The interactions of the plasma with the channel walls can have two main effects on the axial distribution of the plasma properties: on the one hand, the electrons reaching the wall or getting reflected by the sheath lose part of their kinetic energy. This limits the rate of ionization that is directly proportional to the electrons’ energy. Furthermore, secondary emitted electrons from the walls lower the overall temperature of the electrons’ population and, thus, play an additional role in regulating the ionization. Changes in the electrons’ energy and, consequently, the ionization rate affects the plasma density profile and the extent of the ionization zone.
On the other hand, electron-wall collisions and the Near-Wall Conductivity phenomenon can play a non-negligible role in the electrons’ axial mobility [2]. The higher the wall-induced electron transport, the lower would be the self-consistent electric field, which affects both the ionization and the acceleration processes.
As a result, it is important to have a reliable approach to resolve the effects of the wall interactions as well in a predictive kinetic simulation. Accordingly, in this section, we present the results of our pseudo-2D axial-radial simulations and compare the results against 1D axial PIC simulations with and without a model to account for the plasma-wall interactions. To have a point of reference, the setup of all simulations is similar to a recent 2D hybrid fluid/PIC simulation from the literature [32] and the results are compared against those from the hybrid code. In this regard, it is important to highlight that an axial or axial-radial PIC simulation is not self-consistent with respect to electron mobility and, as a result, the same model as that in Ref. [32] to account for the enhanced electrons’ cross-field transport is included in our 1D and pseudo-2D simulations. As such, since the choice of electrons’ mobility model can have a dominant influence on the plasma dynamics, it was expected that the results would be mostly comparable, which will be seen to has been the case in Section 5.2.
It is also important to recall that, as mentioned in Section 1, by performing the single-region pseudo-2D axial-radial simulations, we aim to demonstrate preliminarily that this approach can provide a self-consistent method to capture an average effect of the plasma-wall collisions and the associated particles’ momentum and energy loss in lieu of the ad-hoc models that have been proposed in the literature for this purpose [3][32].
5.1. Simulation setup
The domain of the simulations is a rectangular Cartesian plane, resembling the channel and the immediate near-plume zone downstream the exit plane of the SPT-100 Hall thruster. The x-axis is along the channel axis and the y-axis is along the radial direction. The axial extent of the domain was chosen to be 3 cm compared to 6 cm in Ref. [32] in order to speed up the simulation by reducing the number of macroparticles. In the y-direction, however, the same extent of 1.5 cm as in Ref. [32] was used. The simulated operating condition is the same as that adopted in Ref. [32], with the discharge voltage being 300 V and the anode mass flow rate being 5 mg/s. The axial profile of the radial magnetic field is the one shown in Fig. 9 of Ref. [32] with the peak intensity of 16 mT.
At the beginning of the simulation, the electrons and ions are sampled from a Maxwellian distribution at 10 eV for the electrons and 0.5 eV for the ions. These particles are then injected uniformly throughout the domain at exactly same locations. In order to maintain the discharge, electrons are sampled from a half-Maxwellian at 10 eV and are injected into the domain from the cathode boundary at each timestep. The number of electrons to inject is determined based on the quasineutrality approach [24].
An anode temperature of 750 K and a wall temperature of 850 K were assumed, consistent with the corresponding values in Ref. [32]. The neutral particles are sampled from a Maxwellian at the anode temperature and are loaded into the domain according to an initial profile given in Ref. [8]. The neutral density at the anode at the beginning of the simulation is \(3.22\times {10}^{19} {\text{m}}^{-3}\). The evolution of the neutrals’ population is traced kinetically.
In all simulations, the neutrals created due to ion recombination on the anode are sampled from a half-Maxwellian at the anode temperature. Moreover, for the pseudo-2D simulation, the ion recombination on the channel walls is also considered and the corresponding neutrals are sampled from a half-Maxwellian at the wall temperature.
Concerning the plasma-wall interactions, the SEE phenomenon is resolved self-consistently in the pseudo-2D simulation using the Monte-Carlo-based Vaughan model [20]. The secondary electrons are sampled from a half-Maxwellian at the assumed temperature of 2 eV and are diffusely injected into the domain. For the 1D axial simulation with an ad-hoc wall-collision model, the following approach, originally introduced in Ref. [33], was used: a collision frequency (\({\nu }_{wall}\)) was included in an empirical manner as
\({\nu }_{wall}={\alpha }_{w} {\nu }_{w,exp}\)
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(Eq. 14)
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in which, \({\alpha }_{w}\) is a free parameter (here assumed to be 0.1), and \({\nu }_{w,exp}\) is an experimentally measured electron-wall collision frequency (typically \({10}^{7} {s}^{-1}\)). This determines the frequency with which the electrons lose momentum in colliding with the wall. Another relation is invoked to model the energy loss corresponding to either a wall collision or other “anomalous” mechanisms, expressed as [33]
\(\mathcal{W}=\mathcal{ }ϵ\text{exp}\left(-\frac{\mathcal{U}}{ϵ}\right),\)
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(Eq. 15)
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where \(ϵ\) is the electron kinetic energy in eV, and \(\mathcal{U}\) is a threshold electron energy (empirically set at 20 eV). In the PIC simulation, the above electron-wall collision model was included as a fictitious collision event within the MCC module with the frequency and the energy loss given by Eqs. 14 and 15.
The electrons were also assumed to collide with neutrals and the following collision events were resolved through the MCC scheme: the single ionization, four excitations and the elastic momentum-transfer collision.
Concerning the enhanced electrons’ axial mobility, noting that the simulations reported here do not resolve the azimuthal coordinate, an ad-hoc electron mobility model based on the Bohm collision frequency, similar to that described in Section 3.1, was implemented in the simulations. In this regard, to be consistent with the reference 2D hybrid simulation [32], we used a “two-zone” model of the general form \({\nu }_{Bohm}=\left(\frac{\alpha }{16}\right){\omega }_{c}\), with the tuning coefficient \(\alpha\) being 0.035 inside the channel and 5 in the near-plume.
Table 3 presents a summary of the numerical parameters and plasma conditions used for the pseudo-2D axial-radial simulations. The same parameters and conditions, where applicable, are also used for the 1D simulations.
Table 3
Summary of the numerical and physical parameters used for the pseudo-2D axial-radial simulations
Parameter
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Value [unit]
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Computational Parameters
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Time step (\({\Delta }t\))
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\(2\times {10}^{-12}\) [s]
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Total simulation time (\({t}_{final}\))
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\(30 \times {10}^{-6}\) [s]
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Axial domain length (\({L}_{x}\))
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3.0 [cm]
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Radial domain length (\({L}_{y}\))
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1.5 [cm]
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Cell size (\(\varDelta x={\Delta }y)\)
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\(2\times {10}^{-3}\) [cm]
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Initial number of macroparticles per cell for axial grid (\({N}_{pp{c}_{x}}\))
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35
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Initial number of macroparticles per cell for radial grid (\({N}_{pp{c}_{y}}\))
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65
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Physical Parameters
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Initial plasma density (\({n}_{p,init}\))
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\(1\times {10}^{17}\) [m− 3]
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Initial neutral gas density at the anode (\({n}_{n,init}\))
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\(3.22\times {10}^{19}\) [m− 3]
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Electron injection temperature (\({T}_{e}\))
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10.0 [eV]
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Ion injection temperature (\({T}_{i}\))
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0.5 [eV]
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Anode temperature (\({T}_{anode}\))
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750 [K]
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Wall temperature (\({T}_{wall}\)) [for pseudo-2D simulation]
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850 [K]
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Ad-hoc Bohm mobility coefficient inside the channel (\({\alpha }_{ch}\))
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0.035
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Ad-hoc Bohm mobility coefficient in near-plume (\({\alpha }_{pl}\))
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5.0
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Ad-hoc wall collision frequency coefficient (\({\alpha }_{w}\)) [for 1D axial simulation]
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0.1
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5.2. Results and discussion
Figure 12 shows the axial distribution of the time-averaged plasma properties from the pseudo-2D axial-radial simulation, the 1D axial simulations with and without a wall-collision model, and the 2D hybrid simulation of Ref. [32]. The results shown are the average over the entire simulation time, which amounts to one oscillation cycle (rise and drop) of the discharge current. In this regard, since the primary purpose of the present study has been to assess the capability of the single-region pseudo-2D PIC scheme to self-consistently introduce the overall wall-interactions’ effect into the simulation in comparison with 1D simulations, we deemed simulating only a single current oscillation cycle to be sufficient to obtain representative results for such a comparative analysis.
Looking at the plots of Fig. 12, and comparing the predictions of the pseudo-2D simulation against those from the 1D axial simulations and the full-2D reference case, the fact that the pseudo-2D scheme resolves the walls’ sheath and the SEE self-consistently is observed to have improved the consistency of the predictions of the electron temperature (Fig. 12(c)), the ionization rate (Fig. 12(d)), and the ions’ number density (Fig. 12(e)).
Therefore, it is logical to say that, with respect to simulations with ad-hoc wall-collision model, the single-region pseudo-2D simulation can capture more accurately the effect of the channel walls in moderating the electrons’ energy, which, in turn, leads to an improved prediction of processes governed by the electrons’ energy distribution such as the ionization.
However, the limited applicability of the single-region implementation to simulating the axial-radial coordinates of the thruster is also evident from the plots (a) and (b) in Fig. 12. In this respect, it is observed that an unrealistic anode sheath with high potential drop is formed and, consequently, the axial electric field is also overpredicted compared to the 1D simulations. Of course, it is important to note that this feature does not point to an inherent issue with the pseudo-2D PIC scheme; it is only highlighting that, as one may suspect, the single-region implementation does not provide a very good representation of the Hall thruster domain in the axial-radial configuration. Indeed, as it will be clarified shortly, the overprediction of the anode sheath potential drop roots in the fact that the single-region implementation in the axial-radial coordinates implies that the bulk plasma conditions in the radial direction are an average over the entire axial extent of the domain. Accordingly, noting the strong axial variations existing in the Hall thruster plasma, the wall-induced electron transport is overpredicted near the anode, which results in an artificially high electron current and, thus, an overestimated anode sheath potential. Nonetheless, this result is still physically consistent and is a consequence of how the simulation has been set up.
To elaborate on the above explanation and demonstrate its physical consistency, we show in Fig. 13 the axial profile of the electrons’ drift velocity from the pseudo-2D and the 1D simulations (Fig. 13(a)), alongside the radial distribution of the electron’s axial drift velocity from the pseudo-2D simulation (Fig. 13(b)). Figure 13(c) and (d) are the zoomed-in views on the electrons’ axial drift velocity plot in the near-anode zone and in the rest of the simulation domain, respectively.
From Fig. 13(b), it is observed that, near the walls in the pseudo-2D simulation, the axial drift velocity of the electrons’ population is largely negative, i.e., toward the anode. Moreover, a characteristic oscillation in the axial electrons’ velocity has appeared along the radial direction, which is reported in the literature to be the consequence of the Near-Wall Conductivity phenomenon [34], a mechanism that is demonstrated to play a role in enhancing electrons’ cross-field transport [2]. Consequently, it is seen in Fig. 13(d) that, in majority of the domain, the axial electrons’ drift velocity from the pseudo-2D simulation is more negative compared to the 1D axial simulations in which the effect of the wall interactions on the electrons’ mobility is either not captured or incorporated using an ad-hoc model. In addition, from Fig. 13(c), it is observed that the electrons’ axial velocity near the anode is significantly more negative in the pseudo-2D case than in the 1D axial simulations. It is, thus, as a result of this higher electron axial mobility in the pseudo-2D simulation that the anode sheath potential drop has increased to hinder further electrons’ loss to the anode.
Another interesting observation that we elaborate on in the following is related to the axial distribution of the ions’ axial drift velocity, shown in Fig. 12(f). It is observed in this plot that, despite a larger potential drop in the pseudo-2D simulation, the ions’ axial drift velocity at the cathode side of the domain is seen to be very close between the pseudo-2D and the 1D axial simulations. This is justified by referring to Fig. 12(c) showing the axial distribution of the ionization rate and Fig. 14, in which the ions’ velocity phase plots in the \(x-u\) plane is shown with \(u\) representing the ions’ axial velocity. It is observed in Fig. 12(c) that, compared to the 1D axial simulations, the ionization rate profile in the pseudo-2D simulation shows a downstream shift and is relatively broader in extent. This implies that the ions are also created within the acceleration zone, where most of the acceleration occurs, and, hence, not all ion particles feel the entire potential drop. This, as can be seen in Fig. 14(c), translates into a higher velocity dispersion in the ions’ population in the pseudo-2D simulation with respect to the 1D simulations (Fig. 14(a) and (b)). Accordingly, even though part of the ions’ population has indeed reached higher axial velocities in the pseudo-2D case due to the larger potential drop, the larger population of slow ions has caused the mean ion drift velocity from the pseudo-2D simulation to be similar to that from the 1D simulations.
The results and observations discussed above underline that, despite capturing an average effect of the radial processes, the single-region pseudo-2D PIC simulation can provide a physically consistent picture of the axial and radial interactions in a Hall thruster. This confirms that the pseudo-2D PIC approach has overall the potential to serve as a viable solution to resolve the couplings between the axial and radial phenomena.
In this regard, it is worth highlighting that increasing the number of vertical regions along the axis in the pseudo-2D axial-radial simulations, an effort that has been left for the future work, can allow us to resolve the influence of the axial distributions of plasma properties on the wall-induced mobility and the SEE, thus, to be able to obtain a more accurate picture of the underlying physics.