Effect of anisotropic non-Maxwellian electron distribution function on plasma-wall interaction in Hall thrusters

Plasma-wall interaction (PWI) is always the research focus among the Hall thruster community due to its recognized effects on thruster discharge and performance [1–6]. Commonly, the electron distribution function (EDF) is assumed Maxwellian to investigate the PWI problem inside the thruster channel with either fluid [1–4, 7, 8] or hybrid particle-in-cell (PIC) [9, 10] approaches; however, the numerical outcomings [2, 3] are found to present a significant disparity in quantity with the experimental results [4].

The disparity is attributed to the inappropriate assumption of EDF. As the experiments in the early stage have shown, the EDF in Hall thrusters deviates greatly from the Maxwellian form due to the electron-wall interation [11]. Recent kinetic simulations further indicate that the EDF is anisotropic with its normal-to- the-wall component depleted at high energies [6, 12]. Such a depletion of EDF has also been detected in other low-pressure gas discharge devices [13,14]. Therefore, the Maxwellian assumption on EDF is improper to study the PWI problem in Hall thrusters. To evaluate the non-Maxwellian EDF effect, a comparative research with a kinetic model has been made in this letter as shown below.

Different attempts to simulate the non-Maxwellian behavior in Hall thruster discharge have been done [15–17]. Kaganovich et al. have calculated the influence of non-Maxwellian EDF on PWI in Hall thrusters in detail with a 1D (radial) fully kinetic PIC code [15]; the analytical formulas for calculating the plasma flux to the wall, secondary electron flux, plasma potential, and electron cross-field conductivity have been further derived. However, that 1D kinetic model cannot represent the 2D spatial structure of the sheath near the channel wall. Taccognaet al. have focused the 2D (radial and azimuthal) non-Maxwellian behavior on electron anomalous conductivity and instability [16,17], but it seems that they have not stressed the PWI problems sufficiently. Recently, Yu et al. find that the sheath potential oscillates spatially and temporally in the direction parallel to the wall when the temperature of incident electrons with an isotropic Maxwellian distribution is bigger than a critical value (∼27 eV) [18]. It is further illustrated that the anisotropy of electrons with Maxwellian distribution can lower the critical value to about 18 eV [19]. Therefore, it is preferred to study the effect of non-Maxwellian EDF on PWI in the scope of 2D dynamic sheath as is done in this letter.

The kinetic model that characterizes the 2D dynamic sheath has been addressed in detail elsewhere [18]. In the model, ions are regarded as a fixed background since the propellant (usually xenon) atom mass are much greater than electron mass; the ion velocity is set equal to Bohm velocity; electrons are assumed unmagnetized since their Larmor radius is much larger than the sheath thickness (on the order of Debye length λ_D ) for the parameter scale involved in Hall thrusters.

The simulation domain consists of one lateral wall and part of the radial (r) and azimuthal (z) cross-section of Hall thruster channel, as shown in fig. 1. It is assumed that no variation exists along the axial direction. The azimuthal geometry is further simplified as planar instead of annular. The domain dimensions are 0 ⩽z ⩽ 200λ_D and r₀ ⩽r ⩽ r₀ + 10λ_D , where r₀ = 0.035 m . Ions and electrons are generated on the plasma-sheath boundary where the electric potential Φ = 0 and a small quasi-neutral region with a thickness of 3λ_D is imposed. Different from ref. [18,19], the bilateral periodic boundaries are set more reasonable with $\Phi \left( {r,z = 0} \right) = \Phi \left( {r,z = 200\lambda _D } \right)$. Both ions and electrons are deposited on the wall boundary, and the wall potential is determined by ∂Φ /∂r = σ /2ε₀ , where σ is the surface charge density and ε₀ is the vacuum permittivity [20].

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Secondary electron emission (SEE) from the wall is the key element to study the PWI problem. Here, as the consequence of one incident electron hitting on the wall, four events are considered: 1) absorption of the incident electron into the wall (in which case the wall acquires the electron's negative charge); 2) elastic scattering of the incident electron by the wall (namely primary backscattering); 3) emission of one secondary electron from the wall; 4) emission of two secondary electrons from the wall. The exact numerical description for these four events can be found in ref. [18]. This SEE model is an improved one from that proposed originally by Morozov [20], in which EVENT 2) is not considered. Recent experimental study shows that EVENT 2) becomes dominant as the electron energy becomes low [21]. It has been demonstrated that the improved model has a better accordance with the experimental data than Morozov's model does [18]. On the other side, ions striking on the wall are totally absorbed with no SEE at all. The emitted secondary electrons are further assumed obeying a half-Maxwellian distribution with a temperature of 1/3 times of the incident electron temperature.

The non-Maxwellian distribution of incident electrons adopted here (see fig. 2(a)) originates from ref. [15]. It can be seen that the high-energy tail is present for the energy distribution F(ε_∥) parallel to the wall but it is missed for the energy distribution F(ε_⊥) normal to the wall. Using a curve-fitting method, the corresponding analytical expressions of F(ε_⊥) and F(ε_∥) are

$$\begin{eqnarray} F\left( {\varepsilon _{\bot} } \right) &=& \sum\limits_{i = 1}^3 {a_i {\rm exp}\,\left[ { - \left( {\left( {\varepsilon _{\bot} - b_i } \right)/c_i } \right)^2 } \right],} \nonumber\\ F\left( {\varepsilon _\parallel } \right) &=& 0.{\rm{73}}0{\rm{6exp}}\left( { - 0.0{\rm{6257}}\varepsilon _\parallel } \right) \nonumber\\ &&+ 0.0{\rm{8121exp}}\left( {0.00{\rm{3214}}\varepsilon _\parallel } \right), \end{eqnarray} \tag{ 1 }$$

where, a₁ = 0.11 , a₂ = - 0.302 , a₃ = 1.137 ; b₁ = 17.83 , b₂ = 5.455 , b₃ = 0.912 ; c₁ = 3.11 , c₂ = 4.79 , c₃ = 11.71 . The averaged electron kinetic energies $\bar \varepsilon _{\bot}$ and $\bar \varepsilon _{||}$ are then integrated as

$$\begin{eqnarray*}&&\bar \varepsilon _{\bot} = {{\int_0^{40} {F\left( {\varepsilon _{\bot} } \right)\varepsilon _{\bot} {\rm{d}}\varepsilon _{\bot} }}\bigg/{\int_0^{40} {F\left( {\varepsilon _{\bot} } \right){\rm{d}}\varepsilon _{\bot} } }} = {\rm{7}}{\rm{.7913}}\,{\rm{eV,}} \end{eqnarray*} \noindent$$

$$\begin{eqnarray*}&&\bar \varepsilon _\parallel = {{\int_0^{40} {F\left( {\varepsilon _\parallel } \right)\varepsilon _\parallel {\rm{d}}\varepsilon _\parallel } }\bigg/{\int_0^{40} {F\left( {\varepsilon _\parallel } \right){\rm{d}}\varepsilon _\parallel } }} = {\rm{14}}{\rm{.3737}}\,{\rm{eV,}} \end{eqnarray*}$$

which are found close to the corresponding values in ref. [15]. Let $\varepsilon _0 = 2\bar \varepsilon _{\bot} = {\rm{15}}{\rm{.5826}}\,{\rm{eV}}$, the normalized electron energy distributions $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over F} (\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \varepsilon } _{\bot} )$ and $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over F} (\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \varepsilon } _{\Vert} )$ can be obtained as

$$\begin{eqnarray} &&\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over F} \left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \varepsilon } _{\bot} } \right) = \frac{{F({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \varepsilon } _{\bot} \varepsilon _0 })}}{{\int_0^{40} {F\left( {\varepsilon _{\bot} } \right){\rm{d}}\varepsilon _{\bot} } }}\varepsilon _0,\quad\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \varepsilon } _{\bot} = \varepsilon _{\bot} /\varepsilon _0 , \nonumber \\ && \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over F} \left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \varepsilon } _\parallel } \right) = \frac{{F({\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \varepsilon } _\parallel \varepsilon _0 })}}{{\int_0^{40} {F\left( {\varepsilon _\parallel } \right){\rm{d}}\varepsilon _\parallel } }}\varepsilon _0,\quad\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over \varepsilon } _\parallel = \varepsilon _\parallel /\varepsilon _0,\end{eqnarray} \tag{ 2 }$$

which are further depicted in fig. 2(b). In the simulation, we keep the normalized EDFs unchanged varying $\bar \varepsilon _{\bot}$ and $\bar \varepsilon _{\parallel}$. Some other simulation parameters are set as follows. The size of the meshes that discretize the region is 0.5λ_D × 0.5λ_D . The time step Δt = 0.1ω_ep^{- 1} , where ω_ep is the electron plasma frequency. The electron density on the sheath boundary is n_e = 1 × 10¹⁸ /m³ .

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By implementing the model described above, it is found that the sheath have already turned into the 2D dynamic regime when the EDF follows eq. (2) with both $\bar \varepsilon _{\bot}$ and $\bar \varepsilon _{\parallel}$ only 2.5 eV, as shown in fig. 3. It can be seen that the uniformity of potential distribution in the direction parallel to the wall is destroyed and the spatial oscillation of sheath is raised up. Taking into account that the minimum electron temperature for the appearance of spatial oscillation of the sheath is 27 eV in the isotropic Maxwellian case [18] and 18 eV in the anisotropic Maxwellian case [19], it can be concluded that it is the non-Maxwellian distribution of incident electrons that triggers the dynamical behavior of the sheath.

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To compare the effect of non-Maxwellian EDF with that of Maxwellian EDF on PWI, it is chosen three specific EDFs for simulation, namely, 1) the anisotropic non-Maxwellian EDF as shown in fig. 2(a) with $\bar \varepsilon _{\bot} = 7.79\,{\rm{eV}}$ and $\bar \varepsilon _{\parallel} = 14.37\,{\rm{eV}}$ according to ref. [15], 2) the anisotropic Maxwellian EDF with electron temperature $T_{e{\bot}} = 2\bar \varepsilon _{\bot} = {\rm{15}}.{\rm{58}}\,{\rm{eV}}$ and $T_{e \parallel} = 2\bar \varepsilon _{\parallel} = {\rm{28}}.{\rm{74}}\,{\rm{eV}}$, and 3) the isotropic Maxwellian EDF with electron temperature T_e = (T_e∥ + 2T_e⊥)/3 = 24.35 eV. These values are chosen such that the integrated electron temperatures are equivalent for the three cases.

As only those electrons with energies greater than the sheath potential drop can reach the wall, it is essential to calculate the wall potential Φ_w , the electron flux to the wall Γ_w1, and the electron energy flux to the wall S_w1 . The SEE out of the wall is believed to enhance the electron cross-field conductivity in the discharge channel [2], it is thus necessary to calculate the electron-wall collision frequency ν_ew , the SEE flux Γ_w2 and the SEE energy flux S_w2 . The SEE coefficient $\bar \sigma$ (=Γ_w2 /Γ_w1 ) and the parameters reflecting the electron energy deposition on the wall Λ_w (=(S_w1-S_w2 )/S_w1 ) are also calculated. Furthermore, to evaluate the cooling effect of SEE on plasma bulk, the incident electron energy flux S₀₁ and SEE energy flux S₀₂ , as well as the ratio Λ₀ (=(S₀₁-S₀₂)/S₀₁), on the plasma-sheath boundary are calculated. All the results are listed in table 1.

Table 1:. Calculated parameters on plasma-wall interaction with the three EDFs.

Parameters	Anisotropic non-Maxwellian	Anisotropic Maxwellian	Isotropic Maxwellian
$\bar \varepsilon$, Te (eV)	$\bar \varepsilon _{\bot} = 7.79$, $\bar \varepsilon _{\parallel} = 14.37$	Te⊥=15.58 , Te∥=28.74	Te=Te⊥=Te∥=24.35
Φw	2D dynamic regime	−46.08 V ( - 2.96 Te⊥)	−57.91 V (−2.38Te⊥ )
$\Gamma _{w2} \left( {{\rm{m}}^{ - {\rm{2}}} \,{\rm{s}}^{ - {\rm{1}}} } \right)$	9.61×1021	3.36×1022	7.68×1022
$\Gamma _{w2} \left( {{\rm{m}}^{ - {\rm{2}}} \,{\rm{s}}^{ - {\rm{1}}} } \right)$	6.42×1021	3.02×1022	7.26×1022
νew	4.59×105	2.4×106	5.49×106
$\bar \sigma$	0.67	0.9	0.95
$S_{w1} \left( {{\rm{J}}\;{\rm{m}}^{ - {{2}}} \,{\rm{s}}^{ - {1}}} \right)$	4.02×105	5.19×105	1.14×106
$S_{w2} \left( {{\rm{J}}\;{\rm{m}}^{ - {\rm{2}}} \,{\rm{s}}^{ - {\rm{1}}} } \right)$	2.49×105	1.77×105	3.67×105
Sw1 - Sw2	1.53×105	3.42×105	7.73×105
Λw (%)	38.06	65.9	67.81
S01 (J m- 2 s- 1 )	3.63×106	4.70×106	6.44×106
S02 (J m- 2 s- 1 )	3.55×106	4.59×106	6.19×106
S01 - S02	8×104	1.10×105	2.50×105
Λ0 (%)	2.20	2.34	3.88

It is not surprising that the sheath oscillates temporally and spatially in the specified anisotropic non-Maxwellian case, as shown in fig. 4. It is known that the sheath is formed to balance the ion flux with the electron flux. As the anisotropic non-Maxwellian EDF holds the most proportion of low-energy electrons among the three EDFs, a smallest sheath potential drop can prevent enough incident electrons to the wall when the ion flux is fixed; therefore, one can find in fig. 4(a) that the potential drop |Φ_w | is much lower than those in the Maxwellian cases. On the other hand, the classical theory have indicated that the sheath potential drop |Φ_w | increases as the SEE coefficient $\bar \sigma$ decreases [22]. However, as one can find in table 1, the SEE coefficient $\bar \sigma$ is the smallest in the non-Maxwellian case where the sheath potential drop is also the lowest; therefore, the classical theory is not valid to analyse the non-Maxwellian situation. The depletion of the high-energy tail of the EDF in the direction normal to the wall, and consequently the fewest energetic electrons knocking on the wall, in the anisotropic non-Maxwellian case is the main cause for the lowest SEE coefficient $\bar \sigma$.

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Commonly, on the basis of the same electron temperature, the higher the sheath potential drop |Φ_w | is, the more incident electrons are reflected back, and the less incident electrons reach the wall. However, the calculation here shows that the electron flux to the wall Γ_w1 in the isotropic Maxwellian case is two times bigger than that in the anisotropic Maxwellian case although |Φ_w | in the isotropic Maxwellian case is higher than that in the anisotropic Maxwellian case (see fig. 4(a)). The reason for such a contradiction is that Γ_w1 depends on the temperature component normal to the wall rather than the integrated temperature. It can be deduced that Γ_w1 increases as the normalized sheath potential drop |Φ_w |/T_e⊥ decreases; from this point of view, the calculated result is reasonable according to the calculated |Φ_w |/T_e⊥ shown in fig. 4(b). The same principle can be used to understand the remarkable enhancement of the electron energy flux to the wall S_w1 in the isotropic Maxwellian case compared with that in the anisotropic Maxwellian case. Note that according to the analysis above, Γ_w1 was supposed to be the largest in the anisotropic non-Maxwellian case since |Φ_w |/T_e⊥ is the lowest; however, Γ_w1 is actually the smallest. It is obvious that the fewest energetic electrons reaching the wall in the anisotropic non-Maxwellian case accounts for that confliction. Consequently, the electron-wall collision frequency ν_ew is the smallest.

A further comparative study has been made to find out whether the electron flux to the wall Γ_w1 under non-Maxwellian EDF is reduced due to the shape of the EVDF only or due to the oscillations as well. In the presence of sheath oscillations, our simulation shows that the average wall potential $\bar \Phi _w = - {\rm{25}}\,{\rm{V}}$ (see fig. 4(a)) and the corresponding Γ_w1 = 9.61 × 10²¹ m⁻² s ⁻¹ . Based on the average wall potential $\bar \Phi _w$ and the same non-Maxwellian EDF, the electron flux $\bar \Gamma _{w1}$ in the absence of sheath oscillation is calculated analytically, which is 6.06 × 10²¹ m⁻² s⁻¹. One can see that the electron flux under the oscillation regime is obviously larger than that under the steady-state regime. Therefore, the electron flux to the wall is reduced due to the shape of the EVDF only, and the sheath oscillation can increase the electron flux up to 60%. It can be understood as follows. Compared with the steady-state sheath, the spatial oscillation of the sheath gives a chance to those electrons with relatively low energies (lower than $- e\bar \Phi_w$) to reach the wall and can also repel those electrons with relatively high energies (bigger than $- e\bar \Phi _w$) away from the wall. As the low-energy electron population is remarkably greater than the high-energy electron population for a non-Maxwellian EDF, a greater net electron flux on the wall is obtained.

As SEE flux usually increases with the electron flux to the wall, it is not surprising to find that the SEE flux Γ_w2 in the anisotropic non-Maxwellian case is the smallest among the three cases; as a result, the electron cross-field conductivity due to SEE is the weakest. Furthermore, as the majority of incident electrons in the anisotropic non-Maxwellian case have low energies, they have the greatest probability to be scattered elastically on the wall (EVENT 2)) with no energy loss; consequently, the integrated energy of secondary electrons is the greatest among the three cases. Therefore, one can find in table 1 that the SEE energy flux S_w2 in the isotropic Maxwellian case is only 1.5 times larger than that in the anisotropic non-Maxwellian case, even though the corresponding difference of SEE flux Γ_w2 is nearly three times as large as that in the anisotropic non-Maxwellian case. Consequently, the ratio of electron energy flux Λ_w deposited on the wall in the anisotropic non-Maxwellian case is much less than those in the Maxwellian cases. It can be then understood easily that the cooling effect, referenced by the parameter Λ₀, is the smallest in the anisotropic non-Maxwellian case.

In summary, with a two-dimensional kinetic model, the remarkable difference on plasma-wall interaction between the anisotropic non-Maxwellian EDF and the Maxwellian EDFs has been predicted. Compared with those in the Maxwellian cases, the onset of sheath spatial oscillation in the anisotropic non-Maxwellian case needs a very low critical electron temperature (about 5 eV). Furthermore, the electron flux to the wall, the electron-wall collision frequency, the coefficient of secondary electron emission and the electron energy deposited on the wall are the smallest and the cooling effect to the bulk electrons is the smallest in the anisotropic non-Maxwellian case. The depletion of the high-energy tail in the non-Maxwellian EDF accounts for all of the above findings.

This work is supported by the National Science Fund for Distinguished Young Scholars under Grant No. 50925625, the Foundation for Innovative Research Groups of the National Science Foundation of China (Grant No. 51121004), and the Fundamental Research Funds for the Central Universities (Grant No. 0903005203189).