A simple model for electron temperature in dilute plasma flows

In dilute plasma flows, the continuum flow assumption breaks down, and there are multiple species, multiple time and space scales. For light electrons, the oscillation frequency and speeds are quite high, while heavy ions and neutrals have slow motions. For many situations, multiple physics must be considered, e.g., electric and magnetic fields, rarefication, and collisions. Investigations on plasma flows are usually quite challenging, and many assumptions may be needed in order to perform theoretical, numerical, and experimental studies.

Numerical simulations of dilute plasma are quite helpful to understand the related physics, and there are several simulation approaches. The first approach uses full scale Particle-In-Cell (PIC)¹ simulations, where electrons, ions, and neutrals are treated as different particles. However, these full scale PIC simulations track each movement of each particle, with too many unnecessary details because for many situations, only the collective behaviors of different species are of special concerns. The simulation time steps in PIC simulations are limited by the periods of gyro-motion of electrons, ions, the mean collision time, and the plasma frequency. Due to these reasons, full-scale PIC simulations are quite expensive and not suitable to simulate many practical engineering problems. The second approach solves the magneto-hydro-dynamic equations,² where ions and electrons are treated as a mixture. This method solves for the macroscopic properties, and it cannot offer separated information for electrons and ions. Another practical approach for plasma simulation is a hybrid method³ which simulates heavy ions and neutrals with the direct simulation Monte Carlo (DSMC)⁴ and the PIC methods; electrons are treated as fluids due to their higher mobility. Compared with methods of solving ions and electrons as two fluids,⁵ this hybrid treatment provides detailed collective information for electrons, and a certain degree of accuracy is maintained, while the simulation speed is much faster than full scale PIC simulations. In many applications, the hybrid method is effective and sufficiently accurate. Other simulation approaches include directly solving⁶ the Vlasov equation. However, numerical simulation results are usually case by case.

Experimental studies on plasma flows are quite important and sometimes unreplaceable because they can validate the theoretical and simulation results. However, related experiment costs can be high. By comparison, theoretical investigations and modelling can be quite helpful and can offer some deep insights.

For plasma flows, the electron temperature is one important property. Other than the electron number density and bulk velocity components, electron temperature describes the thermal motion behaviors and can reveal much information for plasma flows. For example, dilute plasma flows from a Hall effect electric propulsion (EP) device can be used for thruster performance evaluations and contamination estimations, and electron temperature can be measured conveniently. The electron thermal temperature and potential increases across the pre-sheath in front of a Tethered-Satellite-System can result in significant current enhancement to the satellite.⁷ It is also reported that new differential equations can be developed to describe the heat fluxes into the wall with two-temperature modelling of the thermal plasmas. Due to the significant temperature differences between the electrons and ions, usually electrons deposit significant energy into plasma (e.g., Refs. 8 and 9).

In this paper, we present some recent modelling work on analyzing electron temperatures and plasma potentials in steady dilute plasma flows. The approach is based on the fluid model for electrons, and the results are semi-analytical. The number density gradients in the dilute plasma flows are assumed to be small, and the magnetic field effects are also assumed negligible.

The next few sections are organized as follows. Section II explains the detailed electron model and formula derivations for electron temperatures and plasma potentials. Section III includes two validation cases and some discussions. Section IV summarizes and concludes this paper.

The continuity equation for electrons is

where n_e and $ve→$ are the electron number density and electron velocity vector and $Sė$ is the local electron generate rate. If the magnetic field is neglected, the generalized Ohm's law states

where $j→$ is the electricity current, σ is the electricity conductivity, T_e is the electron temperature, and k is the Boltzmann constant.

The charge continuity condition is

The electron temperature equation can date back to the 70s'¹⁰ 

where m_e and m_h are the mass for electron and heavy particles, $E→$ is the electricity field vector, ν_e is the electron collision rate, and T_H is the heavy particle temperature. The above relation can be simply understood from the first law: time change rate of the electron total energy (the 1st term at the left hand side) + convection of energy (the 2nd term) + energy conduction via the Fourier's law (the 3rd term) + Ohm' heating (the 4th term) + electron energy loss due to ionization (the 5th term, $Eė$⁠) + allocations of translational energy between heavy particles (neutrals or ions) and electrons (the 6th term) + work done by expansion.

In general, the electron kinetic energy $12meve2ne$ can be assumed negligible than the thermal energy $32kTene$ and removed from the energy relation. In this work, the plasma flow is assumed to be steady and the time change rate of the electron total energy is neglected. After simple derivations, the following partial differential equation is obtained:

The Fourier's heat transfer law states $q=−κe(Te)∇Te$⁠, here the electron heat conductivity κ_e is a function of local electron temperature, then $∇·q=−∇·(κe(Te)∇Te)=−κe(Te)∇2Te−∇κe(Te)·∇Te$⁠, and these two terms are related to the left hand side term and the 1st right hand side term.

To further simplify the above relations for the electron temperature distribution, the following assumptions are adopted:

• Assumption 1. The plasma flow is steady;

• Assumption 2. The normalized electron number density gradient, $1ne∇ne$⁠, is small; hence, Eqs. (2) and (3) degenerate to

  $∇2ϕ=ke∇2Te.$

  (6)
  A simple integration leads to the following relation:

  $∇ϕ=ke∇Te+a→,$

  (7)

  where $a→$ is a constant vector for the whole flowfield, and it is to be determined later.

• Assumption 3. The transport coefficients and electric conductivity σ are constant through the whole flowfield as a first order approximation;

• Assumption 4. Ionization effects can be neglected, i.e., the last term in Eq. (5) is zero. Also, Eq. (1) degenerates as $∇·(neve→)=0$ for the whole flowfield; and

• Assumption 5. Because the large difference between the temperature for electrons and the temperature for ions or neutrals, T_H is considered as a constant in the term for translational energy re-allocation after collisions. Here, the subscript _H represents heavy particles including ions and neutrals.

Assumptions 2 and 4 lead to $∇·(neve→)=0=ne∇·ve→+ve→·∇ne$ and $∇·ve→=0$⁠. Then, the work done by expansion can be neglected.

With the above assumptions, Eq. (5) degenerates as

where

Obviously, $A1<0$ and $A2>0$⁠.

In general, Eq. (8) is difficult to handle with multiple varying coefficients and multi-dimensions. As an initial effort, this work discusses the one-dimensional scenario by concentrating on properties along a streamline, then there is a general solution to this differential equation, coefficients $A→0$⁠, A_1, and A₂ are assumed as local constants

Obviously, λ₁ is positive and λ₂ is negative; C₁ and C₂ are two constants to be determined with proper boundary conditions.

Correspondingly, the plasma potential along a streamline is

where a_s is the projected value of $a→$ along a streamline and b is a constant to be determined with boundary conditions.

It shall be mentioned that in the literature there are other models for the electron temperature (e.g., Ref. 11); however, Eq. (4) is the most convenient one for development due to the simple near linear format. This advantage has not been fully explored. Some past work adopted Eq. (4), but magnetic field (e.g., Ref. 12) is also involved and hence the expressions are rather complex for further simplifications.

As shown, the semi-analytical expression for electron temperature, Eq. (10), is based on the detailed fluid model for electrons. For validations, numerical simulations with the hybrid method are needed, i.e., DSMC/PIC for heavy ions and neutrals, and the electron properties are obtained by solving the detailed fluid model. The numerical simulations are expected to create close results to the analytical results at locations without strong variations. Experimental studies are feasible, but they need special setups, and it may be inconvenient to identify specific curved streamlines along which properties shall be measured.

The primary concern in the validations is whether Eq. (10) holds. To construct the temperature profile can be done conveniently with two steps. The first step is to sample the hybrid numerical simulation results at one specific point, and based on the information, the local coefficients A₁, A₂, λ₁ and λ₂ can be computed. The second step is to choose the electron temperature at another point along the same streamline. By using the temperature values at two points, the coefficients C₁ and C₂ can be determined.

A three dimensional hybrid simulation of a plasma flow from a BHT200 Hall thruster is performed. The magnetic leakage is assumed negligible outside the thruster acceleration channel. The detailed simulation parameters are available in the literature.¹³ Figure 1 shows the electron temperature contours in the XZ plane. The small triangular shape in front of the thruster center is a protection cap, and there is a cathode above the thruster. The cathode exit points to X = 0.05 cm on the centerline. Figure 2 shows the corresponding potentials and streamlines inside the XZ plane. These two figures show similar patterns between the plasma potentials and electron temperatures inside the XZ plane. The properties along the thruster centerline are chosen for comparisons because there are some experimental measurements available at locations $X≤0.2$ cm. As illustrated by Fig. 2, at locations with $X≥0.2$ cm, the centerline can be approximated as a streamline. Another reason to adopt the centerline is because a_s is constant along a straight line; however, along a curved streamline, a_s is usually not constant. The streamlines in Fig. 2 illustrate that electrons flow out of the cathode and the thruster acceleration channel.

This new model assumes that local gradients are small and local properties are almost constant. This assumption can be tested by examining the hybrid simulation results. Figure 3 shows the normalized centerline eigenvalue λ₁. As illustrated by the last expression in Eq. (10), λ₁ is positive and λ₂ is negative. The contribution from the term containing λ₂ to Eq. (10) decreases quickly and becomes negligible; hence, the corresponding figure is not included here. Figure 4 shows the normalized a_s and b along the thruster centerline. The corresponding values at X = 0.2 cm are chosen as reference values for normalization. These two figures indicate within the distance range 0.2 cm $≤$ X $≤$ 0.4 cm, these three normalized parameters are in the order of unity; hence, it is acceptable to apply the new analytical results.

Figure 5 illustrates the electron temperature distributions along the BHT200 centerline, and Fig. 6 illustrates the corresponding potential distributions. Both figures include experimental measurements at very near fields; the hybrid numerical simulation results; and the semi-analytical expressions by using the information at two points (marked with two black dots) from the numerical simulation results. The selections of these two points are relatively arbitrary; the main criterion is that they must be at locations with mild variations, which is one critical assumption to obtain Eqs. (10) and (11). The semi-analytical results involve two exponential functions and one linear function, even though the profiles seem linear. The simulation results are reasonable because they are close to experimental measurements, and the constructed curves follow the same trends of the numerical simulation results.

The above two figures illustrate that from very limited simulation results we can construct the plasma potential and electron temperature distributions which include two exponential functions. Is it possible to recover distributions over a wider range with two exponential functions, by using simulation results without any information on local properties? Figures 7 and 8 present some hybrid numerical simulation results of plasma plume flows out an H6¹⁴ with the hybrid simulation method. The cathode is placed at the H6 thruster center and aligned with the X-axis. As a result, the X-axis is a perfect streamline. These two figures include two smooth curve-fitting results for electron temperatures and plasma potentials, and each curve includes two exponential functions. In the simulations, at locations closer to the acceleration channel, the magnetic field leakage and the cathode effects may be important; hence, the farfield curve is closer to Eqs. (10) and (11).

Some discussions are offered here before we conclude this section. First, at farfield where all the assumptions are more reasonable, the electron temperature is computable from the plasma potential measurements, by using Eq. (7). This is due to the common factors in both formulas, and $a→$ and b are constants through the whole flowfield. The profiles are smooth and the variations are mild at farfield. Second, farfield properties can be obtained by extrapolations, based on the curve-fitting results. This is because Eqs. (10) and (11) predict that electron temperature or plasma potential profiles decrease as exponential functions. Experiments may offer limited amount of measurements, but Eqs. (10) and (11) allow us to extrapolate. Third, Figs. 5 and 7 illustrate that two curves are needed to represent the plasma electron temperature profile in a plasma plume flow from a Hall effect thruster. Plasma flows from the acceleration channel impinge at the thruster centerline in front of the thruster, creating the highest thermal temperature. λ₁ and λ₂ and coefficients are different at different sides of the peak because of the totally opposite trends there. Figs. 6 and 8 illustrate that probably one fitting-curve is sufficient to describe electron potential. However, we emphasize that these observations are based on simulation results for specific plasma plume flow out of a Hall thruster, and more investigations are needed in the future for other general dilute plasma flows. Fourth, the new electron temperature and potential formulas contain more physics and factors than the simple Boltzmann relation; hence, these new formulas shall be relatively more reasonable and more widely applicable.

A study on electron temperatures and plasma potentials in dilute plasma flows is performed. Started from the detailed fluid model for electrons and with several assumptions, compact exact formulas for electron temperatures and plasma potentials are obtained. Both formulas include two exponential functions, and the formula for plasma potentials includes an extra linear distribution term. An assumption of local constant coefficients is adopted which is generally more reasonable at locations without large gradients, such as the farfield plume flows from a Hall effect thruster.

Numerical simulations of plasma flows from two Hall thrusters are performed with the hybrid simulation method. For the first test case, along the thruster centerline, the nearfield potentials and electron temperatures are close to the experimental measurements. The profiles for electron temperatures and plasma potentials at further downstream are constructed by using detailed information at one upstream point, and the values at another downstream point. The profiles develop as exponential functions and are close to simulation results. For the second test case, the electron temperature and plasma potential profiles along the centerline are approximated by using double-exponential functions. In general, the comparisons are satisfying, but more investigations are needed in the future, for example, more specific solutions for plasma flows of different dimensions, and possible new simulation methods.

It shall be reminded that, even though plasma plume flows from EP devices are adopted for validations, the new formulas obtained in this study are generally applicable to many other dilute plasma flows with mild variations.

This work was supported by AFRL with Contract No. FA9550-15-F-0001.