{n}}\right) & ={{n}_{e}}\left({{\nu}_{a}}-\alpha {{\nu}_{d}}\right)\equiv {{S}_{n}}, \end{array}\end{eqnarray} \tag{ 2c }$$

where ${{\nu}_{I}}$, ${{\nu}_{a}}$ and ${{\nu}_{d}}$ are the total frequencies for ionization, attachment and detachment, respectively; ${{\nu}_{{{I}_{i}}}}$ is the ionization frequency for positive ion species i; and $\alpha \equiv {{n}_{n}}/{{n}_{e}}$ is the plasma electronegativity.

Neglecting the non-linear inertia terms with the electrons and the negative ion, the momentum-transfer equations for the charged-particles write

$$\begin{eqnarray}\begin{array}{*{35}{l}} \frac{e}{{{m}_{e}}\nu _{e}^{\text{eff}}}\,{{n}_{e}}\boldsymbol{\nabla} V-\frac{{{k}_{\text{B}}}{{T}_{e}}}{{{m}_{e}}\nu _{e}^{\text{eff}}}\boldsymbol{\nabla} {{n}_{e}} & ={{n}_{e}}{{\mathbf{v}}_{e}} \end{array}\end{eqnarray} \tag{ 3a }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} \frac{e}{{{M}_{i}}\nu _{i}^{\text{eff}}}\,{{n}_{i}}\boldsymbol{\nabla} V+\frac{{{k}_{\text{B}}}{{T}_{i}}}{{{M}_{i}}\nu _{i}^{\text{eff}}}\boldsymbol{\nabla} {{n}_{i}} & =-{{n}_{i}}\left({{\mathbf{v}}_{i}}+{{\mathbf{v}}_{i}}\cdot \frac{\boldsymbol{\nabla} {{\mathbf{v}}_{i}}}{\nu _{i}^{\text{eff}}}\right) \end{array}\end{eqnarray} \tag{ 3b }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} \frac{e}{{{M}_{n}}\nu _{n}^{\text{eff}}}\,{{n}_{n}}\boldsymbol{\nabla} V-\frac{{{k}_{\text{B}}}{{T}_{n}}}{{{M}_{n}}\nu _{n}^{\text{eff}}}\boldsymbol{\nabla} {{n}_{n}} & ={{n}_{n}}{{\mathbf{v}}_{n}}, \end{array}\end{eqnarray} \tag{ 3c }$$

where M_i and M_n are the ith positive ion and the negative ion masses, respectively; T_i and T_n are the ith positive ion and the negative ion temperatures, respectively; and $\nu _{j}^{\text{eff}}$ are the effective momentum-transfer collision frequencies, given by

$$\begin{eqnarray}\begin{array}{*{35}{l}} \nu _{e}^{\text{eff}} & \equiv & {{\nu}_{e}}+{{\nu}_{I}}-{{\nu}_{a}}+\alpha {{\nu}_{d}} \end{array}\end{eqnarray} \tag{ 4a }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} \nu _{i}^{\text{eff}} & \equiv & {{\nu}_{i}}+{{\nu}_{{{I}_{i}}}} \end{array}\end{eqnarray} \tag{ 4b }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} \nu _{n}^{\text{eff}} & \equiv & {{\nu}_{n}}+{{\nu}_{a}}/\alpha -{{\nu}_{d}}, \end{array}\end{eqnarray} \tag{ 4c }$$

with ${{\nu}_{j}}$ representing the momentum-transfer collision frequency between j and neutral species.

Furthermore, the description assumes the quasi-neutrality and the flux conservation conditions, given by

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{n}_{+}}\equiv \underset{i=1}{\overset{p}{\sum}}{{n}_{i}}={{n}_{e}}+{{n}_{n}} \end{array}\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{\Gamma}_{+}}\equiv \underset{i=1}{\overset{p}{\sum}}{{\Gamma}_{i}}={{\Gamma}_{e}}+{{\Gamma}_{n}}, \end{array}\end{eqnarray} \tag{ 6 }$$

which can be combined with (2a)–(2c) to obtain the balance of rates due to the volume kinetics

$$\begin{eqnarray}&&{{S}_{+}}\equiv \underset{i=1}{\overset{p}{\sum}}{{S}_{i}}=\underset{i=1}{\overset{p}{\sum}}{{n}_{i}}{{\nu}_{{{I}_{i}}}}={{S}_{e}}+{{S}_{n}}={{n}_{e}}{{\nu}_{I}}.\end{eqnarray} \tag{ 7 }$$

The unified theory proposed by Self and Ewald [28] considers a set of only four transport equations: the continuity and the momentum-transfer equations for a single type of positive ions, the momentum-transfer equation for electrons (neglecting its inertia term), and the quasi-neutrality condition. Therefore, in order to convert our system of equations (2a)–(6) into Self and Ewald's formulation we need: (i) to rewrite (2b) and (3b) for the positive ion species i, by introducing a single effective ion that describes the transport of the total positive charge in the plasma; (ii) to make assumptions about the negative ion density and flux, thus obtaining independent solutions to (2c) and (3c) that can be combined with (5) and (6) to obtain relations between the densities and the velocities of electrons and positive ions. These relations will then be used in (2b), (3a) and (3b), to retrieve a system of equations formally identical to that of [28]. The next sections will implement steps (i) and (ii).

2.1.1. Ambipolar transport for dense plasma with multiple positive ions.

In order to describe the transport of the total positive charge in the plasma, by using a single effective ion  +, we first assume that all i-ions have similar temperatures ${{T}_{1}}\sim {{T}_{2}}\sim \ldots {{T}_{p}}\equiv {{T}_{+}}$, drift velocities ${{v}_{1}}\sim {{v}_{2}}\sim \ldots {{v}_{p}}\equiv {{v}_{+}}$, ionization frequencies ${{\nu}_{{{I}_{1}}}}\sim {{\nu}_{{{I}_{2}}}}\sim \ldots {{\nu}_{{{I}_{p}}}}\equiv {{\nu}_{{{I}_{+}}}}$, and effective momentum-transfer collision frequencies $\nu _{1}^{\text{eff}}\sim \nu _{2}^{\text{eff}}\sim \ldots \nu _{p}^{\text{eff}}\equiv \nu _{+}^{\text{eff}}$. The validity of this assumption will be checked a posteriori in section 4.1, based on the results obtained, but in general the different positive i-ions are expected to exhibit: (i) similar temperatures, in most cases; (ii) velocities that scale as $1/\sqrt{{{M}_{i}}}$ at low pressures (hence small $\nu _{i}^{\text{eff}}$), in which case equation (3b) yields the conservation of energy in the cold ion approximation (${{T}_{i}}\to 0$); (iii) velocities that scale as 1/M_i at high pressures and for cold ions, in which case we can further neglect the non-linear term in equation (3b); (iv) similar collision frequencies with neutrals, for similar dimensions of the scattering region. Moreover, we introduce the hypothesis of proportionality, $\boldsymbol{\nabla} {{n}_{i}}/{{n}_{i}}\sim \boldsymbol{\nabla} {{n}_{+}}/{{n}_{+}}$ (i  =  1, 2,..., p), which corresponds to assume that all positive ions have approximately the same density profile. With these assumptions, (2b) and (3b) can be summed for all i-ions to yield

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \boldsymbol{\nabla} \cdot \left({{n}_{+}}{{\mathbf{v}}_{+}}\right)={{n}_{+}}{{\nu}_{{{I}_{+}}}}\equiv {{S}_{+}} \end{array}\end{eqnarray} \tag{ 8a }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} \frac{e}{{{M}_{+}}\nu _{+}^{\text{eff}}}\,{{n}_{+}}\boldsymbol{\nabla} V+\frac{{{k}_{\text{B}}}{{T}_{+}}}{{{M}_{+}}\nu _{+}^{\text{eff}}}\boldsymbol{\nabla} {{n}_{+}} & =-{{n}_{+}}\left({{\mathbf{v}}_{+}}+{{\mathbf{v}}_{+}}\cdot \frac{\boldsymbol{\nabla} {{\mathbf{v}}_{+}}}{\nu _{+}^{\text{eff}}}\right), \end{array}\end{eqnarray} \tag{ 8b }$$

where n₊ and M₊ are the effective-ion density and mass, respectively, the latter defined as

$$\begin{eqnarray}&&{{M}_{+}}\equiv \frac{\sum\nolimits_{i=1}^{p}{{n}_{i}}}{\sum\nolimits_{i=1}^{p}\left({{n}_{i}}/{{m}_{i}}\right)}.\end{eqnarray} \tag{ 9 }$$

2.1.2. Low-pressure limit for the transport in weak electronegative plasma.

At low pressure and for a low density of the negative ion, the various collision frequencies ${{\nu}_{n}}$, ${{\nu}_{a}}$ and ${{\nu}_{d}}$ can be taken vanishingly small (our results show that ${{\nu}_{a}}/{{\nu}_{I}}=\alpha {{\nu}_{d}}/{{\nu}_{I}}\simeq {{10}^{-2}}$ for $\alpha \simeq 0.01-0.1$), in which case (2c) and (3c) become (see (1d)) [32, 33]

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \boldsymbol{\nabla} \cdot \left({{n}_{n}}{{\mathbf{v}}_{n}}\right)={{S}_{n}}\simeq 0 \end{array}\end{eqnarray} \tag{ 10a }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} e{{n}_{n}}\boldsymbol{\nabla} V-{{k}_{\text{B}}}{{T}_{n}}\boldsymbol{\nabla} {{n}_{n}}\simeq 0. \end{array}\end{eqnarray} \tag{ 10b }$$

The solution to these equations yields a density that follows a Boltzmann distribution, in equilibrium with the potential V at temperature T_n, and a drift velocity equal to zero, for null boundary-flux at the wall, thus confirming that the negative ion remains confined to the plasma core. In cylindrical geometry, with ${{n}_{{{n}_{0}}}}$ representing the on-axis negative-ion density,

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{n}_{n}}(r)={{n}_{{{n}_{0}}}}\exp \left(-\frac{e\vert V\vert}{{{k}_{\text{B}}}{{T}_{n}}}\right) \end{array}\end{eqnarray} \tag{ 11a }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} r{{\Gamma}_{n}}(r)=r{{n}_{n}}(r){{v}_{n}}(r)=\text{const}=0\,\Rightarrow \,{{v}_{n}}(r)=0, \end{array}\end{eqnarray} \tag{ 11b }$$

which can be combined with (5) and (6) to give

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{n}_{e}} & = & {{n}_{+}}-{{n}_{{{n}_{0}}}}\exp \left(-\frac{e\vert V\vert}{{{k}_{\text{B}}}{{T}_{n}}}\right) \end{array}\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{v}_{e}} & = & \frac{{{n}_{+}}}{{{n}_{e}}}{{v}_{+}}=\left(1+\alpha \right){{v}_{+}}. \end{array}\end{eqnarray} \tag{ 13 }$$

Conditions (12)–(13) can be used in (3a) to express it as exclusive function of the density n₊ and velocity v₊ of the effective positive ion

$$\begin{eqnarray}&&\xi \frac{e}{{{m}_{e}}\nu _{e}^{\text{eff}}}\,{{n}_{+}}\boldsymbol{\nabla} V-\frac{{{k}_{\text{B}}}{{T}_{e}}}{{{m}_{e}}\nu _{e}^{\text{eff}}}\boldsymbol{\nabla} {{n}_{+}}\simeq {{n}_{+}}{{\mathbf{v}}_{+}},\end{eqnarray} \tag{ 14 }$$

with

$$\begin{eqnarray}&&\xi \equiv \frac{1+\alpha \left({{T}_{e}}/{{T}_{n}}\right)}{1+\alpha}\end{eqnarray} \tag{ 15 }$$

assumed spatially constant, which is better satisfied if $\alpha \ll 1$, hence for a low density of the negative ion. This assumption is not to be used in models providing a spatial description of the plasma, in which case one must note that $\alpha \simeq \text{const}$ is a good approximation only for detachment-dominated plasmas and at higher pressures [43].

Finally, the set of equations (8a), (8b), (12) and (14) can be written, for cylindrical geometry and assuming ${{T}_{+}}\simeq {{T}_{\text{g}}}$, in a form identical to that of [28]

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \frac{1}{r}\frac{\text{d}}{\text{d}r}\left(r{{v}_{+}}\right)+{{v}_{+}}\frac{\text{d}\left(\log {{n}_{+}}\right)}{\text{d}r}={{\nu}_{{{I}_{+}}}} \end{array}\end{eqnarray} \tag{ 16a }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{v}_{+}}\nu _{+}^{\text{eff}}+{{v}_{+}}\frac{\text{d}{{v}_{+}}}{\text{d}r}\simeq -\frac{e}{{{M}_{+}}}\frac{\text{d}V}{\text{d}r}-\frac{{{k}_{\text{B}}}{{T}_{\text{g}}}}{{{M}_{+}}}\frac{\text{d}\left(\log {{n}_{+}}\right)}{\text{d}r} \end{array}\end{eqnarray} \tag{ 16b }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{v}_{+}}\nu _{e}^{\star}\simeq \frac{e}{{{m}_{e}}}\frac{\text{d}V}{\text{d}r}-\frac{{{k}_{\text{B}}}T_{e}^{\star}}{{{m}_{e}}}\frac{\text{d}\left(\log {{n}_{+}}\right)}{\text{d}r} \end{array}\end{eqnarray} \tag{ 16c }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{n}_{e}}={{n}_{+}}-{{n}_{{{n}_{0}}}}\exp \left(-\frac{e\vert V\vert}{{{k}_{\text{B}}}{{T}_{n}}}\right)\;\;\;, \end{array}\end{eqnarray} \tag{ 16d }$$

with (see also (7), (4a) and (4b), under the assumption of a low-density negative ion)

$$\begin{eqnarray}\begin{array}{*{35}{l}} \,\,{{\nu}_{{{I}_{+}}}} & = & \frac{{{\nu}_{I}}}{1+\alpha} \\ \,\,\nu _{+}^{\text{eff}} & = & {{\nu}_{+}}+{{\nu}_{{{I}_{+}}}} \\ \nu _{e}^{\star} & \equiv & \frac{\nu _{e}^{\text{eff}}}{\xi}\simeq \frac{{{\nu}_{e}}+{{\nu}_{I}}}{\xi} \\ \,\,T_{e}^{\star} & \equiv & \frac{{{T}_{e}}}{\xi}. \end{array}\end{eqnarray}$$

In the low-pressure asymptotic limit, the numerical solution of (16a)–(16c), assuming ${{M}_{+}}/{{m}_{e}}\gg 1$ and ${{T}_{\text{g}}}/T_{e}^{\star}\ll 1$, yields [28]

$$\begin{eqnarray}&&{{\left(\frac{{{M}_{+}}}{{{k}_{\text{B}}}T_{e}^{\star}}\right)}^{1/2}}\nu _{+}^{\text{eff}}R\simeq {{\left(\frac{{{M}_{+}}}{{{k}_{\text{B}}}T_{e}^{\star}}\right)}^{1/2}}{{\nu}_{{{I}_{+}}}}R\sim 1\Rightarrow {{\nu}_{{{I}_{+}}}}R\simeq {{\left(\frac{{{k}_{\text{B}}}T_{e}^{\star}}{{{M}_{+}}}\right)}^{1/2}}\equiv {{c}_{s}},\end{eqnarray}$$

with c_s the ion-sound speed that blends the space-time scales of the problem. Note that, by using the definitions of $T_{e}^{\star}$ and ξ, the last expression can be written as

$$\begin{eqnarray}&&c_{s}^{2}=\frac{{{k}_{\text{B}}}{{T}_{e}}}{{{M}_{+}}}\frac{{{T}_{n}}{{n}_{+}}}{{{T}_{n}}{{n}_{e}}+{{T}_{e}}{{n}_{n}}}\;\;\;,\end{eqnarray}$$

which corresponds to the Bohm criterion for an electronegative plasma [32, 44]. The expression is used in [44] to estimate the negative-ion density, by measuring the sheath width with a Langmuir probe.

2.1.3. Application.

Self and Ewald's results have been converted by Ferreira and Ricard [29] in the convenient form of an effective diffusion coefficient as a function of pressure, expressed as an abacus for ${{\left({{D}_{se}}/{{D}_{a}}\right)}_{\text{SE}}}$ versus ${{\left(\Lambda/{{\lambda}_{+}}\right)}_{\text{SE}}}$, where the subscript SE refers to Self and Ewald results. The same procedure can be applied to the results issued from the revised transport theory, presented in sections 2.1.1 and 2.1.2 to further consider the presence of several positive ions and a single negative ion with low density. As expected, the new abacus can be obtained from the original one by using adequate normalization factors as follows

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \frac{{{D}_{se}}}{{{D}_{ae}}}\;\;\;\text{versus}\;\;\;\frac{\Lambda}{{{\lambda}_{+}}} \end{array}\end{eqnarray} \tag{ 17a }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \frac{{{D}_{se}}}{{{D}_{ae}}}={{\left(\frac{{{D}_{se}}}{{{D}_{a}}}\right)}_{\text{SE}}}\frac{1+\alpha}{\xi} \end{array}\end{eqnarray} \tag{ 17b }$$

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} \frac{\Lambda}{{{\lambda}_{+}}}={{\left(\frac{\Lambda}{{{\lambda}_{+}}}\right)}_{\text{SE}}}{{\left(\frac{1}{\xi}\right)}^{1/2}}\;\;\;, \end{array}\end{eqnarray} \tag{ 17c }$$

where D_ae represents the classical ambipolar diffusion coefficient for electrons [34], and ${{\left({{D}_{se}}/{{D}_{a}}\right)}_{\text{SE}}}$ and ${{\left(\Lambda/{{\lambda}_{+}}\right)}_{\text{SE}}}$ are obtained using the results of [28], considering the renormalized set of equations (16a)–(16d).

Equations (17a)–(17c) can be used also to deduce the effective diffusion coefficients D_si, for each positive ion species i. First, we note that by summing (1b) for all positive ions, using (7) and (10a), one obtains

$$\begin{eqnarray}&&\underset{i=1}{\overset{p}{\sum}}{{D}_{si}}{{n}_{i}}={{D}_{se}}{{n}_{e}}.\end{eqnarray} \tag{ 18 }$$

Second, because (18) is a general ambipolar expression, it must hold also under high-pressure conditions, in which case

$$\begin{eqnarray}&&\underset{i=1}{\overset{p}{\sum}}{{D}_{ai}}{{n}_{i}}={{D}_{ae}}{{n}_{e}},\end{eqnarray} \tag{ 19 }$$

where the classical ambipolar coefficients D_ae and D_ai are given by [34]

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{D}_{ae}} & = & \frac{{{n}_{+}}\left({{D}_{e}}{{\mu}_{+}}+{{D}_{+}}{{\mu}_{e}}\right)}{{{n}_{+}}{{\mu}_{+}}+{{n}_{e}}{{\mu}_{e}}} \end{array}\end{eqnarray} \tag{ 20a }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{D}_{ai}} & = & \frac{{{n}_{+}}\left({{D}_{i}}{{\mu}_{+}}-{{D}_{+}}{{\mu}_{i}}\right)+{{n}_{e}}\left({{D}_{e}}{{\mu}_{i}}+{{D}_{i}}{{\mu}_{e}}\right)}{{{n}_{+}}{{\mu}_{+}}+{{n}_{e}}{{\mu}_{e}}}. \end{array}\end{eqnarray} \tag{ 20b }$$

In these equations,

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{D}_{+}} & \equiv & \frac{\sum\nolimits_{i=1}^{p}{{D}_{i}}{{n}_{i}}}{{{n}_{+}}} \end{array}\end{eqnarray} \tag{ 21a }$$

$$\begin{eqnarray}\begin{array}{*{35}{l}} {{\mu}_{+}} & \equiv & \frac{\sum\nolimits_{i=1}^{p}{{\mu}_{i}}{{n}_{i}}}{{{n}_{+}}}, \end{array}\end{eqnarray} \tag{ 21b }$$

with D_i and ${{\mu}_{i}}$ the free-diffusion coefficient and the mobility, respectively, of positive ion species i (the expressions adopted for these transport parameters are listed in table 2). Finally, the previous equations can be combined to yield

$$\begin{eqnarray}&&\frac{{{D}_{se}}}{{{D}_{ae}}}=\frac{1}{{{D}_{ae}}}\frac{\sum\nolimits_{i=1}^{p}{{D}_{si}}{{n}_{i}}}{{{n}_{e}}}\simeq \frac{{{D}_{si}}}{{{D}_{ai}}}\,\frac{\sum\nolimits_{i=1}^{p}{{D}_{ai}}{{n}_{i}}}{{{D}_{ae}}{{n}_{e}}}=\frac{{{D}_{si}}}{{{D}_{ai}}},\end{eqnarray} \tag{ 22 }$$

where we have assumed a ratio ${{D}_{si}}/{{D}_{ai}}$ equal for all positive ions i. Relation (22) can be combined with (17a)–(17c) and (20b) to calculate the effective diffusion coefficients D_si for each positive ion species i.

Relations (17a)–(17c) quantify the modifications introduced in the charged-particle transport features, mostly due to the presence of a negative ion in the plasma. These modifications are the outcome of changes in the potential distribution and the size of the space-charge sheath, affecting the loss of charges from the plasma, hence the value of the electric field required for its maintenance. For the typical working conditions considered here, ${{T}_{e}}/{{T}_{n}}\sim {{T}_{e}}/{{T}_{\text{g}}}\simeq 5.5\times {{10}^{4}}/{{10}^{3}}\simeq 50$ and $\alpha \simeq 0.01$–0.1, yielding $\xi \simeq 1$–5.5 and ${{T}_{+}}/T_{e}^{\star}\simeq 0.1\ll 1$, the latter condition being coherent with the assumption used in the numerical solution presented in [28]. For these conditions, figure 2 plots the abacus ${{D}_{se}}/{{D}_{ae}}$ versus $\Lambda/{{\lambda}_{+}}$, using the original results of Ferreira and Ricard [29] and the renormalized values obtained from the generalized transport theory presented here for low pressures, valid for $\Lambda/{{\lambda}_{+}}\lesssim 1$ only.

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2.2. Kinetic model

The solution to the rate balance equations (1a)–(1e) requires the development of a kinetic model, to define the net creation rates S_j (j  =  k,+, n) due to the plasma chemistry. Note that although the initial system corresponds to pure dry air ($80\%{{\text{N}}_{2}}:20\%{{\text{O}}_{2}}$), its high dissociation degree (particularly for oxygen) requires the study of a stable quaternary mixture of N₂:O₂:N:O, namely when solving the EBE.

The kinetic model considers the following neutral / charged species (see table 1): vibrationally excited levels of ground-state molecular nitrogen N₂(X,$\text{v}=0$–59); electronically excited states N₂(A, B, C, a, a', w); ground-state and electronically excited molecules O₂(X, a, b), NO(X, A, B), NO₂(X, A), O₃ and O${}_{3}^{\star}$; ground-state and electronically excited atoms N(⁴S, ²D, ²P) and O(³P, ¹D); positive ions N⁺ , N${}_{2}^{+}$(X, B), N${}_{3}^{+}$, $\text{N}_{4}^{+}$, O⁺ , O${}_{2}^{+}$ and NO⁺ ; and negative ion O⁻. Since O⁻ is expected to be by far the dominant negative ion, no other negative ion species was included in the model. Notice that the molecular ion O${}_{2}^{-}$ is known to be important for medical applications at atmospheric pressure and can reach significant concentrations in the afterglow [45, 46]. However this ion is actually created as a result of the afterglow kinetics, and its concentration in discharge conditions is several orders of magnitude lower than that of O⁻ [46]. In general, the model adopts the very complete kinetic schemes presented in [47, 48] for nitrogen and in [35, 49] for oxygen and for N₂–O₂ mixtures. These schemes were revised mainly for the charged-particle mechanisms, because these reactions are expected to assume a special relevance in the highly-excited plasma.

The losses of neutrals at the walls are treated with the formulation presented in [50]. Each neutral species k is considered to be lost with a characteristic time ${{\tau}_{k}}$ given by

$$\begin{eqnarray}&&{{\tau}_{k}}=\frac{1}{{{D}_{k}}}{{\left(\frac{R}{2.405}\right)}^{2}}+\frac{2R\left(1-{{\gamma}_{k}}/2\right)}{{{\gamma}_{k}}\langle {{v}_{\text{t}{{\text{h}}_{k}}}}\rangle},\end{eqnarray} \tag{ 23 }$$

where D_k is the diffusion coefficient, $\langle {{v}_{\text{t}{{\text{h}}_{k}}}}\rangle$ is the thermal speed and ${{\gamma}_{k}}$ is the destruction probability at the wall. The limiting cases $\tau _{k}^{-1}\simeq {{\gamma}_{k}}\langle {{v}_{\text{t}{{\text{h}}_{k}}}}\rangle /(2R)$ and $\tau _{k}^{-1}\simeq {{D}_{k}}{{(2.405/R)}^{2}}$ are recovered when ${{\gamma}_{k}}\ll 1$ and ${{\gamma}_{k}}\to 1$, respectively. The diffusion coefficients D_k are calculated from the simplified Wilke's formula, as given in [51] and further detailed in [50]. The description of the wall losses for positive ions is presented in section 2.2.2, dedicated to the ion kinetics.

2.2.1. Electron-scattering cross sections.

The model adopts complete sets of electron–neutral scattering cross sections from ground-state molecular nitrogen and oxygen, retrieved from the IST-LISBON database of LXCat [52, 53]. The set for N₂ includes 23 cross sections (effective; excitations to ground-state vibrational levels (X,$\text{v}=1$–10) and to electronic states A${{}^{3}}{\Sigma}_{u}^{+}\left(\text{v}=0\right.$–4; $\text{v}=5$–9; $\text{v}>9$), C${{}^{3}}{{\Pi}_{u}}$, E${{}^{3}}{\Sigma}_{g}^{+}$, a${{}^{1}}{\Sigma}_{g}^{+}$, B${{}^{3}}{{\Pi}_{g}}$, W${{}^{3}}{{\Delta}_{u}}$, B${{}^{3}}{\Sigma}_{u}^{-}$, a${{}^{1}}{\Sigma}_{u}^{-}$, a${}^{1}{{\Pi}_{g}}$, w${}^{1}{{\Delta}_{u}}$ and combined higher-lying levels; and ionization), and it was compiled mostly from [54–58]. The set for O₂ includes 14 cross sections (elastic momentum-transfer; excitations to ground-state vibrational levels (X,$\text{v}=1$–4); excitations to 7 electronic states organized as a${}^{1}{{\Delta}_{g}}$, b${{}^{1}}{\Sigma}_{g}^{+}$, A${{}^{3}}{\Sigma}_{u}^{+}$  +  C${{}^{3}}{{\Delta}_{u}}$  +  c${{}^{1}}{\Sigma}_{u}^{-}$ bound states, A${{}^{3}}{\Sigma}_{u}^{+}$  +  C${{}^{3}}{{\Delta}_{u}}$  +  c${{}^{1}}{\Sigma}_{u}^{-}$ states yielding dissociation into 2O(³P) and continuum Herzberg bands, B${{}^{3}}{\Sigma}_{u}^{-}$ state yielding dissociation into O(³P)  +  O(¹D) and continuum Schumann–Runge bands, and radiative states with thresholds at 9.97, 14.7 eV; dissociative attachment; and ionization), and it was compiled mostly from [49, 59]. These sets, originally limited to 40 eV kinetic energy, were revised and extended to 1 keV, using information mainly from the databases BIAGI-v8.9 and PHELPS [60].

The model considers also electron-impact reactions with N and O atoms, which are or will become shortly available on the IST-LISBON database of LXCat. For atomic nitrogen, it includes the elastic momentum-transfer, the excitation of N(²D) and N(²P) from ground-state N(⁴S), and the ionization also from ground-state, adopting the electron scattering cross sections of Bartschat et al [61], obtained from quantum mechanical calculations. For atomic oxygen, the model includes 8 cross sections from ground-state O(³P) (elastic momentum-transfer; excitation to O(¹D), O(¹S) and O(³P⁰); excitation to the most important Rydberg states lumped together as O(⁴S⁰), O(²D⁰) and O(²P⁰); and ionization), which were retrieved (and in some cases extended up to 1 keV kinetic energy) from the data proposed by Laher and Gilmore [62]. Figure 3 summarizes the electron cross sections adopted here for atomic oxygen, which are particular relevant for the description of plasmas controlled by the electron kinetics and exhibiting high dissociation degrees of O₂. Note that the electron cross section for the direct excitation of O(¹S) is at higher threshold and about one order of magnitude lower than that of O(¹D). Therefore, even if the description of the electron kinetics uses all the cross sections in figure 3, the species O(¹D) is the sole excited oxygen atom considered in the plasma chemistry model.

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The electron scattering cross sections mentioned previously are used in LoKI as input data, to solve the EBE and to calculate electron rate coefficients (by adequate integration of the cross sections over the eedf), used in the source terms S_j with the rate balance equations. The model includes other electron scattering cross sections, for mechanisms not conserving the number of electrons and/or for inelastic/superelastic collisions with excited states, which are not used to solve the EBE because they are expected to have negligible influence upon the eedf. However, some of these mechanisms can play an important role in defining the plasma chemistry, thus the interest in considering their cross sections to calculate the corresponding rate coefficients. Here, we consider the following supplementary electron-impact cross sections: (i) for nitrogen [47], deexcitation to ground-state N₂(X) from states N₂(A,B,C), with superelastic cross section calculated using the Klein–Rosseland formula [63]; excitation/deexcitation between states N₂(A) and N₂(B,C) [64] (N₂(A) contributes only  ∼5% to the excitation of N₂(C) hence the production of N₂(C) from N₂(B) was disregarded, considering the similar densities and excitation thresholds of N₂(A,B)); dissociation to N(⁴S) and N(²D) [65]; stepwise ionization to N${}_{2}^{+}$ from N₂(A) [66] and N₂(a',B,a) [64]; deexcitation to ground-state N(⁴S) from states N(²D,P) [63]; excitation/deexcitation between states N(²D) and N(²P) [67]; stepwise ionization to N⁺ from N(²D,P) [61]; ionization from N₂(X) and N${}_{2}^{+}$(X) to the ionic excited state N${}_{2}^{+}$(B) [68]; (ii) for oxygen [49], deexcitation to ground-state O₂(X) from states O₂(a,b) [63]; excitation/deexcitation between states O₂(a) and O₂(b) [69]; dissociation from O₂(a) and O₂(b) and stepwise ionization from O₂(a), adopting the same cross sections as for O₂(X) with an appropriate shift in the threshold; dissociative ionization, yielding O(³P)  +  O⁺ , from ground-state [70] and from O₂(a), the latter obtained from the former by applying an appropriate shift in the threshold; ozone dissociation, with cross section assumed to be 5 times that of O₂(X) total dissociation [49]; dissociative attachment from O₂(a) [71]; detachment, yielding O(³P) [72]; (iii) for nitrogen oxide, direct ionization yielding NO⁺ [58].

2.2.2. Ion kinetics.

The high excitation of the plasmas studied here, promoted by the combination of small radius with low pressure conditions, increases the relevance of the charged-particle mechanisms even for ionization degrees of only  ∼10⁻⁴.

The appendix lists the different reactions involving plasma ions, sorting them by species. Tables A1–A4 summarize the kinetic schemes of ground-state nitrogen ions (N⁺ , N${}_{2}^{+}$, N${}_{3}^{+}$ and $\text{N}_{4}^{+}$), of oxygen-containing positive ions (O⁺ , O${}_{2}^{+}$ and NO⁺), of oxygen negative ion O⁻ and of excited nitrogen ion N${}_{2}^{+}$(B), respectively. For the ion-wall recombination mechanisms (R37–R40 in table A1, R18-R20 in table A2 and R5 in table A4) we have assumed a loss probability equal to one. Note that the N₂O  +  e products of reaction R7 in table A3 have been split into N₂  +  O  +  e, to avoid including the N₂O species in the model.

The recombination of positive ions at the walls constitutes a source term for the corresponding neutral species, acting as a loss term for the positive charge-density carried out by the ion species. The latter is compensated by setting an adequate ionization rate through the extended Schottky condition (see (1b), (1c), (7) and (18))

$$\begin{eqnarray}&&\frac{{{n}_{e}}{{D}_{se}}}{{{\Lambda}^{2}}}=\underset{i=1}{\overset{p}{\sum}}\frac{{{n}_{i}}{{D}_{si}}}{{{\Lambda}^{2}}}={{S}_{+}}={{n}_{e}}{{\nu}_{I}},\end{eqnarray} \tag{ 24 }$$

where the wall loss frequency $\tau _{i}^{-1}={{D}_{si}}/{{\Lambda}^{2}}$ is coherent with the assumption of a loss probability equal to one (see also (23)). Note that the previous equation embeds the equality of the net creation rates for the charged species. Note further that the total ionization frequency depends on the self-consistent maintenance electric field, calculated as to satisfy the quasi-neutrality condition (5) (see section 2.3 and 3).

The excited ion N${}_{2}^{+}$(B) is the emitter level of the so-called first-negative-system (FNS) of nitrogen, hence justifying the presentation of its kinetic scheme on a separate table. Note that the cross sections for reactions R1–R2 in table A4, obtained from [68], correspond to excitation mechanisms between vibrational ground-states $\text{v}=0$, from N₂(X,$\text{v}=0$) and N${}_{2}^{+}$(X,$\text{v}=0$) to N${}_{2}^{+}$(B,$\text{v}=0$), the total cross sections being calculated using the corresponding Franck-Condon factors q

$$\begin{eqnarray}\begin{array}{*{35}{l}} \sigma _{{{\text{N}}_{2}}\left(\text{X},\text{v}=0\right)}^{\text{N}_{2}^{+}\left(\text{B}\right)} & = & \frac{\sigma _{{{\text{N}}_{2}}\left(\text{X},\text{v}=0\right)}^{\text{N}_{2}^{+}\left(\text{B},\text{v}=0\right)}}{q_{{{\text{N}}_{2}}\left(\text{X},\text{v}=0\right)}^{\text{N}_{2}^{+}\left(\text{B},\text{v}=0\right)}} \\ \sigma _{\text{N}_{2}^{+}\left(\text{X},\text{v}=0\right)}^{\text{N}_{2}^{+}\left(\text{B}\right)} & = & \frac{\sigma _{\text{N}_{2}^{+}\left(\text{X},\text{v}=0\right)}^{\text{N}_{2}^{+}\left(\text{B},\text{v}=0\right)}}{q_{\text{N}_{2}^{+}\left(\text{X},\text{v}=0\right)}^{\text{N}_{2}^{+}\left(\text{B},\text{v}=0\right)}}. \end{array}\end{eqnarray}$$

2.3. The self-consistent maintenance electric field

Simulations use the LoKI numerical code that solves the homogeneous two-term EBE for the mixture N₂  +  O₂  +  N  +  O (hereafter termed Boltzmann solver), coupled to the 0D rate balance equations for the charged and the neutral heavy-species of the plasma (termed Chemistry solver). The code runs iteratively the Boltzmann solver and the Chemistry solver, until convergence.

Figure 4 presents the work-flow adopted in the numerical calculations. Input data, obtained from experimental observations [20], are the tube radius, the gas pressure and temperature, and the electron density. In addition to n_e, p and T_g (the latter two quantities giving the gas density N), the Boltzmann solver requires also the reduced electric field E/N (see section 2.3), the microwave excitation frequency and the mixture composition, as input parameters. Note that a full self-consistent description of the plasma should include a thermal model, in which case the gas temperature could be calculated as part of the numerical solution, instead of being set as an input parameter obtained from the experiment. The development of a thermal model is scheduled as a next step of this research program, being out of the scope of the present paper.

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The solution to the EBE yields the eedf and the various electron transport parameters and rate coefficients required to run the Chemistry solver. The Boltzmann solver takes into account the vibrational distribution function (vdf) of nitrogen, thus including stepwise inelastic and superelastic collisions from the most important vibrational excited states of ground-state nitrogen N₂(X,$\text{v}=0$–10). Although Boltzmann calculations are limited to electron-vibrational collisions for levels $\text{v}\leqslant 10$, the Chemistry solver extends the description of these interactions for levels up to $\text{v}=60$. The corresponding rate coefficients are calculated adopting the following analytical expression [87]

$$\begin{eqnarray}&&{{k}_{\text{v},\text{v}+n}}\left({{T}_{e}}\right)=\frac{{{k}_{0,\text{n}}}\left({{T}_{e}}\right)}{1+a\text{v}}\,,\;\;\;1\leqslant n\leqslant 10\;\;\;;\;\;\;0\leqslant \text{v}\leqslant 50,\end{eqnarray} \tag{ 25 }$$

where we have used a  =  0.15 as recommended in [87]. This expression was recently validated [88], from the comparison with calculations using the comprehensive compilation of electron-vibrational cross sections available in [89].

Electron-electron collisions are not considered, since numerical tests show that these encounters have negligible effect on the eedf for ionization degrees $\lesssim$10⁻⁴. However, because of the high E/N values expected, calculations consider the influence on the electron rate coefficients of secondary electrons, born in ionization events. Note that, despite the high E/N values encountered, the two-term EBE used here gives calculated swarm parameters (mobility and Townsend coefficient) of N₂, O₂ and the $80\%{{\text{N}}_{2}}:20\%{{\text{O}}_{2}}$ air mixture that agree within 35%–45% with the available measured values for these quantities. Note further that the dispersion of the measured values is between $20\%$ and a factor of 15.

The solution to the system of rate balance equations yields the densities and the reactions rates for the different plasma species, including the N₂(X,$\text{v}$) vdf. To solve these equations, the Chemistry solver requires information on the electron parameters, obtained from the Boltzmann solver, and on the rate coefficients for the reactions between heavy species. The Chemistry solver takes into account the dissociation of molecular species from initial densities ${{\left({{n}_{{{\text{N}}_{2}}}}\right)}_{0}}:{{\left({{n}_{{{\text{O}}_{2}}}}\right)}_{0}}$, thus allowing for an adjustment in the relative composition of the mixture, while ensuring a constant pressure value (condition 1. in figure 4).

For a given mixture, the first global iteration loop is closed by the quasi-neutrality condition (5), satisfied by modifying the self-consistent reduced electric field (condition 2. in figure 4), calculated as an eigenvalue to the problem by using a fast algorithm based on a Newton–Raphson scheme. A second final iteration loop changes the mixture composition (condition 3. in figure 4).

The convergence criterion monitors (i) the evolution of the species density, adopting a time-dependent solution until steady-state that satisfies the particle balances within 10⁻¹³%, and (ii) the changes in p, E/N and the mixture composition, between two consecutive iterations, checking for relative errors smaller than ${{10}^{-6}}\%$, ${{10}^{-8}}\%$ and ${{10}^{-4}}\%$, respectively.

The LoKI numerical code is used (i) to analyze the influence of the tube radius on the charged-particle transport features at low pressure, thus checking the behavior of the revised transport theory presented here; and (ii) to simulate typical low-pressure small-radius conditions, of interest for this work, obtaining results for the concentrations and the creation / destruction rates of the main plasma species, as a function of the electron density. The following sections are dedicated to the presentation and discussion of these results.

4.1. Analysis of the charged-particle transport at various radii

The transport theory presented in section 2.1 is supposed to describe the ambipolar diffusion of charged-particles in dense plasmas with multiple positive ions, over a large range of pressures in the absence of negative ions, and at low/intermediate pressures in the presence of a low-density negative ion.

Figure 5 plots, as a function of R, the plasma characteristic ratios $\Lambda/{{\lambda}_{\text{D}}}$ and $\Lambda/{{\lambda}_{+}}$, calculated for constant p  =  300 Pa, ${{n}_{e}}=3.18\times {{10}^{12}}$ cm⁻³ and ${{T}_{\text{g}}}=976$ K (the pair of (n_e, T_g) values is taken from the experiment at R  =  345 μm, see section 4.2 [20]). Calculations are done in the presence of all positive ions (see section 2.2), considering or neglecting the influence of negative ion O⁻. Figure 5 depicts also the relative density of the negative ion, obtained self-consistently from model calculations. The results in this figure show that (i) $\Lambda/{{\lambda}_{\text{D}}}\gg 1$ for all values of R analyzed here, hence confirming the dense plasma behavior; (ii) $\Lambda/{{\lambda}_{+}}\lesssim 1$ at R  =  345 μm, thus confirming a low pressure situation for the typical working conditions considered.

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In principle, the charged-particle transport description presented in section 2.1 is valid for low gas pressure and low negative-ion density. Hence, model calculations in the presence of O⁻ ions should be restricted to $\Lambda/{{\lambda}_{+}}\lesssim 1$ (corresponding to $R\simeq 0.1$ cm), meaning that the solid curves in figure 5, representing the plasma characteristic ratios in the presence of both positive and negative ions, should be limited to the region on the left of the vertical dashed-dotted line. However, because the population of negative ions is shown to decrease with the radius under these conditions, the influence of O⁻ in the plasma characteristic ratios (and more generally in the charged-particle transport, see also figure 2) becomes negligible for $R\gtrsim 0.1$ cm.

Figure 6 compares self-consistent model calculations of E/N and T_e, as a function of R, obtained for the same working conditions of figure 5 using three different descriptions of the charged-particle transport: (i) the theory described in section 2.1, for multiple positive ion species and a single negative ion with low density, i.e. adopting the abacus defined by (17a); (ii) as in (i), but neglecting the influence of negative ions O⁻, i.e. setting $\alpha =0$ and $\xi =1$ in (17b)–(17c); (iii) the classical ambipolar diffusion theory, taking ${{D}_{\text{se}}}={{D}_{ae}}$ and ${{D}_{\text{si}}}={{D}_{ai}}$. Following our previous conclusions that the influence of O⁻ in transport becomes negligible as R increases (see figure 5), we have extended the calculations using the complete transport model up to R  =  1 cm. As expected, figure 6 shows that the different transport descriptions yield similar results at high R, confirming the classical ambipolar diffusion limit at $\Lambda/{{\lambda}_{i}}\gg 1$ (see also figure 5). At low R (e.g. 345 μm), results show the need for a revised transport theory such as the one proposed here, without which the self-consistent reduced electric field and the electron temperature would assume extremely high-values. Note that calculations obtained with the classical ambipolar diffusion theory are taken down to R  =  0.1 cm only (corresponding to the beginning of the low-pressure region), below which the theory is not applicable.

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Recall that the description presented in section 2.1.1, of the ambipolar transport for a dense plasma with multiple positive ions, has assumed similar temperatures, drift velocities, and collision frequencies for all positive ions. This approximation has been checked using the results of simulations for the main positive ions O${}_{2}^{+}$, NO⁺, O⁺ and N${}_{2}^{+}$(X) (see section 4.2), by estimating the differences in the velocities of each species and by comparing their net creation frequencies, calculated as

$$\begin{eqnarray}&&{{\nu}_{{{I}_{i}}}}=\frac{{{D}_{si}}}{{{\Lambda}^{2}}}.\end{eqnarray} \tag{ 26 }$$

Calculations show that (i) the differences in the velocities of the i-ion species, expected to scale as $1/\sqrt{{{M}_{i}}}$, range between $3\%$ (for the most abundant ions) and $30\%$ (between O${}_{2}^{+}$ and O⁺ , exhibiting the highest difference in mass); (ii) for each working condition, the differences between the various ${{\nu}_{{{I}_{i}}}}$ are below $18\%$.

4.2. Results at various electron densities

This paper focuses on the self-consistent model of air plasmas, produced using microwaves (f  =  2.45 GHz) within fused-silica capillaries (R  =  345 μm) at low pressure (p  =  300 Pa). In addition to p, R and f, simulations require also information on T_g and n_e as input data (see section 3), and in the present study the values of these quantities are taken also from the experiment [20]. Figure 7 presents (i) typical measured values of the gas temperature and the electron density, used as input data in the simulations; (ii) the self-consistent reduced electric field, as a function of the electron density, as obtained from the simulations. The values of E/N depicted in this figure correspond to calculations using the transport theory described in section 2.1, considering or neglecting the influence of negative ion O⁻. As observed, the presence of these ions leads to a decrease of  ∼400 Td in the reduced electric field (for all n_e values considered), thus to a reduction in the rates of electron-impact ionization mechanisms in a situation where the charged-particle diffusion losses are also smaller (see figure 2).

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The high intensity of the maintenance reduced electric field, combined with moderate ionization degrees of  ∼10⁻⁴, are responsible for a non-equilibrium distribution of the electron energy. Figures 8(a) and (b) shows the eedf and the N₂(X,$\text{v}$) vdf, respectively, calculated for $600~\text{Td}\lesssim E/N\lesssim 1600~\text{Td}$ corresponding to the working conditions of figure 7. Note the enhancement of the eedf tail, as the reduced electric field increases, which suggests the importance of electron-impact mechanisms in controlling the plasma kinetics. Note further the non-equilibrium features of the plasma, confirmed by the departure of the eedfs from a Maxwellian distribution (check also the insert in figure 8(a), corresponding to a zoom over the low-energy region), and by the vdfs of N₂(X).

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Figures 9–12 present, as a function of the electron density and for the working conditions of figure 7, the calculated densities of the plasmas species associated with the following systems: molecular nitrogen and oxygen neutral species; atomic nitrogen and oxygen neutral species; nitrogen and oxygen ion species; and nitrogen oxide neutral species, respectively. Tables 3–7 present the most relevant kinetic mechanisms included in the model, corresponding to all reactions contributing more than $5\%$ to the creation / destruction rates of each species, for all the working conditions considered here. The wall recombination / deactivation coefficients presented in these tables were adjusted to improve the agreement between model predictions and experimental diagnostics (see companion paper on the experimental investigation of these micro-plasmas [20]), but are within the predictable range for fused-silica [114], considering the higher wall-losses expected under the present low-pressure small-radius conditions.

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Table 3. Summary of the most important creation / destruction reactions of molecular nitrogen.

No.	Reaction	k (cm3s−1)	Reference
MN1	e  +  N2(X)  →  e  +  N2(A,B,C,a,a',w)	f (E/N)	[53]
MN2	e  +  N2(A)  →  e  +  N2(C)	f (E/N)	[64]
MN3	N2(B)  +  N2  →  N2(A)  +  N2	$0.95\times 3\times {{10}^{11}}$	[35, 90]
MN4	N2(a)  +  N2  →  N2(a')  +  N2	$2\times {{10}^{-11}}$	[91]
MN5	N2(w)  +  N2  →  N2(a)  +  N2	10−11	[92]
MN6	N2(a)  +  O2  →  N2(X)  +  2O(3P)	$4.3\times {{10}^{-10}}$	[91]
MN7	N2(B)  +  O2  →  N2(X)  +  2O(3P)	$3\times {{10}^{-10}}$	[74]
MN8	N2(w)  +  O2  →  N2(X)  +  2O(3P)	10−10	[35]
MN9	N2(A)  +  N2(X,$5\leqslant v\leqslant 14$)  →  N2(B)  +  N2(X,$\text{v}=0$)	$2\times {{10}^{-11}}$	[35]
MN10	N2(C)  →  N2(B)  +  hν	$2.74\times {{10}^{7}}$a	[93]
MN11	N2(B)  →  N2(A)  +  hν	$2\times {{10}^{5}}$a	[94]
MN12	N2(A, B, C, a, a', w)  +  wall  →  N2(X)	$\gamma =1$b

^aEmission frequency in s⁻¹. ^bWall deactivation coefficient. Note: The label f (E /N) means that the rate coefficient is calculated by integration of the corresponding cross section over the eedf.

Table 4. Summary of the most important creation / destruction reactions of molecular oxygen.

No.	Reaction	k (cm3s−1)a	Reference
MO1	e  +  O2(X)  →  e  +  O2(a,b)	f (E/N)	[53]
MO2	e  +  O2(a) $\rightleftharpoons$ e  +  O2(b)	f (E/N)	[49, 69]
MO3	e  +  O2(a)  →  e  +  O2(X)	f (E/N)	[53, 63]
MO4	e  +  O2(a)  →  2e  +  O${}_{2}^{+}$	f (E/N)	[53] (see section 2.2.1)
MO5	e  +  O3  →  e  +  O(3P)  +  O2(X)	f (E/N)	[49] (see section 2.2.1)
MO6	e  +  O2(a,b)  →  e  +  O(3P)  +  O(3P)	f (E/N)	[53] (see section 2.2.1)
MO7	O(1D)  +  O2(X)  →  O(3P)  +  O2(b)	$2.56\times {{10}^{-11}}\exp \left(67/{{T}_{\text{g}}}\right)$	[39]
MO8	O−  +  O2(X)  →  e  +  O3	10−12	[82]
MO9	O−  +  O2(a)  →  e  +  O3	$0.75\times 1.9\times {{10}^{-10}}$	[39, 83]
M10	O−  +  O  →  e  +  O2	$1.3\times {{10}^{-9}}$	[82]
MO11	O  +  2O2  →  O${}_{3}^{\ast}$  +  O2	$(2/3)\times 6.4\times {{10}^{-35}}\exp \left(663/{{T}_{\text{g}}}\right)$	[39, 95]
MO12	O2(a)  +  O${}_{3}^{\ast}$  →  2O2(X)  +  O	$5\times 5.2\times {{10}^{-11}}\exp \left(-1287/{{T}_{\text{g}}}\right)$	[96]
MO13	O  +  O${}_{3}^{\ast}$  →  2O2(X)	$8\times {{10}^{-12}}\exp \left(-507/{{T}_{\text{g}}}\right)$	[95]
MO14	O2(b)  +  wall  →  O2	$\gamma =0.02$b
MO15	$\text{O}_{3}^{\ast}$  +  wall  →  O3	$\gamma =0.01$b

^aFor three-body reactions, the rate coefficient is in cm⁶ s⁻¹. ^bWall deactivation coefficient. Note: In this table, T_g is in K. The label f (E /N) has the same meaning as in table 3.

Table 5. Summary of the most important creation / destruction reactions of atomic nitrogen and oxygen.

No.	Reaction	k (cm3s−1)	Reference
A1	e  +  N2(X)  →  e  +  N(4S)  +  N(4S,2D)	f (E/N)	[65]
A2	e  +  O2(X)  →  e  +  O(3P)  +  O(3P,1D)	f (E/N)	[53]
A3	e  +  N(4S)  →  e  +  N(2D,2P)	f (E/N)	[61]
A4	e  +  N(2D)  →  e  +  N(2P)	f (E/N)	[67]
A5	e  +  O(3P)  →  e  +  O(1D)	f (E/N)	[62]
A6	N(2P)  +  N2(X,v $\geqslant 10$)  →  N(4S)  +  N2(A)	${{10}^{-10}}\exp \left(-1300/{{T}_{\text{g}}}\right)$	[73]
A7	O(3P), N(4S)  +  wall  →  1/2O2, 1/2N2	$\gamma =0.1$a
A8	N(2D, 2P)  +  wall  →  N(4S)	$\gamma =0.1$b
A9	O(1D)  +  wall  →  O(3P)	$\gamma =1$b

^aWall recombination coefficient. ^bWall deactivation coefficient. Note: In this table, T_g is in K. The label f(E/N) has the same meaning as in table 3.

Table 6. Summary of the most important creation / destruction reactions of nitrogen, oxygen and nitrogen oxide ions.

No.	Reaction	Obs.
I1	e  +  N2(X)  →  2e  +  N${}_{2}^{+}$(X,B)	R1 in tables A1 and A4
I2	e  +  N(4S,2D)  →  2e  +  N+	R23 in table A1
I3	e  +  O2(X,a)  →  2e  +  O${}_{2}^{+}$	R1, R2 in table A2
I4	e  +  O(3P)  →  2e  +  O+	R7 in table A2
I5	e  +  O2(X)  →  2e  +  O+  +  O(3P)	R8 in table A2
I6	e  +  NO(X)  →  2e  +  NO+	R16 in table A2
I7	N${}_{2}^{+}$  +  N2(A)  →  N${}_{3}^{+}$  +  N	R10 in table A1
I8	N${}_{2}^{+}$  +  N2(X,v  =  0)  +  N2(X)  →  $\text{N}_{4}^{+}$  +  N2	R11 in table A1
I9	N2(A,a')  +  N2(a')  →  e  +  $\text{N}_{4}^{+}$	R16, R17 in table A1
I10	N+  +  N2(X,v  =  0)  +  N2(X)  →  N${}_{3}^{+}$  +  N2	R31 in table A1
I11	N${}_{2}^{+}$(B)  →  N${}_{2}^{+}$(X)  +  hν	R3 in table A4
I12	e  +  O2(X,a)  →  O−  +  O	R1, R2 in table A3
I13	O−  +  O  →  O2  +  e	R6 in table A3
I14	O−  +  N2(X)  →  N2  +  O(3P)  +  e	R7 in table A3
I15	O−  +  O2(X)  →  O3  +  e	R3 in table A3
I16	e  +  O−  →  O(3P)  +  2e	R12 in table A3
I17	O−  +  O+  →  O  +  O	R14 in table A3
I18	$\text{N}_{4}^{+}$  +  O2(X)  →  O${}_{2}^{+}$  +  N2  +  N2	R19 in table A1
I19	N+  +  O2  →  O${}_{2}^{+}$  +  N	R25 in table A1
I20	N+  +  O2  →  NO+  +  O	R26 in table A1
I21	N+ , N${}_{2}^{+}$, N${}_{3}^{+}$, $\text{N}_{4}^{+}$  +  wall  →  N, N2, N2  +  N, 2N2	R37-R40 in table A1
I22	O+ , O${}_{2}^{+}$, NO+  +  wall  →  O, O2, NO	R18-R20 in table A2

Table 7. Summary of the most important creation / destruction reactions of nitrogen oxide species.

No.	Reaction	k (cm3s−1)a	Reference
NO1	NO+  +  wall  →  NO(X)
NO2	N2(X,$\text{v}\geqslant$13)  +  O(3P)  →  N(4S)  +  NO(X)	10−13	[97]
NO3	e  +  NO(X)  →  2e  +  NO+	f (E/N)	[58]
NO4	NO(X)  +  N(4S)  →  N2(X,$\text{v}=3$)  +  O(3P)	$1.05\times {{10}^{-12}}\,T_{\text{g}}^{0.5}$	[97]
NO5	NO(X)  +  N2(a')  →  N2(X)  +  N  +  O	$3.6\times {{10}^{-10}}$	[74]
NO6	NO(X)  +  N(2D)  →  N2(X)  +  O(3P)	$6.3\times {{10}^{-11}}$	[98]
NO7	NO(X)  +  O${}_{2}^{+}$  →  NO+  +  O2	$4.4\times {{10}^{-10}}$	[74]
NO8	NO(X)  +  N2(A)  →  N2(X,v  =  0)  +  NO(A)	$6.6\times {{10}^{-11}}$	[99]
NO9	NO(A)  →  NO(X)  +  hν	$4.5\times {{10}^{6}}$ b	[100, 101]
NO10	NO(A)  +  N2  →  NO(X)  +  N2	10−13	[102–105]
NO11	N(4S)  +  O(3P)  +  O2  →  NO(B)  +  O2	$3.09\times {{10}^{-34}}{{\left(300/{{T}_{\text{g}}}\right)}^{1.4}}$	[106]
NO12	N(4S)  +  O(3P)  +  N2  →  NO(B)  +  N2	$3.09\times {{10}^{-34}}{{\left(300/{{T}_{\text{g}}}\right)}^{1.4}}$	[106]
NO13	NO(B)$\to$ NO(X)  +  hν	$3\times {{10}^{5}}$b	[106]
NO14	NO(B)  +  N2  →  NO(X)  +  N2	$6.1\times {{10}^{-13}}$	[103, 105, 107]
NO15	NO(B)  +  O2  →  NO(X)  +  O2	$1.5\times {{10}^{-11}}$	[103, 105, 107, 108]
NO16	NO(B)  +  NO  →  NO(X)  +  NO	$2\times {{10}^{-10}}$	[103, 105, 108, 109]
NO17	NO(X)  +  O−  →  NO2(X)  +  e	$5\times {{10}^{-10}}$	[82]
NO18	NO2(X)  +  O(3P)  →  NO(X)  +  O2	$6.51\times {{10}^{-12}}\exp \left(998/{{T}_{\text{g}}}\right)$	[110]
NO19	O(3P)  +  NO(X)  +  O2  →  NO2(A)  +  O2	$3.7\times {{10}^{-32}}$	[111, 112]
NO20	O(3P)  +  NO(X)  +  N2  →  NO2(A)  +  N2	$3.7\times {{10}^{-32}}$	[111, 112]
NO21	NO2(A)  +  O2  →  NO2(X)  +  O2	$6\times {{10}^{-11}}$	[112]
NO22	NO2(A)  +  N2  →  NO2(X)  +  N2	$6\times {{10}^{-11}}$	[113]

^aFor three-body reactions, the rate coefficient is in cm⁶ s⁻¹. ^bEmission frequency in s⁻¹. Note: In this table, T_g is in K. The label f (E/N) has the same meaning as in table 3.

The results in figures 9(a) and (b) reveal that N₂(X) and O₂(X) are the most abundant species, as expected. Note that, depending on the electron density, molecular nitrogen exhibits densities 4.5–6 times higher than those of molecular oxygen, the initial air-mixture composition containing only 4 times more N₂ than O₂, as a consequence of the larger dissociation degree for oxygen. Figures 9(a) and (b) shows also that O₂(a), O₂(b) and N₂(A), N₂(B) are the dominant excited species, with densities  ∼  10¹⁴ cm⁻³ and  ∼  10¹² cm⁻³, respectively, and that O₃ has a very low density, 5-6 orders of magnitude below that of O₂(X), as expected keeping in mind the high gas temperatures and the low-pressure discharge conditions under investigation. The kinetics of the dominant molecular neutral species is controlled (i) in nitrogen, mainly by electron-impact excitation from N₂(X) and by diffusion and wall deactivation (check reactions MN1 and MN12 in table 3), and in the case of N₂(B,C) also by radiative transitions (MN10 and MN11); (ii) in oxygen, mainly by electron-impact excitation, ionization and dissociation (check reactions MO1–MO6 in table 4), and in the case of O₂(b) also by quenching O(¹D) with O₂(X) (MO7) and by diffusion and wall deactivation (MO14).

At low E/N (hence low n_e) electron-impact mechanisms become less effective, which explains the increase of N₂(X) and O₂(X) as the electron density decreases. The higher concentrations of these molecules at low electron density enhance the losses of the N₂ excited states via quenching with N₂ and dissociative quenching with O₂ (MN3–MN9). However, although the densities of N₂(B, C, a, w) decrease as n_e decreases, the concentrations of N₂(A) and N₂(a') are maintained through reactions MN3 and MN4, which become important source channels for these species at low electron density (see figure 9(a)). In oxygen at low n_e, the quenching of excited states with O₂ ground-state species is less important. Indeed, the excitation transfer MO7 between oxygen atoms and molecules becomes the most important mechanism as the electron density decreases, being responsible for the slight increase in the populations of both O₂(b) and O₂(a) (also via MO2) (see figure 9(b)).

Figure 9(a) reveals an inversion in the populations of the N₂(A) and N₂(B) states, as the electron density increases. The phenomenon is caused by the highly-active electron kinetics under the present working conditions, for which ${{T}_{e}}\simeq 6$ eV (see figure 6). In fact, numerical tests confirm that it is possible to avoid the population inversion by setting to zero the rate coefficients of the direct electron-impact excitation of N₂(A,B,C), yet keeping their creation via heavy-species collisions. The result can be further explained by analyzing the theoretical prediction for the density ratio of these states, in a situation exclusively controlled by electron-impact elastic, inelastic and superelastic collisions, in which case their populations would follow a Boltzmann distribution at temperature T_e

$$\begin{eqnarray}&&\frac{\left[{{\text{N}}_{2}}\left(\text{A}\right)\right]}{\left[{{\text{N}}_{2}}\left(\text{B}\right)\right]}=\frac{{{g}_{\text{A}}}}{{{g}_{\text{B}}}}\exp \left[-\frac{\Delta {{E}_{\text{A}}}-\Delta {{E}_{\text{B}}}}{{{T}_{e}}}\right],\end{eqnarray} \tag{ 27 }$$

where ${{g}_{\text{A}}}=3$ and ${{g}_{\text{B}}}=6$ are the statistical weights of states N₂(A) and N₂(B), respectively, and $\Delta {{E}_{\text{A}}}=6.2$ eV and $\Delta {{E}_{\text{B}}}=7.4$ eV are the corresponding excitation energies. Equation (27) shows that an inversion in the populations of these states is expected for ${{T}_{e}}>\left(\Delta {{E}_{\text{A}}}-\Delta {{E}_{\text{B}}}\right)/\ln \left({{g}_{\text{A}}}/{{g}_{\text{B}}}\right)\sim 1.73$ eV, i.e. well below the typical  ∼  6 eV characterizing the temperature of electrons in the present situation. Obviously, the complete kinetic modeling of N₂-O₂ low-pressure plasmas in capillaries is influenced by more than just electron-impact mechanisms. As seen, heavy-particle reactions (strongly dependent on the gas temperature, not the electron temperature) and the transport of species both play a fundamental role in the results. For example, the non-equilibrium features induced by these mechanisms contribute to a difference by a factor of about 10⁻⁴ between the populations of N₂(X) ground-state and N₂(A,B) excited states (see figure 9(a)), well below the ratio [N₂(A,B)] / [N₂(X)] ≃1–2 (corresponding to a population inversion also with respect to the ground-state), obtained under an equilibrium situation at ${{T}_{e}}\simeq 6$ eV. Actually, the influence of the heavy-particle dynamics is observed also at low electron density (see figure 9(a)), where electron scattering can no longer induce a N₂(A)–N₂(B) population inversion.

Figure 10 shows that O(³P) is the most abundant atomic species, with a density of  ∼  10¹⁵ cm⁻³, followed by N(⁴S) and O(¹D) with densities  ∼20 times lower. These results confirm the high dissociation degree of oxygen ([O(³P)]/[O₂(X)]  ∼20%–70%, for ${{n}_{e}}\simeq (1-4)\times {{10}^{12}}$ cm⁻³), in contrast with the [N(⁴S)]/[N₂(X)]  ∼0.1% dissociation reported for nitrogen. The main creation channels of the dominant atomic species are electron-impact dissociation and excitation (check reactions A1–A5 in table 5), which explains the increase in their densities with n_e. The main destruction mechanism for these species is by diffusion and wall recombination / deactivation (A7–A9), and for the excited state N(²P) is also by quenching with higher-lying vibrational levels of N₂ (A6).

Figures 11(a) and (b) shows that O${}_{2}^{+}$, NO⁺ and O⁺ (the latter at high n_e) are the most abundant ion species, followed by N${}_{2}^{+}$(X) and the negative ion O⁻ whose populations are already  ∼10 times below the electron density. As expected, the main creation channel for the dominant positive ions is by electron-impact ionization (check reactions I1–I6 in table 6), whereas the main destruction mechanism is by diffusion and wall recombination (I21–I22). The radiative emission (I11) plays an important role in the creation / destruction of N${}_{2}^{+}$(X) and N${}_{2}^{+}$(B). Similarly to the atomic species, the densities of O${}_{2}^{+}$, O⁺ , N${}_{2}^{+}$(X,B), N⁺ and N${}_{3}^{+}$ increase with n_e, due to the ruling of electron-impact mechanisms in the creation of these species. However, the densities of positive ions NO⁺ and $\text{N}_{4}^{+}$ behave differently when the electron density increases: (i) NO⁺ is created mainly by electron-impact ionization I6, but its density remains almost constant due to the decrease in the concentration of the parent neutral species NO(X) with n_e (see figure 12); (ii) $\text{N}_{4}^{+}$ is created mainly via the three-body reaction I8 involving the ground-state N₂(X), whose density decreases with n_e (see figure 9).

The main creation / destruction mechanisms of the negative ion O⁻ are dissociative attachment (I12) and associative detachment (I13), respectively. At low electron density, the higher concentrations of N₂(X) and O₂(X) make the losses of O⁻ to depend also of reactions I14 and I15, but the stronger attachment rate associated with I12 is ultimately responsible for an increase in the density of the negative ion.

The most abundant species with the NO system is NO(X), with a density (2–$\left. 6\right)\times {{10}^{14}}$ cm⁻³, followed by NO₂(X) with a density 300–500 times lower, as can be observed in figure 12. The main creation / destruction mechanisms of these species are: for NO(X), wall recombination and electron-impact ionization (check reactions NO1 and NO3 in table 7); for NO₂(X), associative detachment and dissociation by impact with O-atoms (NO17 and NO18). Note that collisions of ground-state oxygen atoms with N₂(A) and of ground-state oxygen molecules with N(²D) contribute less than $1\%$ to the formation of NO(X), contrary to what is observed for example in microwave plasmas produced in centimeter-radius tubes at pressures of 10²–10³ Pa [113]. In general, the kinetics of the nitrogen oxide system is controlled by collisions between heavy species, namely: (i) ground-state atoms O(³P), N(⁴S) and molecules N₂(X), playing an important role in the creation of NO(B) and NO₂(A) (NO12 and NO20) and in the destruction of NO₂(X) and NO₂(A) (NO18 and NO22); (ii) the nitrogen excited state N₂(A), responsible for the excitation of NO(A) from NO(X) (NO8). The radiative decays NO9 and NO13 are the main destruction channels of NO(A) and NO(B), respectively. As outcome, the populations of the nitrogen oxide species are affected only indirectly by the electron density, exhibiting small variations with changes in n_e.

This work presented the self-consistent modeling of microwave (f  =  2.45 GHz excitation frequency) micro-plasmas, produced in dry air ($80\%{{\text{N}}_{2}}:20\%{{\text{O}}_{2}}$) within capillaries (R  =  345 μm inner radius) at low gas pressure (p  =  300 Pa). The study used the LoKI numerical code, an in-house simulation tool that couples the homogeneous two-term EBE with the 0D rate balance equations for the most relevant charged and neutral species of the plasma. The code uses as input data the experimental values of the electron density (${{n}_{e}}\simeq \left(1\right.$–$\left. 4\right)\times {{10}^{12}}$ cm⁻³) and the gas temperature (${{T}_{\text{g}}}\simeq 600$–1000 K) [20], in addition to f, R and p, calculating the self-consistent reduced electric field E/N by imposing the quasi-neutrality condition, and adjusting the relative composition of the mixture after the dissociation of molecular species, while ensuring a constant pressure value. Note that a full self-consistent description of the plasma should include a thermal model, to calculate the gas temperature as part of the numerical solution, which is scheduled as a next step of this research program. A very complete thermal model should also consider different ion temperatures, as the outcome of the ion recombination and the energy deposition on the small-radius discharge tube, with impact on the formulation of the charged-particle transport model.

The transport of charged particles, under the extreme conditions of small-radius low-pressure adopted here, was described using the bases of the unified transport theory for dense plasmas at various pressures proposed by Self and Ewald [28], and later reinterpreted by Ferreira and Ricard [29]. This theory, originally set for plasmas containing a single positive ion species, was revised here so as to consider the presence of several positive ions and a negative ion with low density. In particular, (i) the transport of the total positive charge in the plasma was described using a single effective positive ion; (ii) the model formulation was limited to low/intermediate pressures, considering vanishingly small attachment/detachment rates, in which case the low-density negative ion has negligible drift velocity and remains confined to the plasma core. Under these hypothesis, we have retrieved a system of equations formally identical to that of [28], whose solution can be worked-out from the original results of Self and Ewald, yielding a correction upon the abacus ${{D}_{se}}/{{D}_{ae}}$ versus $\Lambda/{{\lambda}_{+}}$ obtained in [29]. For the typical low-pressure working conditions considered here, this correction is sufficient to modify the relative characteristic diffusion length, the reduced maintenance electric field and the electron temperature of the air micro-plasmas under study, even at electron to negative-ion density ratios as small as 0.05. At larger discharge radii, the self-consistent population of O⁻ becomes negligible, and the current transport formulation can still be used to give the asymptotic values predicted by the classical ambipolar diffusion theory for $\Lambda/{{\lambda}_{\text{D}}}$, $\Lambda/{{\lambda}_{+}}$, E/N and T_e.

The description of the plasma chemistry required the study of the stable quaternary mixture N₂:O₂:N:O, namely when solving the EBE. We have revised the very complete kinetic schemes presented in [47, 48] for nitrogen and in [35, 49] for oxygen and for N₂–O₂ mixtures, particularly the mechanisms of charged-particle creation / destruction that have special relevance for maintaining the highly-excited plasma. In this context, the electron-impact scattering cross sections from ground-state molecular / atomic nitrogen and oxygen were also revised and extended to 1 keV. These sets of cross sections are or will become shortly available on the IST-LISBON database of LXCat [52]. The model considers also other electron scattering cross sections, for mechanisms not conserving the number of electrons and/or for inelastic/superelastic collisions (including dissociation) with excited states, namely N₂(A,B,C), N(⁴S,²D,P) and O₂(a,b), which can play an important role in defining the plasma chemistry.

Model results at various n_e show that the most abundant neutral species are N₂(X) and O₂(X) (with densities of  ∼(2–$\left. 3\right)\times {{10}^{16}}$ cm⁻³ and  ∼(2–$\left. 5\right)\times {{10}^{15}}$ cm⁻³, respectively); O(³P) (∼(1–$\left. 2\right)\times {{10}^{15}}$ cm⁻³); NO(X) (∼(2–$\left. 6\right)\times {{10}^{14}}$ cm⁻³); O₂(a), O₂(b) and O(¹D), N(⁴S) (∼10¹⁴ cm⁻³ and  ∼$5\times {{10}^{13}}$ cm⁻³, respectively); and N₂(A), N₂(B) (∼10¹² cm⁻³). These results confirm the high dissociation degree of oxygen ([O(³P)]/[O₂(X)]  ∼20%–70%, in contrast with only [N(⁴S)]/[N₂(X)]  ∼0.1% dissociation for nitrogen. Moreover, the model predicts a very low density for O₃, which is found 5–6 orders of magnitude below that of O₂(X), as expected considering the present high-temperature low-pressure conditions. The kinetics of the dominant neutral species with nitrogen and oxygen is controlled mainly by electron-impact reactions (excitation, ionization, dissociation), by radiative transitions and by diffusion and wall deactivation / recombination. At low electron densities, the quenching of excited species by N₂(X) and O₂(X) becomes a competitive mechanism. The main creation / destruction mechanisms of NO(X) are wall recombination of NO⁺ and electron impact ionization, respectively. The most abundant ion species are O${}_{2}^{+}$ and NO⁺ (with densities of  ∼10¹² cm⁻³); O⁺ (∼10¹¹–10¹² cm⁻³, as n_e varies between $(1-4)\times {{10}^{12}}$ cm⁻³); and N${}_{2}^{+}$(X) and O⁻ (∼10¹¹ cm⁻³). The positive ion species are created mainly by electron-impact ionization and are destroyed mainly by diffusion and wall recombination, the FNS radiative emission playing also an important role in the creation of N${}_{2}^{+}$(X) from N${}_{2}^{+}$(B). The main creation / destruction mechanisms of the negative ion O⁻ are dissociative attachment and associative detachment, respectively.

The modeling of air micro-plasmas presented here can contribute to deepen the knowledge on these systems, particularly in what concerns the identification of the most important species controlling the plasma maintenance. The extreme small-radius low-pressure conditions adopted have allowed to study pure (rare-gas-free) $80\%{{\text{N}}_{2}}:20\%{{\text{O}}_{2}}$ plasmas, and have also put forward specific creation / destruction paths dominated by electron-impact collisions and by diffusion and wall deactivation / recombination, respectively. Accordingly, the LoKI numerical code was successfully adapted to describe the transport of charged-particles in ambipolar weakly electronegative plasmas, adopting a strategy easy to be implemented in other studies dealing with similar working conditions. The code is to be validated, by comparing simulation results with experimental diagnostics of the line-emission intensities and the absolute densities of several relevant molecular and atomic plasma species. The conclusions of this validation procedure will be presented in the second part of this work, to be published in a follow-up paper.

This work was partially supported by Portuguese FCT—Fundação para a Ciência e a Tecnologia, under Project UID/FIS/50010/2013, and by Project PLASMA-ADIST. L L Alves benefited from an FCT scholarship (SFRH/BSAB/1360/2013) to carry out this work. The authors thank G Stancu, O Leroy, K Gadonna and T Minea for providing the experimental data used as input parameters in the simulations.