Establishing criteria for the transition from kinetic to fluid modeling in hollow cathode analysis Open Access

Hollow cathodes are efficient plasma sources for numerous applications, including but not limited to, spacecraft plasma propulsion,^1–6 beam injectors for fusion devices and instruments,^7,8 surface processing,^9–12 plasma-material interaction studies,¹³ and plasma switch technologies.^14–16 Thermionic hollow cathodes feature an insert coated with a low-work function material such as lanthanum hexaboride (LaB6) or barium tungsten (BaW). Following an initial heating stage induced by an external energy source, such as a radio frequency (RF) exciter or a heater, the primary emitted electrons acquire sufficient energy for ionization, subsequently triggering plasma ignition. The cathode then maintains heating primarily through ion bombardment at the wall, leading to a self-sustaining discharge. The plasma generated thereafter reaches a high density before spreading out of the tube and producing a plume.

Although extensively studied in the literature, hollow cathodes present a rich physics environment that is not yet fully understood. First, the plasma density, and sometimes the neutral pressure in the case of space propulsion applications, drops by several orders of magnitude between the channel region and the plume, considerably increasing the mean free path of particles.¹ Furthermore, multiple studies have reported that the plume area could be the site of instabilities that may lead to the formation of high-energy ions.^17–19 Finally, the design of hollow cathodes has undergone significant enhancements through a series of incremental improvements. These advances include the adoption of novel emitter materials^2,20 along with the incorporation of low sputtering-yield materials and fine-tuning of cathode geometry for optimal performance. For instance, adjusting the size and shape of the orifice, and separating the channel chamber from the plume area can affect the emitter temperature profile and thus the discharge current.²¹ Thanks to these innovations, both the lifespan and operational efficiency of hollow cathodes have experienced gradual and notable enhancements.^1,22 These improvements were made possible thanks to inputs provided by numerical modeling, which has become increasingly accurate over time. Therefore, a detailed description of plasma flow in both the channel and the plume, and an in-depth understanding of potential instabilities are crucial for future hollow cathode design improvements.

Comprehensive modeling of hollow cathodes is challenging due to the vast range of plasma regimes within the device. In the channel, where the plasma density is high and assumed near equilibrium, a fluid model is often appropriate, whereas, in the plume, kinetic effects may appear. Historically, the channel region has attracted most of the interest, and therefore, most descriptions of hollow cathodes are based on fluid models.²³ Zero-dimensional (0D)^24,25 and 1D models^26,27 focus mainly on the inner channel and orifice area and give quick estimates of plasma production and temperature. A better description of plasma–wall interaction and neutral flow effects at the insert can be obtained with 2D models^28,29 in combination with the Richardson–Dushman equation.³⁰ This equation governs electron emission from the walls and improves the understanding of erosion mechanisms.³¹ Capitalizing on these efforts, 2D models have been extended to the near plume region³² often including self-consistent evolution of the insert temperature.^33–35 One motivation for including the plume area is the potential presence of anomalous resistivity, which is higher than what classical transport and Ohm's law would predict, leading to higher electron temperature,.^36,37 Such an effect seems to originate from ion acoustic turbulence (IAT), identified by Jorns et al.,³⁸ leading to the creation of energetic ions, which ultimately may erode the cathode.^17,31 The IAT was further investigated in Refs. 39 and 40 in the context of hollow cathodes for space propulsion. Fluid models account for IAT via an adjustable parameter to fit experimental measurements.^33,41,42 However, since fluid models cannot self-consistent capture kinetic effects, such as the IAT, this parameter needs to be adjusted for each specific configuration using information obtained from experiments or kinetic simulations.

These kinetic effects can be captured by explicitly accounting for a separate population of fast electrons in the fluid model, distinct from the near-Maxwellian low energy plasma electrons. This approach was successfully demonstrated in Ref. 43, which presented a model of a thermionic cathode that can serve as a current and voltage stabilizer, investigated in experimental studies.^44,45 Additional kinetic effects may be captured by employing hybrid methods, as shown in Refs. 46 and 47. In these studies, electrons are represented using a drift diffusion model, whereas heavy particles, such as ions and neutrals, are treated as particles tracked via a particle-in-cell (PIC) approach. These modeling techniques result in favorable comparisons with experimental data obtained from a LaB6 hollow cathode. A fully kinetic model of a hollow cathode for space propulsion was investigated in Ref. 48. In this work, a 2D-2V PIC simulation modeled a miniaturized orificed hollow cathode. A parametric study over the cathode voltage, gas pressure, and radius size was performed and subsequent effects on the steady-state parameters were reported. The simulation was sped up by using a miniaturized geometry and lowering the mass of the xenon gas by a factor of 100. Non-Maxwellian behavior of the ion and electron velocity distribution functions (IVDF and EVDF) was identified in both the channel and plume area. Two other full PIC simulations with cylindrical geometry were investigated in Refs. 49 and 50, where it was again suggested that non-Maxwellian behavior for electrons exists within the insert region.

These physics and modeling challenges are mostly described in the context of electric propulsion but also apply to hollow cathodes utilized as plasma switches, which differ significantly in two main aspects. First, unlike in space propulsion applications, the neutral pressure is approximately uniform in both the channel and the plume. Second, the plume expands into closed boundaries linked to an external circuit, rather than being emitted into a vacuum. Hollow cathodes designed for plasma switch applications represent a promising technology for the development of future power grids utilizing direct current (DC) instead of alternating current (AC) at medium or high voltage.^51–53 DC transmission is an attractive solution for meeting growing electricity demand and integrating the intermittency of renewable energies. Indeed, it offers lower capacitive and inductive losses for a given right-of-way with respect to AC transmission lines and provides greater control and stability, avoiding the synchronization issues that can affect AC grids. However, as reported in Ref. 16, DC circuit breakers are inherently more costly and complex than AC breakers, as a classic mechanical switch opens at a relatively slow rate, which can result in dangerous and destructive arcing. In this context, the authors in Ref. 16, experimentally explored different geometries, gases, and operating conditions of a hollow cathode that could potentially serve as a DC circuit breaker. In particular, they noted that the discharge voltage across the cathode would drop from $∼$ 30 to 40 V before reaching a plateau corresponding to the ionization or excitation potential of the gas and depended on the gas pressure, which ranged from 30 mTorr to 1.5 Torr, and the discharge current. They also reported that the ionization mechanism is likely changing under the various test conditions. When the voltage drop, corresponds to a lower plasma density, the ionization is mostly dominated by direct ionization from primary electrons stemming from the cathode. In contrast, when the plasma density is high, a regime for which the voltage drop plateau has been reached, ionization is mostly due to the heating of the plasma bulk by the primary electrons.

We pursue two goals in the present paper. The first is to determine a general criterion for when either a fluid or kinetic approach is most suited to model the discharge channel. We then investigate how the plasma expands into the plume, which is a crucial characteristic for plasma switch design. To do so, we perform kinetic modeling of a hollow cathode for plasma switch applications, which, to the best of our knowledge, has not previously been reported. This paper is organized as follows. In Sec. II, we describe in detail the numerical setup and discuss the main assumptions underlying our modeling. In Sec. III, the plasma dynamics at steady state is elucidated, before we conduct an in-depth examination of the electron energy distribution function (EEDF) and develop our criterion for kinetic or fluid modeling.

The configuration considered in this work is based on the hollow cathode experimental setup described in Ref. 16. The computational domain is comprised of a rectangular box of size $Lx×Ly$ with geometry as shown in Fig. 1. The channel of the hollow cathode is located between the symmetry axis on the left of the simulation domain and the conducting cathode surface to the right, with dimensions $r×Lc$⁠. A thin dielectric layer, spanning only a few cells in width, is positioned between the cathode and the anode along the right boundary (shown in green). The anode (shown in blue) is connected to an external circuit for which a constant current, denoted as I₀, is applied. As shown in Fig. 1(c), the computational domain does not model the whole experimental setup but instead intends to model the inner channel and near plume region where most of the physics occurs. Additionally, incorporating a dielectric on the side aligns with the practical usage of hollow cathodes as plasma switches.¹⁴ The simulation is performed using the Cartesian version of the explicit momentum conserving PIC-MCC code EDIPIC-2D,⁵⁴ which has been verified through numerous international benchmarks,^55–57 and used to model a wide variety of low-temperature plasma systems.^58–62 Although the actual hollow cathode is cylindrical, the generic criterion that determines whether the plasma operates in a fluid or kinetic regime does not depend on the geometry. Indeed, while the distribution of the potential or the density differs between a Cartesian or cylindrical geometry, the criterion that will be presented in Sec. III C depends on the plasma parameters in the bulk and on the total potential fall between the bulk and the cathode. Additionally, PIC modeling in cylindrical coordinates can suffer from enhanced numerical noise near the centerline,⁶³ which can significantly distort the results. Therefore, in order to avoid unnecessary modeling uncertainties, Cartesian modeling was chosen for this work.

The plasma consists of electrons and singly charged argon Ar⁺ macroparticles with fixed statistical weight $pw=2.92×107$⁠. A constant neutral background of Ar is maintained at a pressure of $Pn=93 mTorr$⁠, corresponding to a density of $nn=3×1021 m−3$ and temperature of 300 K. We reiterate that this represents a major difference from hollow cathodes used in electric propulsion for which the neutral pressure exponentially decreases from the channel exit into the vacuum. Ions can only interact with neutrals via charge exchange collisions,⁶⁴ whereas electrons collide with the neutral background via elastic, excitation, or ionization collisions using state-of-the-art cross-section data.⁶⁵ Due to high-electron densities, Coulomb collisions between electrons are also implemented via Nanbu's approach.⁶⁶ To our knowledge, Coulomb collisions were not included in previous full PIC models of hollow cathodes. No recombination reactions are considered in the present setup. Neutral metastables are not modeled in this work and so stepwise ionization is not considered. For space propulsion applications, stepwise ionization is usually omitted as it does not seem to affect plasma production significantly.³³ For plasma switch applications, in large cathodes,¹⁶ the presence of metastables may lower the plasma potential in the bulk as stepwise ionization requires a lower acceleration of electrons emitted from the cathode to effectively produce plasma. Although interesting, we reserve this work for a companion paper as it is out of the scope of the present study.

The simulation is initialized with a uniform plasma density of $n0=1018 m−3$ and a Maxwellian distribution with temperature $Te,0=5 eV$ and $Ti,0=0.03 eV$ for electrons and ions, respectively. The cell size $Δx$ and time step $Δt$ are chosen such that they satisfy the accuracy and stability conditions for an explicit momentum conserving PIC scheme,⁶⁷ i.e., $Δx<λDe$ and $Δt<0.2ωpe−1$⁠, where λ_De and the ω_pe are the Debye length and the electron plasma frequency, respectively. In the present simulations, the maximum density is expected to reach approximately $∼5×1019 m−3$⁠, which with an electron temperature of a few electronvolts, and would require a cell size of a few micrometers and a time step less than a picosecond. The subsequent computational cost would be substantial for the considered computational domain and a simulation time of over a few hundred microseconds. To mitigate this cost, the vacuum permittivity is scaled by a factor $εr≈1057$⁠, which increases the allowed time step and cell size by a factor ∼33. Such a value was chosen to enlarge the cells sufficiently to achieve dimensions similar to those used in Ref. 16. This technique to reduce simulation cost has been widely used in low-temperature plasma simulations, particularly for space propulsion^68,69 and also for hollow cathodes.^49,50 In the latter two PIC studies, it was shown that scaling ε_r by a factor of 900 did not significantly affect plasma bulk properties, helping to guide our choice of ε_r. Since this study focuses on plasma properties in the bulk, the plasma sheath expansion that stems from the increase in vacuum permittivity is not a concern in this work. These considerations result in a chosen time step and cell size of $Δt=12.1 ps$ and $Δx=44 μm$⁠, respectively. In the most constrained scenario, utilizing the electron and temperature density from the channel, we obtain $0.2/ωpe≈37 ps$ and $λDe≈$ 100 μm guaranteeing a high spatiotemporal resolution for the simulation.

The system is symmetric with respect to a centerline at $x=0 cm$ with Fig. 1 showing the right half of the cathode and plume regions used in the model. A Neumann boundary condition for the field solver is imposed at the symmetry axis, while particles crossing $x=0 cm$ are reflected back into the domain with the sign of their x-direction velocity flipped. The cathode, whose inner radius is 1 cm, is modeled as a conducting and grounded material that fully absorbs particles hitting its surface. The cathode also emits thermionic 0.15 eV electrons from the surface into the channel region. In contrast to other studies that rely on the Richardson–Dushman law³⁰ to determine the electron flux emitted by the cathode wall, the flux is determined by a preset fixed current I. Therefore, the cathode is not self-consistently heated by ion bombardment. Secondary electron emissions are instead replaced by a constant injection of electrons whose velocity is sampled from a half Maxwellian with a temperature $Te,inj=0.15 eV$⁠. The temperature is therefore uniform along the cathode wall and remains constant. In the actual device, the temperature profile would be determined by ion bombardment, and the latter would also govern electron emission. For hollow cathodes used in space propulsion applications, fluid models^28,33 and to some extent, PIC models⁴⁹ have included such a self-consistent mechanism. However, this work primarily focuses on the electron thermalization process, which does not require a detailed description of electron emission and this aspect will be addressed in future research. The anode, located at y = L_y, is a fully conducting surface connected to an external circuit comprised of a current source I₀, such that the electric potential at the anode adjusts automatically to receive this current. The implemented external circuit model is described in the work by Vahedi and DiPeso.⁷⁰ A thin dielectric layer, enlarged for clarity in Fig. 1, connects the anode and the cathode, similar to the configuration of experimental plasma switches.¹⁴ When a particle hits the dielectric boundary, its charge is added to the nearest grid nodes by linear interpolation. Surface charge accumulating on the dielectric prevents a net current flow from leaking to the side at x = L_x. The plasma potential is obtained self-consistently by solving the 5-point discrete Poisson's equation using the PETSc linear algebra package⁷¹ with the generalized minimal residual (GMRES) solver. The electric field is then calculated from the potential using a second-order finite volume method.

In Table I, the main numerical parameters are reported for our reference case C0. Values of currents I and I₀ are chosen such that the corresponding current densities, J and J₀, computed in the present Cartesian geometry, are similar to those expected in a cylindrical hollow cathode for plasma switch application. J and J₀ are computed by dividing the currents by the area $A=Lc×1 m$ as the off-plane direction defaults to one meter length. Other cases, labeled from C1 to C5, with different values for the external circuit and emitted current densities, J₀ and J, will be considered in Sec. III C.

Figure 2 shows time-averaged 2D contour plots of the hollow cathode switch operating at steady state. Electrons are created at the thermionic surface of the cathode and then accelerate through the $∼16–17 V$ potential increase from the grounded wall to the centerline, as shown in Fig. 2(c). The energy gain is slightly above the ionization potential for argon, 15.76 eV, and so each injected electron may have the opportunity to ionize a single neutral, leading to frequent ionization events in the channel area as illustrated by Fig. 2(d). Similarly to what was observed in plasma switch experiments detailed in Ref. 16, this results in a dense diffuse glow discharge plasma in the channel, which, in the present work, leads to a maximum density of $∼9×1018 m−3$ in the plasma bulk at $x=0 cm, y=2.5 cm$ [Fig. 2(a)]. The plasma density drops near the non-emissive lower boundary as the potential monotonically goes to zero, which is the behavior typically observed in hollow cathode experiments and simulations.^1,41

Electrons from the dense plasma in the channel expand into the plume under the effects of the pressure gradient, and the density profile may be approximated by an analytical law (see  Appendix A for further details). As they accelerate through the narrow channel exit, the total current density J, driven primarily by electrons, dramatically increases, reaching a maximum of $∼25−30 kA·m−2$ [Fig. 2(b)]. The current density present in our system generates a magnetic field of approximately 1 G. This field strength is insufficient to magnetize the electrons, and therefore, electromagnetic effects do not substantially impact the plasma dynamics. As expected for hollow cathode behavior, electrons are effectively extracted from the channel and eventually travel to the anode where they are absorbed. Overall, the plasma density within the plume reduces to one-tenth of the levels found in the channel area. This density drop from $∼1019$ to $∼1018 m−3$ is significantly less than that observed in hollow cathode used for electric propulsion, where plasma expansion into the vacuum leads to density drops from $∼1021$ to $∼1017 m−3$ as shown in Refs. 1 and 33. The plasma expansion is correlated with a slight drop in the total electron temperature as shown in Fig. 2(e). Here, total electron temperature is defined with respect to the pressure tensor and as the sum of the three directional components. Cooling of the plasma in this region is consistent with hollow cathodes used for electric propulsion.^1,33

Figure 3 shows cross sections of the data from Fig. 2, allowing us to scrutinize the discharge behavior more closely. The cathode sheath, which is clearly visible in Fig. 3(e), leads to acceleration of thermionic electrons and a corresponding increase in $Te,x$ as shown in Fig. 3(f). This temperature anisotropy is then relaxed by Coulomb collisions as the electrons propagate into the plasma bulk. These effects cannot be captured by fluid codes, which typically assume quasi-neutrality all the way up to the cathode wall.^1,33,41 Although self-consistent modeling of the sheath can be achieved via multi-fluid moment models,^74,75 such models should also account for an anisotropic pressure tensor,^76,77 in order to accurately reproduce this behavior. Nonetheless, no fluid models are able to fully capture the non-Maxwellian nature of the EEDF; which, as will be shown in Sec. III B, is responsible for the observed anisotropy.

Previous PIC modeling of hollow cathodes has not specifically addressed this temperature anisotropy, since although non-Maxwellian behavior is mentioned, no temperature plots are shown. In the channel, authors in Ref. 48 showed energy distribution functions, albeit limited to less than 20 eV. Since they observed a potential drop between the cathode and plasma bulk greater than 20 eV, the beam electrons, and associated anisotropy, are not represented. In Refs. 49 and 50, an EEDF in the plasma bulk is shown; however, this is limited to below 7 eV, whereas the potential drop is reported as $∼25 eV$⁠. Although not visible in these simulations, we suspect that anisotropy in the EEDF and electron temperature are present but not depicted. In these PIC studies, it is important to note that electron–electron Coulomb collisions were not accounted for, which is the only physical mechanism able to thermalize the beam electrons. In our own simulations, failure to include Coulomb collisions significantly affected the plasma profiles in both the plume and channel, resulting in unphysical highly non-Maxwellian EEDFs.

In the anode region, the external circuit effectively adjusts the potential drop between the plasma and the surface. By ensuring that the electron current arriving at the anode matches the source current strength I₀, the potential drop is kept moderate to prevent excessive repulsion of electrons. Consequently, a maximum potential drop of 3 V is observed in parts (a) and (c) of Fig. 3, which is insufficient to break local quasi-neutrality as the electron density does not decrease rapidly enough. As shown in a more detailed picture provided by Fig. 4, this absence of sheath translates into a mean ion velocity that increases up to $∼800 m·s−1$⁠, below supersonic speeds. Indeed, the Bohm velocity is found to be $uB≈eTe/mi≈1.52×103 m·s−1$⁠, as the electron temperature is locally ∼1–1.5 eV. Therefore, the ion Mach number M_i peaks at $∼0.52$ at the anode $y=10 cm$ whereas it exceeds 2 at the bottom of the cathode at $y=0 cm$⁠. At the top of the cathode, at $y=5 cm$⁠, the ion Mach number is also likely greater than unity due to the presence of a sheath, as illustrated in Fig. 3(c). Since a significant portion of electrons with moderate or high energy are allowed to escape at the anode, the electron velocity distribution function is preferentially depleted in the orthogonal direction to the anode wall (y direction). Thus, $Te,y$ diverges from the two other temperature components as the plasma flow approaches the anode as shown in parts (b) and (d) of Fig. 3. This represents a second instance of temperature anisotropy observed in the system.

As the plasma expands into the plume, an ambipolar electric field is created to ensure quasi-neutrality, while the electrons accelerate under pressure gradient effects. Thus, a slight decrease in plasma potential less than 2 V is visible in Fig. 3(a). This modest voltage drop between the channel and the anode was also reported in plasma switch experiments, in Ref. 16, but differs from what is observed in hollow cathodes used for electric propulsion, where experiments and simulations show an increase in plasma potential between the channel and the plume.^1,33,34 Indeed, in these cases, the electrons can reach a sufficient velocity^38,40 to trigger ion acoustic turbulences (IAT), elevating the now oscillating plasma potential in the plume. However, these effects are not observed in the current study, as the maximum electron drift velocity, $ve$⁠, is approximately $105 m·s−1$⁠. When compared to the thermal speed, $ve,th∼8×105 m·s−1$⁠, the electron Mach number remains below one, a necessary condition for the onset of IAT.

For completeness, the ion and electron densities, plasma potential, total current density, and the components of electron temperatures are shown at $y=8 cm$ in Figs. 3(g) and 3(h).

Figure 5(b) shows the ion energy distribution function (IEDF) in the channel at different x locations, from the centerline up to near the cathode wall. In the plasma bulk, at x = 0 cm, ions follow a Maxwellian distribution. As ions approach the sheath, starting at $x∼0.85$ cm the IEDF becomes skewed due to ion acceleration [Fig. 3(e) at x = 0.80 cm and x = 0.95 cm]. Ions accelerate under the potential drop between the plasma bulk and the sheath with a theoretical terminal energy of $∼16–17 eV$ (not depicted).

Coulomb collisions, responsible for plasma Maxwellization, compete with inelastic processes, ionization, and excitation. Understanding when one is predominant over another is crucial to properly assess energy-related processes such as plasma production. Thus, our study was extended to various hollow cathode operating conditions to identify which regimes require a kinetic treatment or when a fluid approximation may suffice, specifically we investigate how the change in potential drop from the channel to the plume influences behavior. According to Eq. (3), the cold electron plasma density, determined by the injection current I, and the energy of fast particles, governed by the potential drop at the cathode wall are key parameters in this study. The potential drop can be tuned by the external current I₀ set by the external circuit. Indeed, whenever $I0>I$⁠, the hollow cathode must produce sufficient plasma to satisfy the global balance $I0=I+Iioniz−Iloss$⁠, where I_ioniz and I_loss are the current due to ionization and wall losses, respectively. An increase in plasma potential elevates the energy levels of emitted electrons and simultaneously enhances the confinement of cold electrons within the channel. This enhancement effectively raises $Iioniz−Iloss$⁠, thereby supplying adequate current to the external circuit. The span of operating currents and associated potential drops for our different simulation cases are provided in Table II.

The EEDFs for all cases within the bulk of the channel plasma, measured at the centerline $x=0 cm$ and $y=3 cm$⁠, are depicted in Fig. 6(a). Cases C0, C1, C2, and C5 maintain the equality J = J₀, while increasing the discharge current that effectively controls the plasma density level in the channel. The presence of fast electrons is more pronounced in case C1, for which the emitted current and the plasma bulk density have been halved (⁠ $J0=2.06 kA·m−2$⁠) with respect to case C0 (⁠ $J0=4.12 kA·m−2$⁠). This trend correlates with a decrease in the frequency of Coulomb electron–electron collisions, attributable to the diminished electron density. In contrast, cases C2 and C5 present an electron density approximately three (⁠ $ne≈2.65×1019 m−3$⁠) and twenty times higher (⁠ $ne≈1.1×1020 m−3$⁠) than case C0, respectively. The resulting EEDF is closer to a Maxwellian as beam electrons thermalize more rapidly. In cases C3 and C4, the current density J is unchanged; however, the external circuit current density J₀ is increased by 30% and 60%, respectively. In such cases, the beam electrons, accelerated through the cathode sheath, are fast enough to reach the center of the channel before thermalizing via Coulomb collisions. Thus, a clear local maximum in the EEDF is visible for case C3 at an energy close to the cathode voltage drop, i.e., $∼20 V$⁠. In case C4, the scenario is more extreme with the measured cathode voltage drop approaching approximately ∼36 V, and a corresponding EEDF maximum is present at a similar energy. Accelerating through the sheath of this potential, electrons can ionize and excite neutral particles several times before joining the cold population, with the various collision pathways shown in Table III. A secondary local peak, observed at approximately 24.5 eV, can be attributed to 36 eV beam electrons that have undergone an excitation collision, considering that the excitation energy for argon is 11.5 eV. This is referred to as process P1 in Table III. Between 5 and 20.3 eV, the electron population decreases exponentially with higher energies. For these electrons, additional excitation of ionization processes is possible depending on their energy. For instance, 20.3 eV electrons are produced via process P2 and stem from 36 eV beam electrons that have ionized once, suffering a loss of 15.7 eV. Similarly, 24.5 eV electrons lead to the production of 13 and 8.8 eV electrons via excitation (process P3) and ionization (process P4), respectively. Electrons with an energy of 20 eV can ionize or excite (processes P5 and P6, respectively) one last time while 13 eV electrons can participate in one final excitation event, denoted as process P7 in Table III.

In case C4, at low energies, no excitation or ionization event can occur and Coulomb collisions thermalize electrons and the EEDF follows a Maxwellian distribution, with a temperature measured as $∼0.4 eV$⁠. It is observed that as kinetic effects become increasingly pronounced, there is a noticeable reduction in the temperature of cold electrons. Indeed, in this regime, the beam electrons are mostly responsible for ionization and do not heat the bulk electrons. Thus, cases C3 and C4 have a cold electron temperature of 1.4 and 0.4 eV, whereas for cases with a nearly Maxwellian EEDF, such as C2 or C5  $Te>2 eV$⁠. Consequently, estimating the ionization rate based solely on a Maxwellian distribution that depends on the measured electron temperature T_e would yield highly inaccurate results in the cases of C3 and C4.

The proposed criterion offers a valuable method for assessing the operational regime of various hollow cathodes documented in the literature. For a hollow cathode employed in space propulsion with xenon propellant,⁸⁴ we have a density of approximately $5×1020 m−3$ and an energy beam ranging between 10 and 15 V. This results in a β value of roughly 15, under the assumption of a pressure near 1 Torr. Consequently, a fluid treatment of electrons within the channel is appropriate, a conclusion corroborated by extensive research.^28,34 In contrast, thermionic cathodes, with applications as current and voltage stabilizers,^43,45,85 demonstrate a different behavior. These cathodes exhibit dual electron populations: a cold Maxwellian group and a fast electron beam. In experiments involving helium at 1 Torr, the cold electron density was reported as $5.35×1018 m−3$⁠, with a beam energy of 27 eV. This yields a β value of approximately 0.03, justifying the application of a kinetic treatment.

Finally, results from this study echo several findings from the investigation in Ref. 16. First, it is reported that the voltage drop in the cathode channel decreases as the current density in the external circuit increases and reaches a plateau for the tested noble gases, i.e., argon, helium, and xenon. The plateau value converges toward the ionization or excitation potential depending on whether direct or two-step ionization is the dominant mechanism for plasma formation. This trend is reproduced by our cases C1, C0, C2, and C5, for which J₀ goes from $2.06$ to $80 kA·m−2$ and the measured potential drop ranges from 16.7 to 15.8 V. As two-step ionization is not modeled here, the potential cannot drop below the ionization potential and only direct ionization is possible. The authors in Ref. 16 also report that the ionization mechanism seems to vary with the plasma density, suggesting that, at low density, beam electrons directly ionize the neutral gas, whereas when the density is higher, beam electrons first heat the bulk electrons and the tail of the Maxwellian EEDF is then responsible for ionization. Although these two mechanisms are also identified in the present paper, the governing criterion depends on the parameter β instead of solely the plasma density. Depending on the regime they operate, accounting for the non-Maxwellian nature of the EEDF may be necessary for the modeling of hollow cathodes for plasma switches since it is reported in Ref. 16 that failing to do so led to systematic errors in their 0D fluid model.

A cross-comparison of cases C2, C4, and C5 is given in Fig. 7 to highlight the impact of the non-Maxwellian nature of the EEDF on the simulation results. For all cases, and similarly to case C0 described in Fig. 2, a dense plasma is created in the inner channel and then expands into the plume. The ions closely follow electric field lines, whereas, in the plume, the electrons' motion is determined by a balance between the density gradient and the electric field strength as detailed in  Appendix A. It is noteworthy that in case C5, in contrast to other cases, given the magnitude of the current density shown in Fig. 7(l), a fully electromagnetic code would probably be better suited. For this configuration, we would anticipate an induced magnetic field that would pinch the plasma column, making the plasma expansion less uniform. However, this should not influence the subsequent discussion regarding the impact of the ratio β on the various simulations. The non-Maxwellian EEDF impact is clearly visible in Case C4, for which the temperature anisotropy is very high near the cathode wall in Fig. 7(j). Since the potential fall between the plasma bulk and the grounded cathode is approximately 36 V, electrons strongly accelerate in the x direction, leading to a primarily heating of electrons in this direction up to 6 eV and numerous ionization collision events. As mentioned in the analysis of Fig. 6(a), the population of cold electrons in the plasma bulk, which follows a Maxwellian distribution, are not heated by fast electrons in the inner channel and the measured temperature is low at 0.4 eV.

In the plume area, as the EEDF relaxes to a Maxwellian, the electrons warm up and the temperature rises to 2 eV. In contrast, for case C5, for which the EEDF is very close to a Maxwellian, electrons are already heated by the fast electrons stemming from the cathode and the temperature is close to 2.2 eV as shown in Fig. 7(o). In the plume, the temperature only slightly drops as the plasma expands. Case C2 exhibits a similar behavior as case C5: in the inner channel, most ionization events occur in the plasma bulk and the electron temperature is close to 2–2.5 eV. The main visible difference between case C2 and C5 is the plasma density, which is approximately ten times higher. This is not too surprising as the EEDF in case C2 only exhibits a slight deviation from a Maxwellian as illustrated in Fig. 8. However, this disparity is sufficient to lead to slightly wrong estimates of the ionization collision frequency in the plasma bulk. For case C3, according to Eq. (5), the PIC simulation yields an ionization collision frequency of $νPIC≈6.9×104 s−1$⁠. In contrast, the Maxwellian fit derived from Fig. 5(a) estimates $νMaxwell≈9.8×105 s−1$⁠. Therefore, an error of ∼20% would be made in this case, which could probably lead to a noticeable but moderate discrepancy if a fluid model were to be used.

In this paper, 2D3V PIC simulations of an argon hollow cathode for plasma switch application have been performed. Analysis of the energy distribution functions for both ions and electrons revealed that non-Maxwellian behavior can occur in the channel region. It was found that a Maxwellian EEDF is present only when the thermalization of electrons occurs sufficiently fast. This happens when Coulomb collisions are frequent enough to transfer energy from the beam electrons to the cold electron population before they have time to ionize or excite neutral particles. The ratio $β=νee,f/max{νioniz,f,νexcit,f}$⁠, as defined in Eq. (4), compares the electron–electron collision frequency with the maximum collision frequency between direct ionization and excitation for fast beam electrons and serves as a straightforward criterion to determine whether fluid or kinetic modeling approaches are suitable for hollow cathode simulations. Whenever $β≫1$⁠, then beam electrons quickly thermalize and a fluid model is sufficient to describe the system. In contrast, if $β≪1$⁠, then beam electrons play an essential role in ionization processes and the EEDF in the plasma bulk is not Maxwellian, requiring a kinetic treatment. The transition from a kinetic to fluid regime occurs around $β∼1$⁠, as shown in Fig. 6(b). We found that for most hollow cathodes used in space propulsion, a fluid treatment of electrons in the channel is a reasonable assumption, which is reflected by state-of-the-art models.^34,41 In contrast, some applications related to plasma deposition processes⁸⁶ or to a plasma switch¹⁶ may operate in regimes for which a kinetic treatment is required.

To further our understanding of the ionization mechanism, particularly within hollow cathodes utilized in plasma switch applications, several follow-up studies are possible. First, modeling two-step ionization processes by tracking metastable species would reveal under which conditions they become more dominant than direct ionization. Refining such prediction could be a useful input for future design as it was suggested in Ref. 16 that metastables may have an effect on the potential drop in these devices. Finally, in order to help the design of future devices, it would certainly be beneficial to account for a self-consistent thermionic electron injection scheme, one which depends on the cathode temperature, and accounts for self-induced magnetic field generated by high currents.

The information, data, or work presented herein was funded in part by the Advanced Research Projects Agency-Energy (ARPA-E), U.S. Department of Energy, under Award Number DE-AR0001107. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. The authors thank Dr. David J. Smith and Dr. Svetlana E. Selezneva for fruitful discussions. This research was supported by the U.S. Department of Energy, Office of Fusion Energy Science under Contract No. DE-AC02-09CH11466 as a part of the Princeton Collaborative Low Temperature Plasma Research Facility.

The authors have no conflicts to disclose.

Willca Villafana: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). Andrew Tasman Powis: Conceptualization (equal); Formal analysis (supporting); Methodology (supporting); Writing – review & editing (lead). Sarveshwar Sharma: Formal analysis (supporting); Visualization (supporting); Writing – review & editing (equal). Igor D. Kaganovich: Conceptualization (equal); Formal analysis (supporting); Funding acquisition (lead); Investigation (supporting); Project administration (lead); Supervision (lead); Writing – review & editing (equal). Alexander V. Khrabrov: Formal analysis (supporting); Software (supporting).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

In the present estimate, the value of $nc=9×1018 m−3$⁠, corresponding to the maximum density found in the channel is used instead of $∼1018 m−3$⁠, which is the approximate average value measured at the channel exit (⁠ $y=5 cm$⁠). The model described by Eq. (A3) assumes the electron temperature to be uniform, particularly in the y direction. However, the pressure gradient is also due to a slight temperature drop in the plume and this effect further accelerates electrons. Thus, in the PIC simulation, this additional contribution tends to extract more electrons from the cathode and so the plasma density in the plume is therefore expected to be slightly higher than the model estimate. Adjusting the parameter n_c to a higher value, we obtain density magnitudes that are consistent with the PIC data, as illustrated in Figs. 10(a) and 10(b). In Fig. 10(b), with this adjustment the density profile in the x direction is slightly overpredicted at y = 7 cm. This could be attributed to the fact that a uniform density is assumed at the channel exit (y = 5 cm), whereas the density drops to 0 near at the upper left corner of the cathode as shown in Fig. 2(a). At y = 8 cm and y = 9 cm in Fig. 10(b) density profiles are slightly underestimated most likely due to the aforementioned uniform temperature assumption. Despite these discrepancies, the overall profile and trend in both the x and y directions are reasonably predicted, which is a crucial input for hollow cathodes designed for plasma switch applications.