Simulation of glow and arc discharges in nitrogen: effects of the cathode emission mechanisms

We constructed a fluid plasma model with the drift-diffusion approximation as employed in the Plasma Module of COMSOL Multiphysics 6.0 [32]. The EEDF is calculated using BOLSIG+ software. The electron impact cross sections are then integrated over the distribution function and incorporated in the model. Comparison with the experimental data modelled in [13, 18] was performed and the results are presented in appendix B. Although we are using the same kinetic scheme, we observe electron density that is nearly an order of magnitude higher compared to the results in [18]. When comparing to the results from [13], very good agreement is reached for the gas temperature, but deviation of the vibrational temperature is observed as a function of the current. Nevertheless, the aim of our work was different, i.e. to study the effect of the cathode emission on the positive column, and the possible differences between the glow and the arc regime, where we find no differences. We believe this claim is still valid, despite the limitations of our model to reproduce the experiments of [13, 18]. However, in view of the above discrepancy, this claim should still be validated by experiments before drawing final conclusions.

The species considered in the model are ${{\text{N}}_2}\left( {{{\text{X}}^1}{{\Sigma }}_{\text{g}}^ + } \right)$, ${{\text{N}}_2}\left( {{{\text{A}}^3}{{\Sigma }}_{\text{u}}^ + } \right)$, ${{\text{N}}_2}\left( {{{\text{a}}^{{\text{'}}1}}{{\Sigma }}_{\text{u}}^ - { }} \right)$, ${{\text{N}}_2}\left( {{{\text{B}}^3}{{{\Pi }}_{\text{g}}}} \right)$, ${{\text{N}}_2}\left( {{{\text{C}}^3}{{{\Pi }}_{\text{u}}}} \right)$, ${\text{N}}{(^4}{\text{S}})$, ${\text{N}}{(^2}{\text{D}})$, ${\text{N}}{(^2}{\text{P}})$, ${{\text{N}}^ + }$, ${\text{N}}_2^ +$, ${\text{N}}_3^ +$, ${\text{N}}_4^ +$ and the electrons. Excitation of the vibrational degree of freedom (R2, table A1) is assumed to proceed through the first eight levels, and excitation to higher levels is neglected, as in [18].

For each species, the continuity equation is solved:

$$\begin{align}\frac{{{\text{d}}{n_{\text{s}}}}}{{{\text{d}}t}} + \,\nabla \cdot {{\boldsymbol{{\Gamma }}}_{\text{s}}} + \left( {{{\boldsymbol{{u}}}_{\text{g}}} \cdot \nabla } \right){n_{\text{s}}} = {R_{\text{s}}},\end{align} \tag{ 1 }$$

where ${n_{\text{s}}}$ is the species density, ${{\boldsymbol{{\Gamma }}}_{\text{s}}}$ is the species flux, ${{\boldsymbol{{u}}}_{\text{g}}}$ is the gas velocity and ${R_{\text{s}}}$ is the net rate of change, based on the reactions detailed in appendix A. The neutral species flux is given as:

$$\begin{align}{{\boldsymbol{{\Gamma }}}_{\text{n}}} = - {D_{\text{n}}}\nabla {n_{\text{n}}},\end{align} \tag{ 2 }$$

where ${D_{\text{n}}}$ is the diffusion coefficient of the neutral species in ${{\text{N}}_2}$, given in table A6 in appendix A, and ${n_{\text{n}}}$ is the neutral number density.

For the charged species, the flux is expressed as:

$$\begin{align}{{\boldsymbol{{\Gamma }}}_{\text{s}}} = - {D_{\text{s}}}\nabla {n_{\text{s}}} + \frac{{{q_{\text{s}}}}}{{\left| {{q_{\text{s}}}} \right|}}{n_{\text{s}}}{\mu _{\text{s}}}{{E}},\end{align} \tag{ 3 }$$

where$\,{D_{\text{s}}}$, ${\mu _{\text{s}}}$, and ${q_{\text{s}}}$ are the diffusion coefficient, mobility and species' charge, respectively, and $E$ is the electric field vector. The subscript 's' indicates either electrons or ions. The electron transport properties ${D_{\text{e}}}$ and ${\mu _{\text{e}}}$ are calculated by BOLSIG+. The Einstein relation is used to calculate the ion mobility from the ion diffusivity, listed in table A6 in appendix A.

The electron energy balance equation is solved to obtain the averaged electron energy ${\bar \varepsilon _{\text{e}}}$:

$$\begin{align}\frac{{{\text{d}}\left( {{n_e}\overline {{\varepsilon _e}} } \right)}}{{{\text{d}}t}} + \nabla \cdot {{\boldsymbol{\Gamma}} _\varepsilon } + {\boldsymbol{E}} \cdot {{\boldsymbol{\Gamma}} _{\text{e}}} + \left( {{{\boldsymbol{u}}_{\text{g}}} \cdot \nabla } \right){n_{\text{e}}}\overline {{\varepsilon _e}} = {Q_{{\text{bg}}}} + {S_{{\text{en}}}},\end{align} \tag{ 4 }$$

where ${{\boldsymbol{{\Gamma }}}_{{\varepsilon }}}$ is the electron energy flux and ${S_{{\text{en}}}}$ is the average energy lost through electron collisions. An additional background power density ${Q_{{\text{bg}}}}$ is used in order to reduce gradients between the discharge column and the surrounding gas. This improves computational stability and reduces requirements on the mesh. The background power density is kept low (10⁵ W m⁻³) such that it does not significantly influence the results. The electron energy flux is given by:

$$\begin{align}{{\boldsymbol{\Gamma}} _{{\varepsilon }}} = - {D_{{\varepsilon }}}\nabla {n_{\text{e}}}\overline {{\varepsilon _{\text{e}}}} - {\mu _{{\varepsilon }}}{n_{\text{e}}}\overline {{\varepsilon _{\text{e}}}} {\boldsymbol{{E}}},\end{align} \tag{ 5 }$$

where ${D_\varepsilon }$ and ${\mu _\varepsilon }$ are the electron energy diffusion coefficient and electron energy mobility, respectively. The Poisson equation is also solved, and coupled to the above system of equations. Thus, the electric field is determined self-consistently from the space charge density:

$$\begin{align}\nabla \cdot \left( {{\epsilon _0}{\boldsymbol{{E}}}} \right) = - \nabla \cdot \left( {{\epsilon _0}\nabla V} \right) = \mathop \sum \limits_{\text{s}} {q_{\text{s}}}{n_{\text{s}}},\end{align} \tag{ 6 }$$

where ${\epsilon _0}$ is the vacuum permittivity and V is the electric potential. It is important to note that there is no special treatment of the ambipolar electric field in our model. The electric field entering equation (1) originates from the Poisson equation (6) and the externally connected circuit. The equality of the fluxes of ions and electrons is not implied, hence the losses are determined by the electric field vector, which is self-consistently computed.

The gas temperature is calculated using the energy balance equation, assuming the ion temperature is equal to the gas temperature:

$$\begin{align}&\rho {C_{\text{p}}}\frac{{\partial {T_{\text{g}}}}}{{\partial t}} + \rho {C_{\text{p}}}{u_{\text{g}}} \cdot \nabla {T_{\text{g}}} - \nabla \cdot \left( {{k_{\text{g}}}\nabla {T_{\text{g}}}} \right)\nonumber\\&\quad = {Q_{\text{g}}} + {{\text{N}}_2}\left( {\text{X}} \right)\frac{{{e_{\text{v}}} - {e_{\text{v}}}\left( {{T_{\text{g}}}} \right)}}{{{\tau _{{{VT}}}}}},\end{align} \tag{ 7 }$$

where $\rho$ is the gas density, ${C_{\text{p}}}$ is the heat capacity at constant pressure, ${k_{\text{g}}}$ is the thermal conductivity and ${Q_{\text{g}}}$ is the gas heating released from the reactions in the plasma and the Joule heating of the electrons, ${{\text{N}}_2}\left( {\text{X}} \right)\frac{{{e_{\text{v}}} - {e_{\text{v}}}\left( {{T_{\text{g}}}} \right)}}{{{\tau _{{\text{VT}}}}}}$ is the heating due to vibrational-translation (V-T) relaxation, ${e_{\text{v}}}\left( {{T_{\text{g}}}} \right)$ is the equilibrium value of the mean vibrational energy and ${\tau _{{\text{VT}}}}$ is the V-T relaxation time calculated from the expressions in [18]. Estimations of the effect of Joule heating resulting from ion currents resulted in negligible differences in the temperature in the positive column and a small effect near the cathode region. In order to improve the computational stability of the problem, the term was neglected from the calculations.

In addition to the gas temperature balance equation, the vibrational energy balance equation for ${{\text{N}}_2}\left( {\text{X}} \right)$ is also solved:

$$\begin{align}&{{\text{N}}_2}\left( {\text{X}} \right)\frac{{{\text{d}}{e_{\text{v}}}}}{{{\text{d}}t}} + \nabla \cdot \left( {{{{N}}_2}\left( {{X}} \right){D_{{{\text{N}}_2}}}\nabla {e_{{v}}}} \right) + {u_{\text{g}}} \cdot \nabla {e_{\text{v}}}\nonumber\\&\quad = {Q_{{e_{\text{v}}}}} - {{{N}}_2}\left( {{X}} \right)\frac{{{e_{{v}}} - {e_{{v}}}\left( T_{\text{g}} \right)}}{{{\tau _{{{VT}}}}}}\end{align} \tag{ 8 }$$

where ${e_{\text{v}}}$ is the mean vibrational energy, ${Q_{{e_{\text{v}}}}}$ stands for the heating and cooling rate of the vibrational degrees of freedom due to Joule heating and reactions in the plasma. The mean vibrational energy is related to the vibrational temperature with

$$\begin{align}{e_{\text{v}}}({T_{\text{v}}}) = \frac{{\hbar \omega }}{{{\text{exp}}\left( {\frac{{\hbar \omega }}{{k{T_{\text{v}}}}}} \right) - 1}}\end{align} \tag{ 9 }$$

where ${T_{\text{v}}}$ is the vibrational temperature and $\hbar \omega$ is the vibrational quanta of the nitrogen molecule. The vibrational distribution function is assumed to follow the analytical description as in [13, 18] given by:

$$\begin{align}\left. {\begin{array}{*{20}{c}} {{N_2}\left( {X,v} \right) = {N_2}\left( {X,0} \right)\exp\left( { - v\frac{{\hbar \omega }}{{k{T_v}}}} \right),0 < v \leqslant {v_1};} \\[6pt] {{N_2}\left( {X,v} \right) = {N_2}\left( {X,v = {v_1}} \right)\frac{{{v_1} + 1}}{{v + 1}},{v_1} < v \leqslant {v_2}\,} \\[6pt] {{N_2}\left( {X,v} \right) = {N_2}\left( {X,v = {v_2}} \right)\exp\left( { - \left( {v - {v_2}} \right)\frac{{\hbar \omega }}{{k{T_{\text{g}}}}}} \right),\,v > {v_2}} \\[6pt] {{v_1} = 9,\,{v_2} = {v_1} + \left( {35 - {v_1}} \right)\exp\left( { - \frac{{{T_{\text{g}}} - 300}}{{3000}}} \right)} \end{array}} \right\}\end{align} \tag{ 10 }$$

The solution for the gas flow velocity is obtained from the Navier–Stokes equations:

$$\begin{align}\rho \frac{{\partial {{\boldsymbol{{u}}}_{\text{g}}}}}{{\partial t}} + \rho \left( {{{\boldsymbol{{u}}}_{\text{g}}}.\nabla } \right){{\boldsymbol{{u}}}_{\text{g}}} &= \nabla .\left[ - {\text{p}}{\boldsymbol{{I}}} + {\mu _{\text{g}}}\left( {\nabla .{{\boldsymbol{{u}}}_{\text{g}}} + {{\left( {\nabla .{{\boldsymbol{{u}}}_{\text{g}}}} \right)}^T}} \right)\right.\nonumber\\&\qquad\quad\left.\! - \frac{2}{3}{\mu _{\text{g}}}\left( {\nabla \cdot {{\boldsymbol{{u}}}_{\text{g}}}} \right){\boldsymbol{{I}}} \right]\end{align} \tag{ 11 }$$

$$\begin{align}\frac{{\partial \rho }}{{\partial t}} + \nabla \cdot \left( {\rho {{\boldsymbol{{u}}}_{\text{g}}}} \right) = 0\end{align} \tag{ 12 }$$

where ${\boldsymbol{{I}}}$ is the identity matrix, p is the gas pressure, ${\mu _{\text{g}}}$ is the dynamic viscosity (which is a function of temperature) and the superscript T identifies a tensor operation. The pressure is assumed to be close to constant at 1 atm and thus, the gas density $\rho$ is assumed to be a function of gas temperature only. The transport properties ${\mu _{\text{g}}}$, ${C_{\text{p}}}$ and ${k_{\text{g}}}$ are taken as tabulated data from [33].

The model is solved for an axisymmetric geometry comprising two hemispherical electrodes (r_electrode = 0.2 cm) separated by a discharge gap of 1.6 cm placed in a cylindrical tube (r_tube = 2 cm), presented in figure 1. The bottom electrode (AB) is chosen to be the cathode, where a negative voltage is supplied through a ballast resistor connected to a voltage source. The ballast resistor was chosen as 1 MΩ and the applied voltage was varied to control the current.

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The BC on the cathode for the electron density balance equation (1) is given as:

$$\begin{align}{\mathbf{n}} \cdot {{\boldsymbol{{\Gamma }}}_{\text{e}}} = \frac{1}{2}{v_{{\text{e}},{\text{th}}}}{n_{\text{e}}} - \left( {\mathop \sum \nolimits {\gamma _{\text{i}}}({{\boldsymbol{{\Gamma }}}_{\text{i}}} \cdot {\mathbf{n}}) + {{\boldsymbol{{\Gamma }}}_{\text{t}}} \cdot {\mathbf{n}}} \right),\end{align} \tag{ 13 }$$

where ${v_{{\text{e}},{\text{th}}}} = \,\sqrt {\frac{{8{k_{\text{B}}}{T_{\text{e}}}}}{{\pi {m_{\text{e}}}}}}$ is the electron thermal velocity, ${{\boldsymbol{{\Gamma }}}_{\text{i}}} \cdot {\mathbf{n}}$ is the ion flux normal to the cathode and ${\gamma _{\text{i}}}$ is the secondary electron emission coefficient for each ion. It is important to note that the secondary electron emission coefficients are not well known for these systems and are often calculated as fitting parameters of a specific experiment. The secondary electron emission coefficients in this work were taken from [34] as ${\gamma _{\text{i}}} = 0.032$ and assumed to be the same for the four different ions. ${{\boldsymbol{{\Gamma }}}_{\text{t}}}$ is the electron flux due to field-enhanced thermionic emission, given as [24]. To distinguish between the arc and glow case, the thermionic emission flux (${{\boldsymbol{{\Gamma }}}_{\text{t}}}$ in equation (13)) is only included when considering the arc case, whereas the secondary electron emission flux (${{\boldsymbol{{\Gamma }}}_{\text{i}}}$ in equation (13)) is only calculated in the glow case:

$$\begin{align}{{\boldsymbol{{\Gamma }}}_{\text{t}}} \approx \left( {{{\text{A}}_{\text{T}}}{T_{\text{C}}}^2 + {{\text{A}}_{\text{F}}}{{\boldsymbol{{E}}}_{\text{C}}}^x} \right){\text{exp}}\left[ { - {{\left( {\frac{{{T_{\text{C}}}^2}}{{{{\text{B}}_{\text{T}}}}} + \frac{{{{\boldsymbol{{E}}}_{\text{C}}}^2}}{{{{\text{B}}_{\text{F}}}}}} \right)}^{ - 1/2}}} \right].\end{align} \tag{ 14 }$$

Equation (14) accounts for the combined effect of the thermionic and field emission on the electron emission. Although our model assumes a cold cathode (${T_{\text{C}}} = 300\,K$), and thus thermionic emission is negligible, we keep the more general expression (14) to allow the future use of our model in more diverse cases. Equation (14) has a rather complicated nature and a detailed discussion regarding the exact derivation of the constants is outside the scope of this work. The coefficients ${{\text{A}}_{\text{T}}}$, ${{\text{A}}_{\text{F}}}$, ${{\text{B}}_{\text{T}}}$, ${{\text{B}}_{\text{F}}}$ and x are separate functions of the cathode material work function ${W_{\text{f}}}$, with x being a factor close to unity. ${{\text{A}}_{\text{T}}}$ (A m⁻² K⁻²) and ${{\text{A}}_{\text{F}}}$ (AV^−x m^{x −2}) are the pre-exponential factors in the Richardson–Dushman and Fowler–Nordheim formula, respectively. ${{\text{B}}_{\text{T}}}$ (K⁻²) and ${{\text{B}}_{\text{F}}}$ (V⁻² m⁻²) can be approximated as exponential factors in both formulae. To account for the surface roughness (which can amplify the electric field through sharp edges in microprotrusions) an effective electric field is defined as ${{\boldsymbol{{E}}}_{\text{C}}} = {\text{FEF}} \times \left( {{\boldsymbol{{n}}} \cdot {\boldsymbol{{E}}}} \right)$, where FEF is the field enhancement factor accounting for amplification of the electric field. This parameter is a similar fitting parameter to ${\gamma _{\text{i}}}$. In [35], a FEF value of 55 is given for a smooth surface. As the purpose of our model is to compare the effects of the cathode phenomena for conditions that will create an arc, we have chosen a FEF factor of 200, which relates to a relatively rough surface without special conditioning. The work function used in our work is chosen to be ${W_{\text{f}}} = 4.36$ eV, which corresponds to Mo [36].

The BC for the electron energy balance equation (4) on the cathode is given as:

$$\begin{align}{\boldsymbol{{n}}} \cdot {{\boldsymbol{{\Gamma }}}_{{\varepsilon }}} = \frac{5}{6}{v_{{\text{e}},{\text{th}}}}{n_{{\varepsilon }}} - \left( {\mathop \sum \nolimits {\gamma _{\text{i}}}{\epsilon _{\text{i}}}({{\boldsymbol{{\Gamma }}}_{\text{i}}} \cdot {\boldsymbol{{n}}}) + {\epsilon _{\text{t}}}\left( {{{\boldsymbol{{\Gamma }}}_{\text{t}}} \cdot {\boldsymbol{{n}}}} \right)} \right),\end{align} \tag{ 15 }$$

where ${\epsilon _{\text{i}}}$ is the average energy of secondary electron emission, given as ${\epsilon _{\text{i}}} = \,{\varepsilon _{\text{i}}} - 2{W_{\text{f}}}\,$, in which $\,{\varepsilon _{\text{i}}}$ is the ionization energy of the bombarding species and ${W_{\text{f}}}$ is the work function of the material. Furthermore, ${\epsilon _{\text{t}}}$ is the average energy of the electrons emitted by field-enhanced thermionic emission. Although the average energy of electron emission has to be a distribution function, for the sake of numerical stability, we have chosen a low constant value of 0.02 eV for ${\epsilon _{\text{t}}}$.

The BCs for the ions are the same for the cathode and the anode, given as:

$$\begin{align}{\boldsymbol{{n}}} \cdot {{\boldsymbol{{\Gamma }}}_{\text{i}}} = {n_{\text{i}}}{K_{{\text{i}},{\text{surf}}}} + {\text{max}}\left( {\frac{{{q_{\text{s}}}}}{{\left| {{q_{\text{s}}}} \right|}}\frac{{{D_{\text{i}}}e}}{{{k_{\text{B}}}T}}{n_{\text{i}}}{\boldsymbol{{E}}} \cdot {\boldsymbol{{n}}},0} \right).\end{align} \tag{ 16 }$$

The expression ${K_{{\text{i}},{\text{surf}}}} = \left( {\frac{{{\gamma _{\text{s}}}}}{{1 - {\gamma _{\text{s}}}/2}}} \right)\frac{1}{4}\sqrt {\frac{{8R{T_{\text{g}}}}}{{\pi {M_{\text{s}}}}}}$ where ${\gamma _{\text{s}}}$ is the sticking coefficient and ${M_{\text{s}}}$ is the species mass, taken from [37], with all surface reactions employing a sticking coefficient of 1. In order to get an adequate comparison between the glow and the arc regime, the temperature of the cathode and the anode was kept constant at 300 K. The boundary (BC) (see figure 1) was chosen as an inlet BC with a given parabolic gas velocity profile, which has a maximum in the middle of the boundary, giving a fully developed flow profile. The specified velocity ${u_{{{\text{g}}_0}}}$ is then the average velocity on the boundary. The outlet BC for boundary (DE) is given as:

$$\begin{align}\left[ { - {\text{p}}{\boldsymbol{{I}}} + {\mu _{\text{g}}}\left( {\nabla \cdot {{\boldsymbol{{u}}}_{\text{g}}} + {{\left( {\nabla \cdot {{\boldsymbol{{u}}}_{\text{g}}}} \right)}^T}} \right) - \frac{2}{3}{\mu _{\text{g}}}\left( {\nabla \cdot {{\boldsymbol{{u}}}_{\text{g}}}} \right){\boldsymbol{{I}}}} \right]{\boldsymbol{{n}}} = {{\text{p}}_0}{\boldsymbol{{n}}}.\end{align} \tag{ 17 }$$

The rest of the BCs are listed in table 1, with indication to figure 1.

Table 1. Boundary conditions.

	${n_{\text{e}}}$	${n_{\text{i}}}$	${n_{\text{n}}}$	V	${\overline \varepsilon _{_{\text{e}}}}$	Tg	${u_{\text{g}}}$	Tv
(AB)	(13)	(16)	${\boldsymbol{{n}}} \cdot {{\boldsymbol{{\Gamma }}}_{\text{n}}} = {n_{\text{i}}}{K_{{\text{n}},{\text{surf}}}}$	$V = {V_{\text{c}}}$	(15)	${T_{\text{g}}} = 300$K	${u_{\text{g}}} = 0$	Tv = 300 K
(BC)	${\boldsymbol{{n}}} \cdot \alpha = 0,\,\left( {\alpha = {{\boldsymbol{{\Gamma }}}_{\text{e}}},{\boldsymbol{{\,}}}{{\boldsymbol{{\Gamma }}}_{\text{i}}},{\boldsymbol{{\,}}}{{\boldsymbol{{\Gamma }}}_{\text{n}}},{\boldsymbol{{\,}}} - \nabla V,\,{{\boldsymbol{{\Gamma }}}_{{\varepsilon }}}} \right)$					${T_{\text{g}}} = 300$K	${u_{\text{g}}} = {u_{{{\text{g}}_0}}}$	Tv = 300 K
(CD)	${\boldsymbol{{n}}} \cdot \alpha = 0,\,\left( {\alpha = {{\boldsymbol{{\Gamma }}}_{\text{e}}},{\boldsymbol{{\,}}}{{\boldsymbol{{\Gamma }}}_{\text{i}}},{\boldsymbol{{\,}}}{{\boldsymbol{{\Gamma }}}_{\text{n}}},{\boldsymbol{{\,}}} - \nabla V,\,{{\boldsymbol{{\Gamma }}}_{{\varepsilon }}}} \right)$					${T_{\text{g}}} = 300$K	${u_{\text{g}}} = 0$	Tv = 300 K
(DE)	${\boldsymbol{{n}}} \cdot \alpha = 0,\,\left( {\alpha = {{\boldsymbol{{\Gamma }}}_{\text{e}}},{\boldsymbol{{\,}}}{{\boldsymbol{{\Gamma }}}_{\text{i}}},{\boldsymbol{{\,}}}{{\boldsymbol{{\Gamma }}}_{\text{n}}}, - \nabla {\boldsymbol{{\,}}}V,\,{{\boldsymbol{{\Gamma }}}_{{\varepsilon }}}} \right)$					${\boldsymbol{{n}}} \cdot {\text{k}}\nabla {T_{\text{g}}} = 0$	(17)	${\boldsymbol{{n}}} \cdot {\text{k}}\nabla {T_{\text{v}}} = 0$
(EF)	${\boldsymbol{{n}}} \cdot {{\boldsymbol{{\Gamma }}}_{\text{e}}} = \frac{1}{2}{v_{{\text{e}},{\text{th}}}}{n_{\text{e}}}$	(16)	${\boldsymbol{{n}}} \cdot {{\boldsymbol{{\Gamma }}}_{\text{n}}} = {n_{\text{i}}}{K_{{\text{n}},{\text{surf}}}}$	$V = 0$	${\boldsymbol{{n}}} \cdot {{\boldsymbol{{\Gamma }}}_{{\varepsilon }}} = \frac{5}{6}{v_{{\text{e}},{\text{th}}}}{n_{\text{e}}}$	${T_{\text{g}}} = 300$K	${u_{\text{g}}} = 0$	Tv = 300 K

The choice of the BCs on (CD) was evaluated to ensure the wall is not influencing the plasma. To test this condition, the position of the wall was varied at r = 2, 4 and 6 cm. The results showed no influence on the gas temperature and electron density, showing that the gas flow is strong enough to insulate the plasma from the wall.

In figure 2 the axial variation (r = 0) of several quantities in the cathode region for the arc and the glow case are presented. The results are plotted as a logarithmic function of $z$ in order to highlight the deviations of the plasma parameters in the cathode layer. A nonquasineutral layer is observed for both the glow and arc cases. For the case of electron supply by field emission and thus the arc discharge (figure 2(a)), the size of the nonquasineutral cathode layer is in the order of 10⁻⁴ cm. The ion and electron density are within the same order of magnitude within this layer, owing to the high efficiency of electron emission.

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The high value of the electron temperature observed in both cases (arc and glow) is attributed to the acceleration of electrons in the high space charge field created by the difference between the electron and ion densities. For the arc regime (figure 2(a)) ${T_{\text{e}}} \cong 3.8{\text{ eV}}$ in the cathode region, while for the glow case (figure 2(b)) ${T_{\text{e}}} \approx 9 - 17{\text{ eV}}$. In the latter case, when the cathode emission mechanism is changed to secondary electron emission, the efficiency of electron emission from the surface is greatly reduced. Therefore, the electron density in the vicinity of the cathode surface becomes significantly lower than the ion density, and the resulting difference in the opposing charges leads to a substantially stronger electric field and thus a much higher ${T_{\text{e}}}$ compared to the arc regime. The size of the space charge imbalance layer is also considerably larger than in the arc regime: above 10⁻³ cm. Beyond the space charge imbalance layer, we can define the negative glow in the glow regime, and a transitional layer in the arc regime (as indicated in figure 2).

Figure 3 represents the axial density profiles of the various positive ions, as well as the gas temperature, for both emission mechanisms, i.e. the arc (a) and glow (b) regime. In the arc case, the dominant ion in the cathode layer is N₂ ⁺ (figure 3(a)), whereas in the glow case (figure 3(b)) the dominant ion changes to N₄ ⁺. The change in the dominant ion is due to the larger increase in gas temperature close to the wall in the arc compared to the glow regime. Indeed, the maximum gas temperature and electron density in the arc regime is reached much closer to the wall (within the transitional layer), while in the glow regime, no maximum gas temperature or electron density is observed in the negative glow region, and the gas temperature rises steadily as a function of axial position towards the positive column.

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As shown in figure 2, the electron density in the vicinity of the cathode is significantly larger in the arc than in the glow regime. There are enough electrons to create a conductive connection between the positive column and the electrodes, and hence the electric potential drop across the non-quasineutral layer in the arc regime is rather small (ca. 15 V, see figure 4; inset). The electron density in the glow regime is significantly lower, and thus, the electric field has to be much higher, for the plasma to sustain the current in this area, i.e. to accelerate the ions, which provide the same total current. This results in a very steep and very large voltage drop across the non-quasineutral layer, of 300 V (see figure 4). However, even though the mechanism of electron emission differs for the arc and glow regimes, we can see in figure 4 that the electric potential across the positive column sustains a linear dependence beyond the near-cathode region. In addition to this, the differences in the electric potential (and thus in electric field) in the positive column are small.

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Here we should stress that the additional voltage at the cathode layer in the glow discharge means additional deposited power, which in this case is significant and in the same order as the rest of the discharge, i.e. ca. 24 W for 80 mA current, while the total power is 56 W. Due to the significant ion current near the cathode in the glow regime, the gas is additionally heated by the ions. Despite that, our calculations suggest that the gas temperature in the cathode fall of the glow discharge is only slightly elevated (see figure 3(b)), which is due to the intense cooling from the cathode, assumed to be at 300 K in our configuration. The gas cooling is effective due to the large surface area of the cathode region in the glow discharge and the small distance to the cathode surface. If the cathode is not cooled, the power deposited there could be sufficient for considerable gas and cathode heating and transition to an arc.

Figure 5 illustrates the 2D electron density profiles in the arc and glow regime for different currents. The cathode region looks clearly different in both regimes, for all currents investigated, i.e. it is characterized by a very high electron density, but confined in a very small region in the arc regime, while the electron density profile is quite wide with much lower values in the glow regime. Moreover, increasing the discharge current leads to qualitatively different behavior in the glow and the arc in the near-cathode region. In the glow regime, the cathode region is sustained by secondary electron emission initiated by the ion flux normal to the cathode surface. Because of this, increasing the current does not lead to more electron emission, but the current density is preserved and the cathode region enlarges radially. This effect is clearly observed in figure 5. The area of the cathode region determined from the current density increases from 0.55 × 10⁻² cm² to 1.2 × 10⁻² cm² and is directly proportional to the current, following the well-known fact that the current density at the cathode surface in normal glow discharges remains constant. In the arc regime, increasing the current results in even stronger field-enhanced thermionic emission. In contrast to the glow regime, the current density in the arc cathode region increases upon increasing current. The area of the arc cathode region determined from the current density remains approximately constant as a function of the current, at a value as small as 7.07 × 10⁻⁶ cm², which corresponds to a radius of 1.5 ×10⁻³ cm. As mentioned above, this results in electron densities that are almost two orders of magnitude higher (see figure 5).

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The significantly higher electron densities in the arc compared to the glow give rise to an increase in the vibrational temperature near the electrode. The axial profile of the vibrational temperature for 80 mA and ${u_{{{\text{g}}_0}}}\; = \;1\;{\text{m}}{{\text{s}}^{ - 1}}$ in the near cathode region is presented in figure 6. Within the glow, the gas temperature and vibrational temperature are nearly in equilibrium, because the electron density in the negative glow is nearly the same as in the positive column. This results in a smooth transition between the near cathode region and the positive column, leading to V-T equilibrium in the whole region. In contrast, in the arc spot, the electron density experiences a maximum (see figure 2(a)) and a sharp decline towards the positive column. Within the high electron density region, electron impact vibrational excitation (R2) deposits a significant amount of energy into the vibrational degrees of freedom, which do not have enough time to relax to the translational degrees of freedom. As a result, a maximum of the vibrational temperature is observed in the high electron density region.

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The key difference between the glow and the arc cathode region identified in the previous subsection is the significantly higher electron density in the arc regime at the discharge axis. This effect enhances the vibrational heating, leading to vibrational temperature much higher in the arc spot than in the glow, as shown in figure 6 above. In figure 7, the axial and radial dependence of the gas temperature, at the three different currents for both the glow and arc regime are presented. Within the positive column, the vibrational and gas temperature are in near equilibrium with each other; thus, the vibrational temperature is not shown in figure 7. The contracted nature of the plasma within the investigated current range results in high gas temperatures ($\approx 4700\,{\text{K}}$) and negligible differences between the arc and glow discharge. In the axial direction, there is no strong gradient of gas temperature, resulting in a nearly constant axial distribution. The only differences observed between the glow and the arc are in the near-cathode region, where the arc exhibits an increase in gas temperature with rising vibrational temperature, as shown in figure 7(a). This leads to strong V-T nonequilibrium in the arc spot.

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The radial gas temperature distribution at the three different currents (at z = 1 cm) is presented in figure 7(b). Upon increasing the current from 80 mA to 160 mA, the radius of the region with elevated gas temperature increases by 12.5% (from 0.24 cm to 0.27 cm), while the cross section increases by 27% (from 0.18 cm² to 0.23 cm²).

The heating and cooling rates in the positive column for 80 mA and z = 1 cm are presented for the arc regime in figure 8. Due to similarities between the glow and arc column, only the result for the arc is shown. The heat conductivity (black curve) transports the energy radially from the middle of the column towards the side, until the convective heat transfer (red curve) stops the expansion. As a function of the current there is an increase in the gas temperature which can be seen in figure 7. The maximum gas temperature increases by 433 K, from 4483 K to 4916 K. The radial distribution of the temperature is then determined by the balance between the heat conductivity and the convective heat losses. Our analysis shows that the radial conductive component has the largest contribution to the loss mechanisms in the plasma column and the dominant heating process is V-T relaxation.

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As indicated in [13], this effect insulates the plasma from the wall, acting as a virtual cold wall. It is then logical that at a constant gas flow velocity, the positive column expands upon increasing current. The expansion of the positive column can also be seen in the radial electron density profile. The radial dependence of the electron density at three different currents, for the glow and the arc regime, at z = 1 cm and ${u_{{{\text{g}}_0}}} = 1\,{\text{m }}{{\text{s}}^{ - 1}}$, is presented in figure 9. Similar to figure 7, because of the contracted nature of the plasma, no differences are observed in the electron density between the glow and arc for all currents. Upon rising current from 80 to 160 mA, the electron density in the positive column experiences a small increase, from $5.8\; \times \;{10^{19}}\;{{\text{m}}^{ - 3}}$ to $6.5\; \times \;{10^{19}}\;{{\text{m}}^{ - 3}}$.

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Generally speaking, our results show that in the positive column and the anode region, the differences in all quantities between the arc and the glow regime amount to negligible values. This effect is highlighted in figure 10, where 2D distributions of the electron density of the arc and glow case are compared to each other. Although the glow cathode region covers a significantly larger area than the arc cathode region, this does not translate to a larger area of the positive column. It is important to note that these results are not universal: they depend strongly on the chemistry and probably on the BCs and the cathode properties.

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Upon increasing the current from 80 to 160 mA, the radius of the current density profile increases by 64% (from 3.8 × 10⁻² cm to 6.3 × 10⁻² cm), while the cross section rises by 167% (from 4.5 × 10⁻³ cm² to 1.2 × 10⁻² cm²). The electron density balance was determined by analyzing equation (1). In figure 11, the total rates of production and destruction, as well as the rate of flux losses and the rate of gas convection losses, determining the electron density balance, are presented for the arc regime at 80 mA and z = 1 cm. As previously stated, due to the negligible differences in the glow and arc column, only the result for the arc is shown. The electron density balance is determined to be local and the diffusive, migrative and gas convective fluxes have a negligible contribution (see figure 11). The absolute value of the loss terms was taken in order to make the comparison between the dominant terms.

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Our reaction analysis revealed that the dominant mechanism of electron production, for both arc and glow regime, is Penning and associative ionization (R28, R29, R30—table A3)

$$\begin{align}{{\text{N}}_2}{ }\left( {{{\text{a}}^{{\text{&#x0027;}}1}}{{\Sigma }}_{\text{u}}^ - { }} \right) + { }{{\text{N}}_2}{ }\left( {{{\text{a}}^{{\text{&#x0027;}}1}}{{\Sigma }}_{\text{u}}^ - { }} \right) \to {\text{N}}_4^ + + {\text{e}},\end{align} \tag{ R28 }$$

$$\begin{align}{{\text{N}}_2}\left( {{{\text{A}}^3}{{\Sigma }}_{\text{u}}^ + } \right) + {{\text{N}}_2}{ }\left( {{{\text{a}}^{{\text{&#x0027;}}1}}{{\Sigma }}_{\text{u}}^ - { }} \right) \to {\text{ N}}_2^ + + { }{{\text{N}}_2} + {\text{e}},\end{align} \tag{ R29 }$$

$$\begin{align}{{\text{N}}_2}{ }\left( {{{\text{a}}^{{\text{&#x0027;}}1}}{{\Sigma }}_{\text{u}}^ - { }} \right) + { }{{\text{N}}_2}{ }\left( {{{\text{a}}^{{\text{&#x0027;}}1}}{{\Sigma }}_{\text{u}}^ - { }} \right) \to {\text{N}}_2^ + + { }{{\text{N}}_2} + {\text{e}},\end{align} \tag{ R30 }$$

which is balanced by dissociative recombination (R32—table A4). The electronically excited ${{\text{N}}_2}\left( {{{\text{A}}^3}{{\Sigma }}_{\text{u}}^ + } \right)$ and ${{\text{N}}_2}{ }\left( {{{\text{a}}^{{\text{&#x0027;}}1}}{{\Sigma }}_{\text{u}}^ - { }} \right)$ are produced primarily from quenching of the electronically excited ${{\text{N}}_2}\left( {{{\text{B}}^3}{{{\Pi }}_{\text{g}}}} \right)\,$and ${{\text{N}}_2}\left( {{{\text{C}}^3}{{{\Pi }}_{\text{u}}}} \right)$ states. Further analysis of the production and loss processes for the positive ions and excited states revealed that they are also locally balanced. The radial distribution of the various ion densities for the same conditions is presented in figure 12. It is clear that the dominant positive ion is N₂ ⁺ throughout most of the positive column. Near the edges of the current conductive region, where the gas temperature is lower, N₃ ⁺ becomes the dominant ion. This is attributed to the charge exchange reaction (R19—table A2). The density profile of the atomic ions N⁺ closely follows that of N₂ ⁺. Upon increasing current, the density of N⁺ approaches the density of N₂ ⁺, as also found in [18].

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In section 3.2 we demonstrated that the cathode region, and hence the electron emission mechanism, has no significant effect on the positive column. To investigate the effect of the gas flow rate on the discharge, we chose a single current condition (80 mA) and a single electron emission mechanism, i.e. thermionic field emission, hence reflecting the arc regime. We varied the inlet flow velocities as 1, 2 and 4 m s⁻¹. The 2D velocity distributions are shown in figure 13.

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We can see that the cathode is effectively shielded by the high velocity region of the flow. A velocity gradient is present in both the axial and radial direction. As the flow velocity rises to 4 m s⁻¹, a small recirculation zone appears above the cathode surface. The velocity in the recirculation zone is rather low and has no significant effect on the arc properties near the cathode.

In figure 14 the electron density and gas temperature profiles at the three different gas flow velocities are presented. In section 3.2 we already demonstrated, in agreement with [13], that the gas flow effectively insulates the arc from the walls through convective heat transfer in the axial direction. The increased axial cooling upon higher gas flow rate leads to further contraction of the area of the elevated gas temperature. The radius reduces by 37.5% (from 0.22 to 0.16 cm) between 1 and 4 m s⁻¹ (see figure 14(b)), and the cross section area reduces by 87.5% (from 0.15 to 0.08 cm²). The arc radius defined from the current density profile changes from 3.8 × 10⁻² cm to 3.3 × 10⁻² cm, corresponding to a 15% decrease, and the cross section area drops by 32% (from 4.5 × 10⁻³ to 3.4 × 10⁻³ cm²). The radial temperature gradient also becomes significantly steeper upon higher gas flow velocity, which further enhances the cooling of the arc due to heat conduction and results in slightly lower temperatures in the center of the positive column. The radial profiles of gas temperature and electron density at three different gas flow velocities, for z = 1 cm, are shown in figure 15. It is interesting to see that the increase in the gas flow velocity does not influence the core of the positive column. The effects of the gas flow are primarily on the periphery of the plasma indicating the electrodes are shielding the column from the gas flow. This leads to a reduction in the radial size of the positive column without significantly affecting the core. A constant V-T nonequilibrium of ∼600 K is observed in the middle of the positive column where the electron density has it highest values (figure 15(a)). The electron density also does not experience a significant change in its maximum value as function of the gas flow (figure 15(b)).

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The dominant ion remains N₂ ⁺ with an increasing population of N⁺ upon higher flow velocities. Naturally, as the plasma is cooled down, a higher voltage is required to sustain the discharge because of the lower gas temperature of the positive column. Because the electrodes are protecting the plasma main conductive channel increasing the velocity does not lead to significant increase in the burning voltage 405 V for 1 m s⁻¹–412 V for 4 m s⁻¹.

Figure 16 shows the axial profile at r = 0 for the radius of the arc defined from the current density j and the gas temperature T_g. The strong coupling between the gas temperature and the electron density is observed as a function of the axial position. Following the discussion in the previous section, the radius of the region with elevated temperature is defined by the balance between the convective and conductive components, meaning that there is a strong connection between the gas velocity and the gas temperature. The electron density balance was determined to be local, indicating that the radius of the current density profile depends strongly on the gas temperature profile and indirectly on the gas velocity distribution. The gas entering the domain from the cathode towards the anode is at 300 K and gradually heats up along the axial direction. The axial transport of heated gas leads to a less steep temperature gradient in the radial direction as function of the axial position, which in turn results in expansion of the arc. This effect is highlighted in figure 16 (b) where the gas temperature radius increases as a function of the axial position for any flow rate together with the current density profile (figure 16(a)).

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