Latest progress in Hall thrusters plasma modelling

Abstract

In the last 30 years, numerical models have revealed different physical mechanisms involved in the Hall thruster functioning leading to a bridge between analytical prediction/empirical intuition and experiments. For this reason, the need for a model to study Hall thruster operation continues to increase. Two basic approaches exist: one based on fluid/hybrid simulation where the velocity distribution of electrons is prescribed and the plasma inside the thruster, considered as quasineutral, is described with macroscopic quantities (density, velocity and energy), with unmagnetized ions being considered as collisionless; the second approach is based on a kinetic description for charged particles where no approximations are made regarding their velocity distributions. Fluid or hybrid approaches offer the advantages of computational efficiency with modest hardware requirements. They are very useful to perform parametric studies but actually the anomalous phenomena believed to be responsible for electron transport across the magnetic field barrier have not been self-consistently modeled using a fluid approach. A kinetic approach is able to better capture phenomena originating on the Debye scale length like the lateral sheaths, E × B electron drift instability, and it is important to explain the anomalous electron transport, but kinetic simulations require very long run times. For the latter, the advances in computer hardware over the past years have allowed researchers to perform simulations under conditions closer and closer to the actual thruster operation. In this review, we will present two approaches, with emphasis on numerical schemes used with assumptions and approximations and the main results obtained. Future directions in the Hall thruster modeling will finally be outlined.

Appendices

Appendix 1: Secondary electron emission models

Secondary electron emission (SEE) induced by electron impact on dielectric surfaces is a complex phenomenon which involves elastic and inelastic scattering of electron in its transport inside the material. The emission of secondaries can also involve not only the primary electron (coming from the plasma) itself, but by an electron cascade process also electrons belonging to the material. Due to the fast timescale of the process, its implementation in PIC model can be done by a phenomenological approach. It is important to note that in PIC models the secondary electrons are generated instantaneously when a primary electron hits the surface. This assumption is well justified, since the time lag of secondary emission is estimated to be 10⁻¹³ to 10⁻¹⁴ s, i.e., much shorter than the timestep used for PIC relevant to HT plasma parameters. The phenomenological models of SEE are based on the two main quantities representing SEE (Villemant et al. 2017): (1) the yield SEY σ, number of electrons emitted per incident electron; (2) energy and angular spectrum of secondaries emitted d²σ/dEdΩ. Both are mainly functions of the incident electron energy E_p, angle of impact θ_p, wall temperature T_w and electron irradiation (aging).

The dependence of the yield from the primary electron energy follows an universal (the same for all materials) behavior that can be represented by five parameters: (1) σ₀ value of SEY at E_p = 0; (2) maximum value of SEY σ_max; (3–4) first and second crossover energy E* and E** corresponding to the lower and higher, respectively, primary energy giving σ(E_p) = 1; (5) incident electron energy corresponding to the maximum yield E_max. The following order always fulfills: E* < E_max < E**. For HT typical regimes (E_p < 1 keV) and dielectric materials used, only two parameters are important, σ₀ and E* (see Fig. 21a, b and Table 3). For a given primary energy, SEY increases with increasing angle of incidence θ_p. Concerning the wall temperature and electron irradiation dependences, σ₀ decreases while E* increases with electron irradiation and T_w (Tondu et al. 2011; Belhaj et al. 2015), making the SEE process more negligible for hotter and highly irradiated surfaces.

Fig. 21

a Total SEE yields as a function of primary electron energy (Tondu et al. 2011) for some HT wall materials with b zoom view around the first crossover energy (indicated with arrows). c Components of SEY as a function of impact electron energy and d energy spectrum dσ/dE of secondary electrons emitted by a beam of electrons with energy E_p = 50 eV. The energy range of the three different secondary populations, (1) elastic backscattered, (2) true secondaries and (3) inelastic backscattered, is evident. SEYs and energy spectrum are computed using the Furman and Pivi (2002) model for Al₂O₃ surface

Table 3 Typical values of SEY at E_p = 0 and first crossover energy for wall materials relevant for HTs (Barral et al. 2003; Dunaevsky et al. 2003; Tondu et al. 2011; Villemant et al. 2017)

Concerning the energy spectrum of the emitted electrons (see Fig. 21c, d), three different populations can be distinguished: (1) high-energy electrons corresponding to primary electrons backscattered from the surface (their energy is slightly below the incident electron energy); (2) true secondary electrons belonging to the material and representing the low-energy part of the spectrum; (3) primary electrons diffused inside the material and having suffered inelastic collisions (their energy range is between the true secondaries and the peak of backscattered electrons). The yield of each secondary electron has a proper behavior as a function of the incident energy. The backscattering and inelastic SEYs σ_e and σ_r grow with the decrease of E_p, while the yield of true secondary electrons σ_ts decreases and reaches zero at an energy of about the width of the potential gap between vacuum and the upper level of the valence band. Therefore, the total yield σ = σ_e + σ_r + σ_ts could have a distinguishable minimum in the low-energy region (E_p < 10 eV). The angular spectrum of emitted electrons shows always an isotropic distribution over the azimuthal angle θ, while the polar angle φ has a cosine (Lambertian) distribution for true secondaries and a more complex distribution depending of the angle of impact for the elastic and inelastic backscattered electrons.

Depending on the number of parameters used, three different phenomenological models have been proposed:

• 1. Linear law model

  It represents the simplest model where only σ₀ and E* are used to represent the total yield following a linear relation

  $$ \sigma \left( E \right) = \sigma_{0} + \frac{E}{{E_{*} }}\left( {1 - \sigma_{0} } \right). $$

  (40)

  All electrons are emitted with a half-Maxwellian distribution with T_see = 2 eV. Their angular distribution is isotropic over the azimuthal angle θ and cosine law over the polar angle φ, and independent of the primary electron angle of incidence.

• 2. Modified Vaughan model

  It was proposed by Sydorenko et al. (2006) and makes use of the Vaughan (1989) fitting formula

  $$ \sigma_{\text{Vaug}} \left( {E,\theta } \right) = \sigma_{ {\max} } \left( \theta \right)\left[ {v\left( {E,\theta } \right){\text{e}}^{{1 - v\left( {E,\theta } \right)}} } \right]^{k} , $$

  (41)

  where $v\left( {E,\theta } \right) = \frac{{E - E_{0} }}{{E_{{\max} } \left( \theta \right) - E_{0} }}$, $E_{{\max} } \left( \theta \right) = E_{{\max} ,0} \left( {1 + \frac{k}{\pi }\theta^{2} } \right)$, $\sigma_{{\max} } \left( \theta \right) = \sigma_{{\max} ,0} \left( {1 + \frac{k}{\pi }\theta^{2} } \right)$ and $k = \left\{ {\begin{array}{*{20}l} {0.62} \hfill & {\quad E < E_{{\max} } } \hfill \\ {0.25} \hfill & {\quad E > E_{{\max} } } \hfill \\ \end{array} } \right.$, to define the partial yields for the different secondaries:

  $$ \sigma_{\text{e}} \left( {E,\theta } \right) = r_{\text{e}} \sigma_{\text{Vaug}} \left( {E,\theta } \right) + \sigma_{{{\text{e}},{\max} }} \left\{ {\begin{array}{*{20}l} {v_{1} \left( E \right){\text{e}}^{{1 - v_{1} \left( E \right)}} } \hfill & {\quad E_{e,0} < E < E_{{{\text{e}},{\max} }} } \hfill \\ {\left[ {1 + v_{2} \left( E \right)} \right]{\text{e}}^{{ - v_{2} \left( E \right)}} } \hfill & {\quad E > E_{{{\text{e}},{\max} }} } \hfill \\ \end{array} } \right., $$

  (42.a)
  $$ \sigma_{\text{r}} \left( {E,\theta } \right) = r_{\text{r}} \sigma_{\text{Vaug}} \left( {E,\theta } \right), $$

  (42.b)
  $$ \sigma_{\text{ts}} \left( {E,\theta } \right) = \left( {1 - r_{\text{e}} - r_{\text{r}} } \right)\sigma_{\text{Vaug}} \left( {E,\theta } \right), $$

  (42.c)

  where $v_{1} \left( E \right) = \frac{{E - E_{{{\text{e}},0}} }}{{E_{{{\text{e}},{\max} }} - E_{{{\text{e}},0}} }}$ and $v_{2} \left( E \right) = \frac{{E - E_{{{\text{e}},{\max} }} }}{\Delta }$. The energy spectrum of secondaries is prescribed as follows: the backscattered electrons keep the primary energy, the energy of inelastically backscattered electrons is considered to be uniformly distributed between zero and the energy of the incident electron and the true secondaries are emitted with a half-Maxwellian distribution with T_see = 2 eV. The same angular spectrum of the linear model is used.The total number of parameters used is 9 and their values for BN are reported in Table 4.

  Table 4 Parameters used in the modified Vaughan model for BN material

• 3. Furman–Pivi modelIt represents the most sophisticated SEE model able to fit in detail the three partial SEY behaviors as a function of impact energy and angle (with the possibility of emission of n > 1 secondaries) (Furman and Pivi 2002). The three yields for normal incidence (in Fig. 21b their behavior for BN in the energy range 0-100 eV is reported) are

  $$ \sigma_{\text{e}} \left( {E,0} \right) = \sigma_{{{\text{e}},\infty }} + \left( {\sigma_{0} - \sigma_{{{\text{e}},\infty }} } \right){\text{e}}^{{ - E/E_{\text{e}} }} , $$

  (43.a)
  $$ \sigma_{\text{r}} \left( {E,0} \right) = \sigma_{{{\text{r}},\infty }} + \left( {1 - {\text{e}}^{{ - E/E_{\text{r}} }} } \right), $$

  (43.b)
  $$ \sigma_{\text{ts}} \left( {E,0} \right) = \frac{{\sigma_{{\max} } sE^{\prime}}}{{\left( {s - 1 + E^{{\prime {\text{s}}}} } \right)\left( {1 - \sigma_{\text{e}} \left( {E,0} \right) - \sigma_{\text{r}} \left( {E,0} \right)} \right)}}, $$

  (43.c)

  where E′ = E/E_max. The incident-angle dependence is implemented assuming the same form for all three components of SEY. Specifically, for the backscattered and redifussed components,

  $$ \sigma_{\text{e}} \left( {E,\theta } \right) = \sigma_{\text{e}} \left( {E,0} \right)\left[ {1 + e_{1} \left( {1 - \cos {\text{e}}^{{e_{2} }} \theta } \right)} \right], $$

  (44.a)
  $$ \sigma_{\text{r}} \left( {E,\theta } \right) = \sigma_{\text{r}} \left( {E,0} \right)\left[ {1 + r_{1} \left( {1 - \cos {\text{e}}^{{r_{2} }} \theta } \right)} \right], $$

  (44.b)

  and for the true-secondary component

  $$ \sigma_{\text{ts}} \left( {E,\theta } \right) = \sigma_{\text{ts}} \left( {E,0} \right)\left[ {1 + t_{1} \left( {1 - \cos {\text{e}}^{{t_{2} }} \theta } \right)} \right], $$

  (44.c)
  $$ E^{\prime}\left( {E,\theta } \right) = E^{\prime}\left( {E,0} \right)\left[ {1 + t_{3} \left( {1 - \cos {\text{e}}^{{t_{4} }} \theta } \right)} \right]. $$

  (44.d)

The emitted energy (see Fig. 21c) and angular spectrum are also functions of impact energy and angle and their calculation requires a complex implementation [see Furman and Pivi (2002) for details]. This precision is at the expense of the computational complexity and the high number of free parameters (42) necessary to implement the model and that makes it not very useful for a parametric study of the effect of the walls on the plasma behavior inside HT.

Appendix 2: Ion erosion model

One critical issue of Hall thruster is the wall erosion since with the existence of a radial electric field inside the channel, ions generate downstream interact with walls (see Sect. 2.5). The erosion rate in eroded thickness by units of time can be expressed as:

$$ R = \frac{{m_{\text{w}} }}{{{\mathcal{N}}_{\text{a}} \rho_{\text{w}} }}{{\varGamma }}_{i, \bot } Y\left( {\varepsilon_{{i,{\text{w}}}} ,\theta_{{i,{\text{w}}}} } \right), $$

(45)

where the properties of the wall materials are characterized through mass $m_{\text{w}}$ and mass density $\rho_{\text{w}}$, ${\mathcal{N}}_{\text{a}}$ is the Avogadro’s number, and ${{\varGamma }}_{i, \bot }$ is the incident ion flux including multi-charged ions. The contribution of unionized propellant atoms on sputtering processes is found to be minor (Sommier et al. 2005). The function Y is the sputtering yield (number of atoms sputtered by incident ions) and depends on the ion impact energy $\varepsilon_{{i,{\text{w}}}}$ and angle $\theta_{{i,{\text{w}}}}$. The energy of ions impinging the walls is the sum of the ion energy gained in the plasma before the sheath entrance and a supplementary energy resulting from the sheath potential drop (see Sect. 2.3). Obviously, for fluid description of ions, additional assumption about the ion energy distribution at the sheath entrance must be done. The function Y can be a complex function depending on energy and angle and is most of time split into two separated functions:

$$ Y\left( {\varepsilon_{{i,{\text{w}}}} ,\theta_{{i,{\text{w}}}} } \right) = Y_{\varepsilon } \left( {\varepsilon_{{i,{\text{w}}}} } \right)Y_{\theta } \left( {\theta_{{i,{\text{w}}}} } \right), $$

(46)

where $Y_{\varepsilon }$ determines the energy dependence at normal incidence and $Y_{\theta }$ acts as a correcting factor to account for angle incidence effect. There is a large uncertainty associated with the sputtering yield for energies of interest in the context of Hall thrusters (between few tens to few hundreds of eV). Most of time semi-analytical laws coming from the original work of Yamamura and Tawara (1996) for monoatomic solids are used in which parameters are adjusted to fit measurements of sputtering yield at normal incidence for high-energy ions. For BN material, measurements are taken from works of Garnier et al. (1999) and Yalin et al. (2007). Figure 22 shows the $Y_{\theta }$ and $Y_{\varepsilon }$ analytical fitting curves in the context of the study of the 6 kW H6 thruster with BN walls. The Yamamura and Tarawa formulation has been completed with additional functions based on works of Bohdansky (1984) where an additional parameter linked to the sputtering energy threshold $Y_{{\varepsilon ,{\text{th}}}}$(unknown) appears. In sputtering models, under the $Y_{{\varepsilon ,{\text{th}}}}$ limit, $Y_{\varepsilon } = 0$. Mikellides et al. found that $Y_{{\varepsilon ,{\text{th}}}}$ taken in the 25–50 eV range is a reasonable choice to match measurements at high energy (Mikellides et al. 2014a). In the study of the SPT100 with BNSiO₂ wall material, Garrigues et al. have fixed the unknown sputtering energy threshold between 30 and 70 eV (Garrigues et al. 2003).

Fig. 22

Analytical fitting curves (left) angular, (right) energy dependencies of sputtering yield for BN walls. $f1_{\text{Y}}$ and $f2_{\text{Y}}$ fitting functions assuming a sputtering energy threshold of 25 eV and 50 eV, respectively (from Mikellides et al. 2014a)

An extensive review on ion erosion models can be found in Boyd and Falk (2001).

Appendix 3: Poisson equation solvers

Usually in HT PIC models, the electrostatic approximation is used: the current density involved (order of 1000 Am⁻²) is quite low to change the externally imposed magnetic field. The generalized form of Poisson equation which can take into account the change of the dielectric permittivity across the boundary between plasma channel and lateral dielectric surfaces takes the following form:

$$ \nabla \, \cdot \,\left( {\epsilon \nabla \phi } \right) = - \rho . $$

(47)

After discretization onto a finite differenced spatial grid, it can be in essence transformed to a system of linear equations

$$ \varvec{A}\phi = \varvec{b}, $$

(48)

where A is a sparse matrix (most of its elements are zeros) containing the information on the geometry used and of the boundary conditions. The latter can be of two types: Dirichlet conditions (fixed electric potential) as at the anode and cathode location and Neumann conditions (fixed electric potential derivative) as at the outflow in the plume region. The different methods used to solve Eq. (48) can be divided into two groups:

• iterative methods (successive overrelaxation, conjugate gradients, multigrid) obtain the solution through a series of iterative steps which decrease the error in the estimated solution; these methods need an initial estimation and the solution from the previous time step is used as initial estimation of the new solution in the current timestep; this makes the performance of these methods strongly dependent on fluctuations of the density between successive time steps and therefore on cell size and time step used;

• direct methods (Thomas tridiagonal, cyclic reduction, fast Fourier transform, LU decomposition) reach the solution in a single step based on the actual factorization of the matrix A.

While the first methods are easily parallelized and vectorized, the second ones are often much faster but require substantially more computational resources than iterative methods. The best compromise for very large grid nodes (N_g > 10⁶) is represented by combinations of iterative and direct solver algorithms (Pekárek and Hrach 2006).

Nowadays, different numerical free software package are available as Poisson’s equation solver: FISHPACK, HYPRE, MEMPS, PARDISO, PETSc, SuperLU, UMFPACK and WSMP.