Coupling of Fluid and Particle-in-Cell Simulations of Ambipolar Plasma Thrusters

Abstract

Ambipolar plasma thrusters are an appealing technology due to multiple system-related advantages, including propellant flexibility and the absence of electrodes or neutralizer. Understanding the plasma generation and acceleration mechanisms is key to improving the performance and capabilities of these thrusters. However, the source and plume regions inside are often simulated separately, and no self-consistent strategy exists which can couple these different simulations together. This paper introduces the MUlti-regime Plasma Equilibrium Transport Solver (MUPETS), a self-consistent coupled model integrating a fluid solver for the plasma dynamics in the source, which are collision-driven, with a kinetic Particle-In-Cell (PIC) code for the plasma dynamics in the magnetic nozzle, which involve expansion across a diverging magnetic field. The methodology begins by solving the plasma source with the classical Bohm condition at the thruster’s throat. The resulting plasma profiles (density, temperature, speed) are input into the PIC code for the magnetic nozzle. The PIC code calculates the plasma plume expansion and determines the electric field at the thruster’s throat. This electric field is then used as a boundary condition in the fluid code, where it replaces the Bohm assumption, and the fluid simulation is repeated. This iterative process continues until convergence. In comparing the MUPETS results with those for an experimental thruster, the plasma densities at the thruster’s throat differed by less than 2–5% between the fluid and PIC regions. The thrust predictions agreed with the experimental trend, and were kept well within the measurement’s uncertainty band. These results validate the effectiveness of the coupling strategy for enhancing plasma thruster simulation accuracy.

1. Introduction

Ambipolar Plasma Thrusters (APTs) represent a class of electric propulsion systems that create and accelerate a plasma using electromagnetic fields rather than electrodes. Among the most promising variants of APTs are Helicon Plasma Thrusters [1,2,3,4,5,6] (HPTs) and Electron Cyclotron Resonance Thrusters [7,8] (ECRTs). These devices generate plasma within a source chamber, where an antenna operating in the Radio Frequency (RF) or microwave range sustains the discharge. A magnetostatic field produced by electric or permanent magnets drives the power coupling of the RF waves into the plasma [9], enhances plasma confinement, and boosts propulsive performance via the magnetic nozzle effect [10,11]. APTs offer multiple system-level advantages, including propellant flexibility, simpler design, and longer lifetimes, thanks to their omission of any anodes and cathodes. These advantages make them highly appealing for applications in the SmallSat market [12], interplanetary missions [13], and Air-Breathing Electric Propulsion (ABEP) missions [14,15].

Despite their potential, the performance of APTs has not yet fulfilled their promise, as current thrusters are marked by only moderate propulsive efficiencies. For instance, the efficiency of HPTs has not yet exceeded 30% [16]. To enhance the performance of APTs, a deeper understanding of the plasma generation and acceleration mechanisms is required. To this end, numerous modeling efforts [17,18,19] have sought to accurately predict the operational behaviors and theoretical limits of these thrusters. However, this is a challenging effort due to the existence of different flow regimes in different sections of the thruster. Within the plasma source, the flow behaves as a continuum governed by the propagation of electromagnetic waves, plasma transport, and their mutual coupling. Downstream in the expanding plume of the magnetic nozzle region, the plasma density drops sharply and the main phenomena at play become the acceleration and magnetic detachment of the plasma.

Codes that describe the plasma state in the source chamber are most often fluidic [20,21,22], assuming that the many collisions inside the high-density source [2] lead to thermal equilibrium and Maxwellian velocity distribution functions [23]. Meanwhile, for description of the plasma plume in the magnetic nozzle, where far fewer or even no collisions take place, a kinetic approach is favored, often in the form of Particle-in-Cell (PIC) simulations [24,25]. PIC codes, which group the plasma particles into macroparticles and then solve the motion of these particles individually, are marked by high computational cost compared to fluid codes. This computational cost increases with the plasma density; consequently, few models employ PIC for the high-density source chamber.

Thus, numerical APT studies can either treat each region with different solvers appropriate to each, or use a hybrid model [26,27] across the entire thruster. Simulating the source and plume separately has the advantage of codes that can be more specific in handling the peculiar phenomena of each region [28,29,30]. Simulating the regions as one with a hybrid model that selectively uses fluid or kinetic methods for different plasma particles has the advantage of simulating the full thruster with one model and simulating the transition between sections; however, it provides a less specific approach in each region. On the other hand, when using a specialized solver for each section [17,22,28], the transition and required interfacing between the two regions presents a challenge for simulating the full thruster. For example, problems in continuity between separated models can occur across the flow regime transition [28]. Instead, approaches often focus on a single region [31] and handle the other region through a global model [19], although this reduces the accuracy of the overall solution. As of yet, no self-consistent strategy exists for coupling different simulations of source and plume together.

This work proposes an innovative self-consistent methodology for coupling a fluid solver [20] to handle the plasma source region and a Particle-In-Cell (PIC) code [25] dedicated to the magnetic nozzle region. This methodology is compared against established numerical approaches and validated through experimental data [28]. By self-consistently coupling the plasma state across flow regimes, the proposed approach aims to improve the ability to predict propulsive performance by treating each regime using a specialized solver. Within the scope of the present paper, this strategy is exploited for modeling APTs; however, it could also be useful for improving the modeling of broader plasma applications, such as other electric propulsion systems or material etching tasks [32,33].

2. Methodology

The model coupling methodology is based on a 2D-3V axisymmetric representation of an APT design, shown in Figure 1. The coupled model simulates the plasma source section and the free space outside of the thruster, where a diverging magnetic field forms the magnetic nozzle, thereby still forming part of the propulsive system. A constant propellant mass flow of neutral gas is injected at the back-wall. Figure 1 also shows the fluid and PIC domains; the fluid domain encompasses the entire source chamber, extending radially from the thruster’s centerline to the chamber walls and axially from the back-wall to the outlet. The plasma plume expanding into free space is modeled as a kinetic flow, and as such is covered by the kinetic PIC domain. The transition between these domains occurs at the throat of the thruster (at $z=z_{throat}$), consistent with approaches where the regions are treated separately [17,28].

2.1. Models

The coupling methodology focuses on the steady-state conditions achieved by the fluid and kinetic models. Both of these models consider the following plasma species: neutrals, singly-charged positive ions, and electrons. The fluid model separates the neutrals further into neutrals in the ground state and neutrals in multiple excited states [21], while in the kinetic model the excited species are assumed to immediately decay to the ground state.

The fluid model governs plasma dynamics using three fundamental equations: conservation of mass for each species, conservation of energy, and the Poisson equation for plasma potential ($\varphi$). In this paper, the methodology is presented only for a Drift Diffusion [34] (DD) implementation; however, generalization to fluid models based on the solution of the full momentum equations is straightforward. The use of the DD formulation for the transport equations of each plasma species is justified if its collision frequency is higher than the estimated velocity gradient [35]. For typical discharges applied on HPTs, this condition is respected if the neutral pressure is above 1 mTorr [36].

The kinetic model tracks plasma species macroparticles, solves their motion under electromagnetic forces, and uses the Poisson equation for $\varphi$. The fluid and kinetic models are further expanded upon in Appendix A.

2.2. Coupling

The coupling scheme is shown in Figure 2. The scheme is iterative, running h parent iterations until convergence. For each iteration h, the fluid solver runs until achieving the steady state. The fluid plasma properties at the throat are then used as input to the kinetic solver, which also runs until achieving the steady state. The kinetic results then are used to correct the initial throat BCs of the fluid simulation, after which a new parent iteration $h+1$ starts. Initially ($h=0$), the plasma source is solved assuming the classical Bohm condition at the throat. For subsequent parent iterations ($h+1$), the fluid model uses the corrected BCs, replacing the Bohm condition with the kinetic potential gradient to ensure self-consistent coupling. Convergence is achieved when the axial electron density profiles between iterations differ by less than $10\%$, which is the uncertainty in measuring the electron current density [37,38].

Coupling Interface Boundary Conditions

Figure 3 shows the applied BCs for the thruster domain. This section describes the BCs used at the interface between the two models, i.e., the throat. The rest of the BCs used in each model are further described in Appendix A.

At the interface, the fluid model assumes a developed flow for ions and neutrals, whie the BC, with $\widehat{k}$ being the unit vector normal to the throat surface, is provided by

$${\displaystyle \frac{\mathrm{d}{n}_{i,0,ex}}{\mathrm{d}\widehat{k}}}{|}_{Fluid}=0.$$

(1)

For the initial fluid simulation at $h=0$, Poisson’s equation is solved by imposing a zero-current condition across the outlet:

$${\Gamma }_{e_{\widehat{k}}}{|}_{Fluid}={\Gamma }_{i_{\widehat{k}}}{|}_{Fluid}.$$

(2)

Rewriting ${\Gamma }_e$ in its drift-diffusion formulation [20] and substituting the electric field E with $-\nabla \varphi$ provides the initial plasma potential BC for $h=0$:

$${\displaystyle \frac{\mathrm{d}\varphi }{\mathrm{d}\widehat{k}}}{|}_{Fluid,h=0}={\displaystyle \frac{{\Gamma }_{i_{\widehat{k}}}+D_e\frac{\mathrm{d}{n}_e}{\mathrm{d}\widehat{k}}}{n_e{\mu }_e}}.$$

(3)

For $h=0$, a BC is still required for the behavior of the electrons. To ensure the stability of the fluid solver, the electron flux is modeled as Equation (4); the Bohm sheath criterion [39] is used by enforcing the equality of fluxes (${\Gamma }_e={\Gamma }_i$) and quasi-neutrality. This approach has been proven to be stable for bootstrapping the simulation in a stable manner, as reported in [20,40,41], and is corrected in subsequent iterations $h+1$.

$${\Gamma }_{e_{\widehat{k}}}{|}_{Fluid,h=0}=v_B{n}_e$$

(4)

Finally, the electron energy density follows the electron density flux [20]:

$${\Gamma }_{{\epsilon }_{\widehat{k}}}{|}_{Fluid}={\displaystyle \frac{2}{3}}\left({\displaystyle \frac{1}{2}}\left(1+ln\left({\displaystyle \frac{m_i}{2\pi m_e}}\right)\right)+2\right)·{\Gamma }_{e_{\widehat{k}}}{|}_{Fluid}.$$

(5)

For subsequent iterations $h+1$, the plasma potential BC is corrected by the kinetic simulation results of h:

$${\displaystyle \frac{d\varphi }{d\widehat{k}}}{|}_{h+1,Fluid}={\displaystyle \frac{d\varphi }{d\widehat{k}}}{|}_{h,PIC}.$$

(6)

The earlier assumed zero-current condition provided by Equation (2) are now directly applied to the electron density for $h+1$. Thus, in all iterations where $h>0$, the assumption of the Bohm condition at the throat is removed, allowing the electron outflow to evolve self-consistently.

In summary, the coupling of the fluid model’s BCs to the kinetic model is achieved by obtaining the kinetic potential gradient and assuming steady-state operation. Only the interface BCs for the potential ($\varphi$) and electron density ($n_e$) are updated. Instead of being predetermined by the Bohm speed, the particle speed at the interface is computed self-consistently, with the ion flux driven by the previous iteration’s electric field and the electrons’ response determined under the current-free assumption.

The kinetic model requires the number density, velocity, and temperature ($n_I$,$v_I$,$T_I$) of the plasma species at the throat in order to construct the sampling distribution function for each particle species (PDF) [29] and velocity distribution function for the PIC solver (VDF). Macroparticles are sampled from the PDF and VDF, and are injected into the numerical kinetic domain from the interface. To maintain a current-free global plasma flow as determined by the steady-state operation of an ambipolar plasma thruster [25,42], the kinetic model balances the injected electron flux against the total outgoing charge of species leaving the kinetic domain through the free-space boundary. More details on this balance are provided in Appendix A. Considering the boundary condition on the potential field, a Dirichlet (i.e., ${\varphi }_{\mathrm{b}}=0$ V) condition is imposed. This can be performed as in the kinetic routine calculation, with the potential included only as a derivative; hence, any constant offset will not affect the results. The potential field offset is then taken into account during postprocessing.

2.3. MUPETS

The coupling strategy is implemented in the MUlti-regime Plasma Equilibrium Transport Solver (MUPETS), a Python code developed at the Alma Propulsion Laboratory of the University of Bologna in cooperation with the University of Padova. MUPETS manages the fluid and kinetic solvers as well as the coupling methodology. The fluid model uses the plasmaFoam solver [40,43], based on the OpenFOAM framework [44] and derived from the 3D-VIRTUS code [41], while the kinetic model uses an adapted version [29] of the Starfish PIC code [45]. Further details on the plasmaFoam and Starfish codes are provided in Appendix B. The modularity of MUPETS allows for easy replacement or updating of solvers; thus, the proposed coupling methodology can be used together with a range of different solvers, allowing for the choice of a solver that best suits the thruster design and operating conditions. MUPETS responsibilities include:

• Preprocessing simulation cases:
  • For fluid, $h=0$: Setting up the initial case with starting values.
  • For fluid, $h>0$: Updating case with BC values from kinetic simulation.
  • For kinetic, h: Constructing PDFs and VDFS for macroparticles.
• Running simulations with relevant solvers and parameters.
• Postprocessing the converged solutions into a common data structure.
• Adjusting conditions for the next simulation based on the current solutions.
• Loop control until the convergence criteria are met.
• Handling numerical domain division.

2.4. Propulsive Performance

The performance of a thruster can be measured by its thrust level, specific impulse, power consumption, and associated efficiencies. The thrust can be calculated during postprocessing of the combined fluid–kinetic data of the entire thruster domain. After this, the other metrics can be derived from the thrust and other postprocessed metrics or input variables, such as the injected propellant mass flow and input power. The thrust is taken to be the momentum flux of the plasma jet [46]:

$$F=\sum _{I=i,e,0}{\int }_A(p_I+m_I{n}_I{u}_{Iz}^2)dA.$$

(7)

Several assumptions stemming from the coupled model simplify the thrust calculation. First, the heavy particles are considered ‘cold’, with $T_e>>T_i,T_0$, allowing the pressure terms for heavy species to be neglected. The electron momentum term is also neglected due to $m_e<<m_i,m_0$. The thrust contribution from neutrals is assumed to be negligible due to their lower velocity [47] and temperature [48]. Additionally, momentum losses to the wall are considered negligible [46] due to magnetic shielding [10,19,49] of the source chamber walls. The momentum flux then simplifies to the sum of the pressure force from the source chamber and the momentum force enhanced by the magnetic nozzle [46]:

$$F_{cham}\approx (1+C^2)\pi R_{\ast }^2{n}_{e\ast }q{T}_{e\ast }$$

(8)

$$F_{mag}={\int }_V{J}_{\theta }{B}_{r}dV$$

(9)

$$F_{thrust}=F_{cham}+F_{mag}$$

(10)

where $F_{cham}$ represents the thrust from the source chamber due to static electron pressure against the back plate inside the source, shown in Figure 4. It is approximated using the average electron density $n_{e\ast }$ and temperature $T_{e\ast }$ at the outlet [19], with $R_{\ast }$ being the outlet radius and C the ratio of the ion drift velocity to the Bohm velocity. $F_{mag}$ represents the magnetic thrust, calculated as the volume integral of the Lorentz force due to the radial magnetic field $B_r$ and the azimuthal current, shown as $F_z$ in Figure 4.

3. Experiment

Through a numerical study, we aimed to determine whether the presented coupling methodology results in improved accuracy by comparing experimental thrust measurements with numerical thrust predictions from MUPETS and other numerical approaches [28] for a laboratory thruster [3]. Thus, in this study we focus on predicting and comparing the thruster plasma state of the resulting thrust against experiments; convergence studies of the MUPETS code have been presented in an earlier work [51].

3.1. Experimental Thruster

The laboratory RF ambipolar thruster developed by the University of Padova [52,53,54] features a source chamber with radius and length of 7 mm and 60 mm, respectively. Neutral Xenon atoms are injected into the thruster at the inlet at a constant mass flow rate of 150 μg s⁻¹. The magnetic field is generated by two sets of permanent magnets arranged in opposing directions. Thrust measurements were taken at six power settings ranging from 23 W to 69 W.

3.2. Numerical Simulations

The thruster geometry was maintained as described for the experimental thruster in Section 3.1. The minimum cell size in the fluid domain was set to 50 μm, while that of the kinetic domain was set to 500 μm. The mesh is shown in Figure 5. Further details on the mesh and run parameters of each code can be found in Appendix B. The magnetic field created by the permanent magnets was modeled separately and provided to OpenFOAM and Starfish for the source chamber and expanding magnetic nozzle section, respectively. The orientation of the magnets and the resulting geometry are shown in Figure 6a. Collisions in the rarefied plume were included for increased fidelity, with collision rates and chemistry coefficients for the Xenon propellant sourced from the literature [21,43,55,56,57,58,59,60,61]. The RF coupled power $P_w$ was simulated at points spaced 15 $\mathrm{W}$ apart, ranging from 15 $\mathrm{W}$ to 57 $\mathrm{W}$. The 57 $\mathrm{W}$ point was used instead of 60 $\mathrm{W}$ for better comparison with previous simulations of this thruster. The space surrounding the thruster was modeled as a vacuum.

The hardware used to run MUPETS consisted of four Intel(R) i7-7700 Cores(TM) @ 3.60 GHz and 64 Gb of RAM, which resulted in a convergence time of 2–3 weeks. The majority of this time was consumed by running the PIC code. A PIC code optimized to reduce this time is currently under development.

4. Results and Discussion

4.1. Plasma Profiles

The full thruster 2D profiles of the magnetic field and plasma at 57 $\mathrm{W}$ solved by the coupled model are shown in Figure 6. Additionally, the species density profiles zoomed in at the source chamber are presented in Figure 7. The interaction of the expansion plumes with the magnetic field is evident in the plume region near and slightly outside of the outlet (Figure 6b–d). The magnetized electrons expand along the magnetic field lines, while the non-magnetized ions, driven by the resulting electric field, follow the electrons closely along the magnetic field lines [49]. Meanwhile, the neutral particles, lacking any charge, are unaffected by the magnetic field and expand into free space. Inside the source (Figure 7), the electrons also follow the magnetic field topology closely. The magnetic cusp regions show enhanced plasma transport to the walls [4], causing increased local plasma losses. The neutral density increases near the walls, showing that plasma recombination into neutrals at the walls dominates the neutral chemistry in this region, which is in agreement with other model and experiment findings [39]. In the bulk of the plasma near the axial center line, ionization dominates and the neutral density decreases. The peak neutral density is located at the magnetic cusp impingement sites on the wall, where the neutral density increases to a few percentage points above its original injected density through recombination, confirming the increased local plasma loss areas. The source results are qualitatively consistent with measurements performed on helicon plasma sources [62,63].

The area-averaged profiles of the charged species densities are shown in Figure 8, demonstrating near-continuity across the regime transition along the thruster’s axial length. At the outlet, which separates the fluid and kinetic domains, discontinuities between the models in predicted ion and electron densities are kept below 2–$5\%$.

4.2. Thrust Predictions

The predicted thrust from MUPETS at various power levels is reported in

Table 1. Propulsive performance metrics.

Power [W]	Thrust [mN]	Error [%]	${\mathit{I}}_{\mathit{sp}}$ [s]	$\mathit{\eta }$ [%]
15	0.17	20	117	0.7
30	0.45	0.3	308	2.3
45	0.64	6	436	3.1
57	1.01	16	689	6.0

, together with the specific impulse and thrust efficiency obtained through $I_{sp}={\displaystyle \frac{T}{g_0\dot{m}}}$ and $\eta ={\displaystyle \frac{T^2}{2\dot{m}{P}_{in}}}$, respectively. The table also includes the error of the prediction with respect to the experimental trend, calculated as follows:

$$Error={\displaystyle \frac{|T_{sim}-T_{exp,trend}|}{T_{exp,trend}}}·100\%.$$

(11)

The thrust is seen to increase almost linearly with the power, consistent with ionization ratios below $10\%$ [29]. The measured propulsive performance across the tested low-power ($Pw\lesssim 100$ W) range is in line with that of commercial low-power RF APTs [13,64]. The thrust predictions were compared to the experimental measurements [3] in Figure 9. Initially, MUPETS underpredicts the thrust, similar to earlier separated approaches [65]. However, iterative corrections to the plasma flow through the thruster yield increased thrust, aligning more closely with the experimental trends. Overall, MUPETS demonstrates self-corrective behavior at all power levels. It occasionally overshoots the experimental trendline (57 W) at high power levels, while at lower power levels it exhibits a more steady but slow incremental increase towards the experimental fit (15 W). The coupling strategy appears to work best for the middle-power range, as for the simulations at 30 W and 45 W the initial large errors (≈50%) in thrust prediction are reduced to just 0.3% and 6%, respectively.

4.3. Convergence

To showcase the convergence of the iterative loop, the differences between iterations in the cross-sectional averaged electron densities are plotted along the axial direction in Figure 10, together with the maximum difference after each iteration. Here, the difference is calculated as follows:

$$diff={\displaystyle \frac{\int n_e{(z,r)}^{h}rdr}{\int rdr}}-{\displaystyle \frac{\int n_e{(z,r)}^{h-1}rdr}{\int rdr}}.$$

(12)

The process converges rapidly; after the second iteration, the electron density profile already differs less than $10\%$ everywhere along the thruster’s longitudinal axis, meeting the convergence criteria set out in Section 2.2. The largest differences exist in the kinetic plume.

5. Conclusions

A self-consistent coupling method between fluid and Particle-In-Cell (PIC) solvers has been developed and integrated into a multi-regime plasma model, namely, the MUlti-regime Plasma Equilibrium Transport Solver (MUPETS), to simulate plasma transport and propulsion performance in an Ambipolar Plasma Thruster (APT). MUPETS controls two distinct plasma transport models: a fluid model based on the plasmaFoam code [40,43], and a kinetic model based on the Starfish code [29,45], running them iteratively to self-consistently solve the full thruster domain that includes both the fluid and kinetic regions. Information exchange between the models at the regime boundary ensures continuity, with the fluid model providing plasma species densities, velocities, and temperature to the kinetic model and the kinetic model supplying the electric field to the fluid model.

The MUPETS coupling strategy shows significant accuracy when validated against experimental thrust data, with discontinuities in the predicted ion and electron densities at the outlet separating the fluid and kinetic domains minimized to 2–$5\%$. The prediction error fell between $0.3\%$ and $20\%$ for the tested Xenon-fed laboratory thruster [3] operating between 15 W and 57 W, which is well within the measurement’s uncertainty band.

The MUPETS code and its coupling methodology present improved predictions of a thruster’s propulsive performance without the need for the high computational cost incurred by simulating the source chamber region through PIC. The proposed methodology is applicable for further studies in electric propulsion and other plasma applications involving multiple regimes. It imposes no additional assumptions on plasma properties at the coupling interface beyond current-free flow; in addition, its relevance extends to various solvers, making it adaptable to numerous cases and codes beyond those considered in this study.