5.2.8.3 Example of TAL simulation: high-power Bismuth TAL

In this section, we describe the simulation of a particular TAL design that uses Bismuth as the propellant. The Bismuth thruster was originally developed in the 1960s in the former USSR. It was demonstrated that specific impulse in the range of 2000–5000 s at power levels of 10–34 kW can be achieved. This thruster has a two-stage configuration in order to separate the ion production and acceleration zones. In this analysis, we will consider only the second stage. In particular, plasma density is used as a parameter. The simulations correspond to the following case: mass flow rate is 20 mg/s, magnetic field is uniform and equals 0.2 T. In this case, the calculated discharge current is about 6 A. The particular design is considered that has following geometry: R₁=6 cm, R₂=10 cm, and L=5 mm. The 2D hydrodynamic model described in the previous section is used to calculate plasma flow in the second stage channel.

The computational domain is shown schematically in Figure 5.34. The electron temperature distribution along the channel is shown in Figure 5.35 in the case of φ_a=3 kV. One can see that the electron temperature peaks at about 120 eV near the channel exit plane and then decreases toward the anode. In order to validate the model, some integral characteristics of the thruster are calculated and compared with experiment.

The steady-state thrust can be calculated at the thruster exit plane as follows: The calculated thrust increases linearly with the current (ion beam current) in agreement with experiment as shown in Figure 5.36. It can be seen that the model predicts thrust levels close to those measured experimentally over the entire range of the beam current (or equivalently mass flow rate).

5.2.8.4 Example of calculation: TAL—analysis of the space-charge sheath near the channel wall

In the channel of the second stage, the guard ring has the cathode potential as shown schematically in Figure 5.34. When high voltage across the acceleration channel is considered, one should take into account the importance of the sheath development near the screening walls (guard ring) of the channel. When a negative voltage is applied to a surface immersed in a plasma, electrons are repelled from the surface, leading to sheath formation. Electrons drift away from the surface due to the presence of the high electric field. In the steady state, the ions are then accelerated toward the surface by the electric field of the sheath. In the 1D steady-state case, the sheath thickness can be estimated according to the Child–Langmuir law [156,157]:

(5.103)

where V_s is the ion velocity at the sheath edge, U is the voltage across the sheath, L_s is the sheath thickness, ε is the permittivity of vacuum, N_s is the plasma density at the sheath edge, and m_i is the ion mass. In a TAL, the channel wall has a potential equal to the cathode potential and therefore the voltage across the sheath, U, is equal to the plasma potential and varies along the channel. Similarly, the steady-state sheath thickness in a partially magnetized plasma was calculated in a plasma immersion ion implantation system [158]. One can see that the steady-state sheath thickness is determined by the plasma density and ion velocity at the sheath edge for a given bias voltage (Equation 5.103).

Based on the 2D hydrodynamic model, the plasma parameters at the plasma–sheath interface (N_s and V_s) as well as ion flux to the wall are calculated. It should be noted that the model used for sheath thickness calculation is one-way coupled, since the 2D plasma flow model does not account for the sheath thickness, while account for the voltage drop in the plasma and the sheath. The sheath thickness is several centimeters in typical high-power TAL conditions and occupies a significant portion of the channel as shown in Figure 5.37. The sheath thickness varies significantly along the channel wall due to variation of the potential drop between the plasma and the wall, which has cathode potential. Note that in typical high-power TAL design, the channel width is about several centimeters. It can be seen that when the discharge voltage is about 10 kV, almost the entire channel becomes nonquasi-neutral (sheath) and the quasi-neutral plasma is confined in the middle of the channel. Therefore, one can conclude that the discharge voltage limits the regime of quasi-neutral acceleration in a TAL with given geometry. Considering the interaction of the highly energetic (due to acceleration in the sheath toward the wall) ion flux with wall materials, one can expect significant erosion.

It was shown above that the effect of the sheath expansion near the acceleration channel wall is very significant in the case of a high-power TAL. It affects mainly the cross-sectional area of the quasi-neutral plasma as shown in Figure 5.39. In turn, the change of the cross-sectional area of the quasi-neutral plasma affects the current density in the axial direction. It is natural to study this effect using a 1D approach since the axial current density (Eq. 5.102) is calculated in a 1D manner along the channel centerline. To this end, we developed a simplified quasi-1D model of the discharge in crossed E×B fields. The original model of this discharge was developed by Zarinov and Popov [59]. Recently, a modified version of that model that took into account plasma–wall interactions by introducing a more detailed electron energy equation was developed by Choueiri [63]. We present here a model [79] in which we adopt the formulation of Refs [59,63] except that sheath expansion near the channel wall in the acceleration region is taken into account as it occurs in a high-power TAL. In order to compare with previously published results [59,63], we adopt here a rectangular coordinate system shown in Figure 5.38. The z-axis is along the channel axis and the y-axis is along the thruster radius, i.e., along the magnetic field. The x-axis corresponds to the azimuthal direction in the Hall thruster. In this case, the current density in the quasi-neutral region will vary along the channel (z-axis). The characteristic time of sheath formation near the channel wall is about the plasma frequency and therefore a steady-state sheath will be considered. In the case considered, the cross-sectional area A(z) of the quasi-neutral plasma is varied along the channel. We start with the following set of equations for the quasi-neutral plasma:

(5.104)

(5.105)

(5.106)

(5.107)

where μ_⊥ is the classical cross-field mobility, j_i(z) is the ion current dependent on the cross section, j_z is the electron current, ν_iz is the ionization frequency, ν_e is the electron collision frequency, and ω_e is the electron cyclotron frequency. The change of the cross-sectional area of the quasi-neutral plasma due to sheath expansion can be calculated as follows:

(5.108)

where R₁ and R₂ are the inner and outer radii of the channel and s is the sheath thickness (dependent on z) that can be calculated from Eq. (5.103). Since the wall has cathode potential (set to 0 for simplicity), the potential drop between the plasma and the wall is simply equal to the plasma potential φ. The channel cross section at the exit plane corresponds to the case of zero sheath thickness, since the wall potential is equal to the exit plane potential and therefore cathode potential.

In addition, following Ref. [59,63,81], a simplified energy equation can be used: T_e=βφ. It should be noted that similar linear behavior of electron temperature with discharge voltage near the channel exit plane was measured very recently [159]. On the other hand, the electron temperature profile in the entire channel is much more complicated and is a result of the balance between electron Joule heating and cooling due to interaction with the channel walls and ionization. For example, a typical profile is shown in Figure 5.35. Therefore, linear behavior of the electron temperature with plasma potential is generally only valid near the channel exit. In this particular 1D model, we are interested primarily in the effect associated with near-wall sheath expansion. Therefore, in order to simplify the problem, we will consider electron temperature as a parameter, which is constant along the channel.

Let us normalize the system of equations (5.104)–(5.107): ϕ=φ/φ_a, ξ=z/l*, θ=T_e/φ_a. The characteristic quantities can be defined as follows:

The derivative of the channel cross section A(z) can be calculated as follows:

(5.109)

Finally, the following normalized system of equations is obtained:

(5.110)

(5.111)

(5.112)

This system of equations is similar to the original formulation of Zharinov and Popov [59] with one exception that change of the cross-sectional area of the quasi-neutral channel A(z) is taken into account.

Below, the following particular case of a high-power TAL is considered: mass flow rate of 10 mg/s, Bismuth as the propellant (i.e., R₁=6 cm and R₂=10 cm). As was mentioned above, in the calculation presented here, we assume a constant electron temperature close to the peak electron temperature (T_e=100 eV, see Figure 5.35).

The effect of the near-wall sheath expansion on the anode layer potential profile is shown in Figure 5.39. One can see that the sheath expansion strongly affects the potential profile and the anode layer thickness decreases from 0.7 down to 0.15 when the sheath expansion effect is taken into account. This happens because the quasi-neutral plasma region is confined in the middle of the channel and as a result the current density increases. In turn, this leads to a higher axial electric field and therefore a smaller anode layer thickness.

The anode layer (acceleration channel) thickness dependence on the discharge voltage is shown in Figure 5.40. One can see that the anode layer thickness significantly decreases with the discharge voltage increasing due to the sheath expansion effect. It is interesting to note that shrinking of the anode layer thickness is important as it leads to a smaller area of contact of the plasma with the walls and therefore smaller total erosion of the channel walls.

It should be noted that the normalized length, l*, is proportional to which leads to a nonmonotonic behavior of the anode layer thickness with the discharge voltage in considered range of discharge voltages. This happens as a result of the near-wall sheath expansion effect. Without considering the near-wall sheath, the dimensional anode layer thickness will monotonically increase with the discharge voltage. Near-wall sheath expansion in the thruster channel leads to the current density increase (see above) and reverses the dependence of the dimensional anode layer thickness on the discharge voltage. The dimensional anode layer thickness will monotonically decrease with the discharge voltage in the large discharge voltage range (>10 kV).

• Magnetic field line: On the spatial scale of a magnetic field line, dynamics are driven by the cyclotron motion of electrons. Electrons are magnetized, and individual field lines can be considered independent of each other. Heavy particle and properties normal to the field lines are assumed frozen. This approach allows us to rapidly simulate electrons and recover mobility self-consistently. Leveraging modern multicore architectures via multithreading allows us to study multiple field lines simultaneously.

• Thruster channel: On the spatial scale of the thruster, plasma is assumed to be quasi-neutral, and electron density can be obtained from kinetic ions. Electron temperature and plasma potential is obtained by solving the quasi-1D equations.

• Plasma plume: Outside the thruster exit, the magnetic field plays a negligible role. The plume is quasi-neutral except in low-density sheath regions around the spacecraft. Of interest here is the formation of charge-exchange (CEX) ions and their impact on spacecraft components. Electron density is obtained from Boltzmann relationship, and potential can be solved by direct inversion or by solving Poisson’s equation.

5.3.1.1 Ablative PPT

PPTs are currently considered as an attractive propulsion option for mass and power limited satellites that require μN-s to mN-s impulse bits [162–167]. One of the most extensively analyzed propulsion functions has been attitude control. For missions that require precision pointing, the PPT offers a unique advantage over other technologies as it delivers a small impulse “bit” with high specific impulse (exhaust velocity). A second major functional area is stationkeeping, which includes drag compensation and formation flying. These functions require less than 1 mN of thrust. For instance, interferometer missions [168] such as ST-3 require submillimeter relative positioning precision. In particular, the US Air Force has a growing interest in highly maneuverable microsatellites to perform various missions, such as space-based surveillance, on-orbit servicing, inspection, and space control [162,163]. Recently, an electromagnetic PPT was successfully operated for pitch axis control on the EO-1 spacecraft [168,169]. It was shown that PPT can be easily scaled down in power and size. A μ-PPT is the miniature version of the traditional PPT has been designed at the AFRL for delivery of very small impulse bit [170,171]. AFRL μ-PPT was developed for a demonstration mission on TechSat21 [172] and it was recently employed at FalconSatIII. The μ-PPT can deliver a thrust in the 10 μN range to provide attitude control and stationkeeping for microsatellites.

In μ-PPT, the discharge across the propellant surface ablates a portion of the propellant, ionizes it, and then accelerates it predominantly electromagnetically to generate the thrust (schematically this thruster is shown in Figure 5.43) [25,173]. It is expected that the use of electromagnetic acceleration to create thrust will also lead to relatively high specific impulse.

It was demonstrated [174] that both discharge energy (peak current) and thruster size affect significantly the discharge uniformity (azimuthal or radial). Azimuthal nonuniformity relates to the current constriction and anode spot formation phenomena. This happens when the discharge current or thruster size exceeds some critical value. Discharge nonuniformity leads to a much higher ablation rate and causes degradation of the specific impulse. On the other hand, small discharge current leads to strong Teflon surface carbonization (charring) and radial nonuniformity, which in turn leads to thruster failure. The primary mechanism of the charring formation was identified and it is related to carbon backflux. Thruster size and discharge energy can be optimized by trading between two conflicting requirements of a large pulse energy (to prevent charring) and a small discharge energy (to prevent current constriction) [174]. Because the Teflon ablation rate grows nonlinearly with the surface temperature, the model predicts a lower rate of ablation in the areas where the surface temperature has a minimum. By taking this into account, the effect of the temperature distribution may be related to the preferential charring of the Teflon surface observed experimentally, as shown in Fig. 5.43 (b). It is interesting to note that comparison of the calculated temperature field and ablation rate with the photograph of the Teflon surface (Fig. 5.43 (b)) shows that the area with surface temperature and ablation rate minimum corresponds to the charred area in the case of the 3.6-mm-diam thruster. Because the charring phenomenon is completely intolerable and leads to thruster failure, the optimal discharge energy should be chosen somewhere near the spot formation limit.

5.3.1.2 Microlaser plasma thruster

Another example of the ablative type of micropropulsion system is the micro-laser plasma thruster (μ-LPT). Most notably, μ-LPTs have been developed by Phipps and Luke [175] and Gonzalez and Baker [176]. In Ref. [176], a Q-switched microchip laser, pumped by a CW diode laser, was used to ablate an aluminum target generating thrust in the range of 0.3 nN to 3 μN with power consumption of about 5 W and pulse frequency in the range of 1 Hz to 10 kHz. The wide dynamic range of thrust levels provided by these devices is one of their most attractive features.

The μ-LPT developed by Phipps and Luke [175] uses a 1–10 W, high-brightness diode laser irradiating various absorbing material and substrate combinations (e.g., black ink on paper, black PVC on Kapton™). Laser coupling coefficients on the order of 60 μN/W and specific impulses on the order of 500 s with a 1 W laser are achieved. One of the major advantages of the μ-LPT is its large dynamic range of impulse bit that can be varied between 0.4 nN-s to 16 μN-s by simply increasing the laser pulse duration. In addition, selection of absorber and substrate materials allows the specific impulse and the laser characteristics to be tailored for specific mission requirements.

The μ-LPT can be operated in two different modes. In reflection mode (R), the laser is incident on the target and the ablated material “reflects” from the surface. This mode has the potential problem of leading to deposition of plume effluent on the laser optics. In transmission mode (T), the laser passes through a transparent substrate film from the back. The substrate is coated on the other side with an absorbing material that is ablated. This approach circumvents the problem of optics contamination found with the R-mode. However, the dynamic range of impulse bit available in T-mode is more restrictive. Coating of laser optics by plume deposition is one of the major lifetime limitations of the μ-LPT in R-mode. Therefore, there is a certain preference for development of T-mode operation. Preliminary computational results for R-mode μ-LPT were presented [177]. A lens focuses the laser diode output on a 25 μm diameter spot on the transparent side of a fuel tape. The beam heats an absorbing coating to high temperature, producing a miniature ablation jet. The material that is ablated is usually PVC or Kapton. Typical parameters of operation are power of 2–14 W, pulse duration of 3–10 ms. The fuel tape thickness is about 185 µm, composed of 125 µm of transparent backing (usually cellulose acetate) and about 60 µm of absorbing coating. Typically Q* (energy of laser light required to ablate 1 kg of target material) is about 2×10⁷ J/kg and the momentum coupling coefficient is about C_w=60–100 μN/W. Recent computational work on T-mode μ-LPT was also developed and compared with experiment [178]. Generally LPT models satisfactorily describe the main features of the thruster such as ablated mass and plume expansion.

5.3.1.3 Microvacuum arc thruster

One possibility to improve some aspects of PPTs without compromising its efficiency is to use metal as a propellant. Success in development vacuum arc technology attracted much attention to this technology [179]. Metal propellant will have the following benefits: lower energy consumption per mass ionized (due to the low ionization potential of metals), high ionization degree, operation with higher repetition rates since the metal melting temperature is higher than that of polymer dielectrics used in some current thruster, and high pulse-to-pulse stability. For this reason, a vacuum arc source was developed and used for an ion thruster [180]. This work led to development of a coaxial vacuum arc plasma thruster [181]. Schematically this thruster is shown in Figure 5.44. However, the measured efficiency was reported to be only approximately 1.6% which is strongly different from the estimations, leaving ample room for improvement. On the other hand, several problems related to this technology were reported, including lifetime and ion current degradation with cathode recession [182]. Therefore, understanding of these problems and the efficiency limitations of the vacuum arc thruster (VAT) technology is an important issue. In addition, basic aspects of VAT such as thrust mechanism, effect of the magnetic field of the thruster operation and its plume, droplet generation were not studied in detail. It was shown that using an axial magnetic field (the so-called magnetically enhanced VAT) helps dramatically to improve thruster characteristics, such as specific impulse (plasma velocity) and to decrease plasma plume divergence [183,184].

In VAT, the thrust is created in the minute spot on the cathode surface called cathode spots (having size of ~10–100 μm) [185]. The high-pressure plasma in the cathode spot creates the reaction force (thrust) on the cathode as schematically shown in Figure 5.44. As earlier as in the 1930s, it was found experimentally that the plasma stream exerts a reaction force on the cathode, which is about 17±3 dyn/A (or ~0.2 mN/A) [186,187]. It is very important to note that since the reaction force is proportional to the arc current, the thrust can be controlled easily. Recent measurements confirmed those conclusions [188]. It should be pointed out that the force generated in the short vacuum arc exceeds that of the conventional cathodic arc [252].

In the late 1960s and early 1970s, several experimental groups investigated the use of vacuum arcs in electric thrusters for spacecraft propulsion (see review paper [189] and references therein). The main reason for considering vacuum arcs for electrical propulsion was the nature of the plasma plume expanding from the cathode, which exhibits high velocity and is highly directional. In general, a vacuum arc is established by using a coaxial geometry with a relatively small diameter cathode which is surrounded by an insulator and an anode. Vacuum arcs can be produced from any electrically conductive material. The suitability of various materials for vacuum arc propulsion was first studied by Dethlefsen [190]. In the case of a magnesium cathode, a plasma velocity of 50000 m/s was demonstrated, which is certainly very appealing to the propulsion community. Pulsed operation was demonstrated and the thrust efficiency was found to be dependent on the electrode material, the amplitude, and duration of the current pulse.

One reason for the material dependence is the fact that metals with low melting point and low thermal conductivity erode rather heavily, with a large part of erosion resulting in macrodroplets. These droplets tend to reduce the thrust efficiency because they are not accelerated to as high velocities as are the ions. While this reduces thrust efficiency, it has to be pointed out that the droplet emission does not depend on material properties alone and hence can be minimized through the action of thermal and mechanical inertia if a current pulse of sufficiently short duration on the order of 10 μs is used in a pulsed VAT. Apart from the high exit velocity and the fact that the cathode material can be chosen from a large variety of candidates, pulsed VATs were found to have other appealing characteristics, which include the fact that a solid fuel is used, the low-voltage operation, as the burning voltage of vacuum arcs does not exceed 100 V, and the simple control by adjusting arc current and repetition rate.

Interest in the vacuum arc source for propulsion applications was renewed [191] in 90s. The measured energy efficiency was about 80% which is comparable to that of xenon ion thruster. The development of the vacuum arc source for an ion thruster led to design of a thruster based on the pulsed vacuum arc without any ion extraction [192]. For later configuration, a specific impulse of about 1000 s was measured and 21% of efficiency was estimated. Further development of this technology is attributed primarily to Alameda Applied Science Corporation (AASC) [185].

The VAT is a simple, yet efficient electric propulsion device that combined with an inductive energy storage (IES) power processing unit (PPU) results in a low-mass (<300 g) system. The basic mechanism underlying the VAT is the production of micrometer-size fully ionized microplasmas (cathode spots) on the cathode surface, which expand into vacuum at high velocities, with the ions producing thrust. Every cathode spot carries a limited amount of current (~1–10 A) and exists for a few nanoseconds, which makes the VAT very scalable with respect to pulsed operation and/or amount of plasma produced.

The need for a low-mass propulsion system motivated the development of the VAT system. A PPT that uses conductive cathode materials as propellant is combined with an energy storage PPU that takes 5–24 V from the bus and converts it into an adequate power pulse for the thruster. It is a system well suited to provide small impulse bits (≈1 µNs) at high specific impulse, I_sp of about 1000–3000 s. Applications include positioning and drag makeup for small, power, and mass limited satellites.

The performance of the VAT is determined by the propellant mass, the degree of ionization of the plasma, the angle of expansion, the average charge state, and the ion velocity. All these parameters have been measured repeatedly in the past and verified for numerous materials and operating conditions [166–169]. Typical values for the ion velocity vary between 10 and 30 km/s, the average arc/ion current ratio has been shown to be of the order 8%, and a cosine distribution has been found to emulate the plasma plume expansion very well.

In order to produce a low-mass system, the VAT was constructed using an IES circuit PPU and simple thruster head geometry. In the PPU, an inductor is charged through a semiconductor switch. When the switch is opened, a voltage peak L dI/dt is produced, which breaks down the thin metal film coated anode cathode insulator surface at relatively low-voltage levels (≈200 V). The current that was flowing in the solid-state switch (for ≤1 ms) is fully switched to the vacuum arc load. Typical currents of approximately 100 A (for ~100–500 µs) are conducted with voltages of approximately 25–30 V. Consequently, most of the magnetic energy stored in the inductor is deposited into the plasma pulse. The efficiency of the PPU may thus be greater than 90%. By varying the length of the trigger signal, the level of the current in the switch and thereby the energy stored in the inductor can also be adjusted. This in turn changes the amount of energy transferred to the arc and the impulse bit of the individual pulse. Obviously, the repetition rate of the individual pulse can be changed by varying the input signal as well.

The mass of the PPU is small (<300 g) resulting in a low-mass system. The plasma output is quasi-neutral; therefore, no additional neutralizer is needed. An electromagnetic interference (EMI) filter might be necessary due to the noisy characteristics of the discharge, high peak currents, and fast switching. A low-mass feed mechanism is available; therefore, even long missions can use this technology.

A picture of the VAT system with an equivalent circuit is shown in Figure 5.45. Additional information about the principle of the VAT can be found elsewhere [193–195].

While the system is simple and can be realized with an extremely low mass, the divergence of the plasma plume reduces the efficiency of the system due to the variations in the thrust vector and also increases the chances of contamination of essential parts of the spacecraft. Based on prior work by Gilmour and coworkers [196,197], who, by using a coaxial arc-diode configuration with an additional magnetic coil, achieved a total efficiency of 30% with a copper cathode and a magnetic field of 500 G and the fact that a magnetic field can be used to direct a vacuum arc plasma [194,198], an axial magnetic field was used to control the plasma flow. The magnetic field is produced by a coil wrapped around the anode of a coaxial thruster head. For the purpose of performing initial experiments, the currents in the coil and in the thruster are controlled independently, while a final design will utilize the magnetic field produced in the inductor of the IES PPU to control the plasma plume [199].

5.3.1.4 Microcathode arc thruster

Recently, a novel thruster design, the microcathode arc thruster (µ-CAT), was developed and investigated. This thruster improves on the vacuum arc discharge thruster by applying specially designed an external magnetic field. The unique magnetic field conditions achieved in a vacuum arc offer several potential advantages in these devices.

The µ-CAT is a simple electric propulsion device that combined with a magnetic coil and an IES PPU results in a low-mass (<100 g) system. A picture of the µ-CAT system and two types thruster are shown in Figure 5.46A–C. Figure 5.46B shows the schematic design of the ring electrodes µ-CAT (RE-µ-CAT), which consist of an annular titanium cathode and a same diameter annular copper anode with 1 mm width. The annular ceramic insulator tube having same inner and outer diameters and a width of about 1 mm was used as separator between the arc electrodes. Figure 5.46C shows the schematic design of the coaxial µ-CAT. Instead of the RE, this design employs cylindrically shaped cathode and anode.

Figure 5.47A shows a schematic of the thruster and the PPU system. The mass of the PPU is small (<100 g), resulting a low-mass system. PPU equipped with an IES system has been designed as shown in the Figure 5.46A and B. When the trigger pulse is applied to a semiconductor insulated gate bipolar transistor (IGBT) switch, the energy is accumulated in the inductor, while when trigger pulse ends, a surge voltage with the magnitude proportional to L dI/dt is generated on the inductor and applied to the electrodes. This leads to a breakdown and initiation of arc discharge between the electrodes. A coil has been applied to the outside of the thruster to produce magnetic field as indicated in Figure 5.47B and C. The direction of the magnetic field could be simply reversed by reversing the coil current.

The µ-CAT operates by producing a fully ionized plasma at the inner surface of the electrode The plasma is formed in the cathode spots and expands into the vacuum zone under the applied magnetic field gradient. The plasma accelerates and expands into vacuum at a high velocity as the magnitude around tens thousands meter per second, which results in the impulse bit. The effect of the magnetic field on the thruster operation is clearly visible in Figure 5.47B. This figure shows a CCD camera observation of the RE µ-CAT firing in the vacuum chamber without and with an added magnetic field (B=0.3 T). The yellow lines in the second image of Figure 5.47B indicate the simulation results of the magnetic field distribution.

The plasma formed by a vacuum arc is created on the cathode surface spots. Using optical methods it was observed that the spots consist of either a homogeneous bright region or consist of cells and fragments with a typical total size of about 10–100 µm [200–202]. The observation of cathode spots motion under magnetic field was studied in 1960s [203,204]. It is known that presence of a transverse magnetic field at the cathode surface produces cathode spot motion in the −J×B direction. Theoretical explanation of this effect was given by Beilis [200]. The observed vacuum arc cathode spot rotation has important implications for propulsion since the cathode spot rotation leads to a uniform cathode erosion, which is critical for assuring long thruster lifetime. The µ-CAT cathode spot rotation was measured by the four-probe assembly Langmuir probes [205]. Four single probes were located along the azimuth direction inside the thruster channel and the four probes ion current measured results were shown in Figure 5.48. The rotation speed was calculated using a quarter of circumference of the thruster inner surface divided by the delay time between each of two neighbor peaks. The average rotation speed is shown in Figure 5.49. It was found that the spot rotation speed increased by factor of 5 (from 20 to 100 m/s) as the magnet field strength increased from 0 T to 300 mT.

Magnetic field affects the total ion current (or jet current) collected in the thruster plume. The dependence of the ratio of jet current I_jet over arc current I_arc (f=I_jet/I_arc×100%) on the magnetic field is shown in Figure 5.50. It was found that due to losses to walls, the total ion current was very low (f=0.06%) in the case without the magnetic field. Increase of the magnetic field resulted in significant increase in the total ion current (up to 50 times) and saturated at about f~3–3.5%. Note that f cannot exceed the (f_max=I_max/I_arc×100%), where I_max is the total ion current generated by the cathodic spot. According to Kimblin [206], the f_max in the case of titanium is about 8%. Thus f~3–3.5% observed in the experiment corresponds to the efficiency of cathodic jet transport through the thruster channel of about 37.5–43.8%.

Since total ion current increases with a magnetic field, one can expect increase in the impulse bit. Indeed recent measurements [207] indicated that the impulse bit increases with a magnetic field reaching about 1 µNs at 0.3 T as shown in Figure 5.51.

Figure 5.52 presents average ion velocity as a function of axial distance without and with magnetic field strengths of 0.17 and 0.3 T. It is seen that without magnetic field, the ions propagate at almost constant velocity as indicated in Figure 5.52 by the lower curve, while application of magnetic field leads to ion acceleration in the axial direction as demonstrated by two upper curves. The ion velocities saturate on the level of around 3–3.5×10⁴ m/s [208].

Plasma flow is subject to the electromagnetic force in the divergent magnetic field region:

where M is the ion mass, n is the plasma density, j_θ is the azimuthal electron current, and B_r is the radial component of the magnetic field. Azimuthal electron current density can be estimated as

where j_r is the electron radial current density. In the plasma jet outside of the interelecrode gap, the total current is zero and therefore the electron current is equal to the ion current. Thus

where V_r is the radial component of the ion current. Radial component of the plasma velocity is developed in course of plasma expansion [147] and could be about 0.3 of the axial velocity component in a high magnetic field. Based on the above scenario, one can estimate the ion velocity change in the axial direction as

Using the following typical parameters of the plasma jet: B (i.e., both B_r and B_z)~0.1 T, n~10²⁰ m⁻³ V_r~10⁴ m/s and Δz~0.01 m, one can estimate that ΔV_z~10⁴ m/s. This estimation is in good agreement with experimental measurements.

5.3.2.1 Liquid PPT

In a microscale device with a high throughput, a solid propellant feeding mechanism may become impractical (unless propellant recession is organized in a proper manner, see previous section as an example). In that case, a liquid-based thruster can take advantage of fluid handling possibilities, such as capillary effect and adsorption. Several PPTs using liquid propellant were demonstrated over the last decade. Scharlemann and York [209] show that water is very effective propellant producing much higher specific impulse than Teflon. Microliquid PPT was demonstrated at Johns Hopkins University (Applied Physics Lab) [210]. JHU/APL thruster prototype weights about 13.5 g (without PPU) and also uses water as a propellant. This thruster accumulates water vapor in a small cavity as it diffuses through a membrane. An orifice at the opposite end of cavity leads to the main thruster chamber. Discharging a spark across two electrodes heats the vapor and injects it into the main chamber causing sustained discharge.

5.3.2.2 Field emission electric propulsion

Another type of thruster that may be relevant to nanopropulsion is field emission electric propulsion (FEEP). FEEP uses field emission ionization to extract and accelerate ions from the propellant. The propellant, which is typically a liquid metal, is stored in a reservoir. To initiate thruster operation, the propellant is first heated to its melting temperature in preparation for propellant extraction. A needle utilizes pressure gradients and propellant surface tension to extract the propellant from the reservoir and brings it in proximity of an extractor electrode. A large potential is applied between the needle and an extractor electrode. As the voltage increases, the propellant forms a cone (called a Taylor cone) that protrudes toward the extractor. The small radius of curvature at the protrusion tip increases the local electric field there, which causes further propellant displacement from the needle orifice. Once an electric field threshold of approximately 10⁹ V/m is reached, atoms at the tip of the Taylor cone are ionized and accelerated through the gate electrode. Recently micro-FEEP thruster was demonstrated [211]. By utilizing electrostatic acceleration principles, FEEP has electrical efficiencies exceeding 90% [212,213]. However, the field emission process requires high voltages (10 kV range), leading to very high specific impulse up to 10,000 s and thus relatively high specific power values. In addition, the use of liquid metal propellant raises concerns regarding spacecraft contamination. While pure ion generation and acceleration are often sought, FEEP charge extraction instabilities can result in the formation of charged microscopic droplets. The amount of droplets formed is a strong function of emission current. A high droplet content decreases propellant utilization efficiency (called mass efficiency) and overall thruster specific impulse and electrical efficiency. In addition, a key limitation of FEEP has been the neutralization cathode. Typically, heated filament cathodes are employed for ion beam neutralization. In addition to their rather short lifetimes, filament cathodes consume an enormous amount of power.

5.3.2.3 Colloid thruster

Colloid thrusters are similar to FEEP featuring emitter tips and accelerating electrodes. However, unlike FEEP, colloid system does not accelerate individual ions but rather accelerate fine charged droplets. A key component of colloid thrusters is the phenomenon referred to as “electrospray.” Among the different spraying modes that can be attained, the so-called cone-jet produces a stable spray of monodisperse droplets. Droplets with diameters ranging from hundreds of microns to a few nanometers may be generated by tuning the physical properties of the electrosprayed solution. A strong electric field applied between the sharp-edged exit of the capillary and an external electrode causes charge separation inside the liquid propellant, which is doped with an additive to increase its electric conductivity. Various instabilities causes jet breakup into small liquid droplets, and by the action of the electric field on the conductive liquid charged droplets are extracted from the capillary at high voltages, producing thrust.

Colloid thrusters are able to deliver a very stable, fine thrust (of the order of 1 µN per cone-jet) with high specific impulse, I_sp up to 1500 s. This type of electrostatic propulsion makes possible and enhances the use of microsatellites. Colloid thrusters also enable satellite flight formation and other missions requiring very precise positioning.

In principle, some combination of FEEP and colloid thruster is possible to provide flexibility in specific impulse and thrust. The idea of the hybrid approach is described in the next section.

5.3.2.4 FEEP operation in the colloid mode with low I_sp range

To extend the range to lower I_sp and higher thrust while preserving high efficiency, the FEEP thruster can be purposely operated in a droplet extraction mode.

Droplet emission is a phenomenon inherent to the operation of liquid metal ion source [214]. Several mechanisms of droplet emission were identified. One possible mechanism is related to the fact that at higher ion currents, the liquid jet becomes unstable, the so-called Rayleigh instability [215]. Another possibility is excitation of capillary waves at the sides of the Taylor cone [216]. Both effects form droplets with size of about 1–100 nm. Typically in liquid metal ion sources, droplets are generated when Rayleigh instability is set that requires electric field higher than some critical one (or current higher than critical) [217,218].

In the indium FEEP, it was shown that the mass efficiency, which is the ratio of the ion mass fraction to the total flux, decreases from 99.5% at 10 μA down to 10% at 200 μA in the case of indium. This means that when current greater than 200 μA, about 90% of the propellant mass is emitted in the droplet form.

In addition, Crowley [219] showed that the temperature rise at the emitter tip enhances droplet emission. According to Crowley, the temperature rise depends upon the electrical resistivity to thermal conductivity ratio. The smaller ρ/k ratio corresponds to smaller temperature rise and therefore smaller droplet fraction. Increasing the propellant temperature would lead to higher ρ/k and therefore enhance the droplet emission. Again, in the indium FEEP, it was shown that the droplet mass fraction increases up to 90% when indium temperature increases from melting point (156.6°C) up to about 280°C.

The approach may be to use a combination of fine-tuning emission current and temperature control to operate the micron-scale FEEP in charged droplet mode. In this case, specific impulse produced by the thruster can be adjusted dependent on mission requirement from being very high of about 10,000 s in the ion extraction mode down to 1000 s in the droplet emission mode.

5.3.2.5 Other concepts

Other electrostatic and electromagnetic schemes can be considered for micropropulsion, such as Hall thruster and ion thruster.

A microion engine is under development at Jet Propulsion Laboratory. This is a MEMS-hybrid device in which the bulk of the engine is fabricated using conventional machining, while several components are machined using MEMS technology. The engine has a discharge chamber of 3 cm. However, further miniaturization is not feasible because thruster becomes extremely inefficient or cannot maintain a plasma within given power limit [220]. Small-scale Hall thruster with power level of 50–200W was also demonstrated recently by MIT [221] and Princeton University PPPL [222]. In this respect, Princeton cylindrical Hall thruster has promise due to decreased surface-to-volume ratio.

5.4.1.1 Ion beam generated from the thruster

Lagrangian approach will be employed here for ion simulations. It should be pointed out that the Lagrangian technique reduces the numerical noise associated with PIC method. Moreover, PIC technique is not accurate in the case of the problem with large-scale variation such as from millimeter to meter, which is the case of the thruster plume.

Ion momentum equation can be expressed as follows:

(5.113)

where V_i is the ion velocity and E is the electric field. Mass elements or macroparticles are ejected from the thruster exit and are tracked using trajectory analysis. The location of each macroparticle is determined by the following system of equations:

(5.114)

Each time the ion density is updated using calculated V(t) with the conservation of mass:

(5.115)

Density is updated on each simulation time step as well as electric field. The iteration process ensures that ion trajectories are consistent with local electric field.

Typically the calculations of the ion density in the plume relay on the condition at the exit plane from the thruster.

5.4.1.2 Electron flow

Electron density distribution is given by a balance of electrostatic force and electron pressure while the magnetic field effects can be neglected. Recall that in the near-plume region of the electromagnetic thruster (such as Hall thruster), a magnetic field may affect the electron motion. However, it was found that this effect is small in a typical case since magnetic field decreases strongly outside of the Hall thruster channel [152]. Effect of the magnetic field on the plume in the near-plume field will be described in Section 5.4.4.1. Mathematically, the balance for electrons under considered conditions can be written as follows:

(5.116)

where P is the electron pressure. Equation (5.116) comes from the general electron momentum conservation taking into account the currentless and unmagnetized electron flow. For densities and temperatures of the plasmas, which are typical for the plumes of Hall thrusters, the electron–ion collision term is negligible. The ratio of the collision frequency to the plasma frequency is much less than one. Further assuming isothermal conditions yields the Boltzmann relation:

(5.117)

where n_ref is the reference electron density defined where potential is zero.

5.4.1.3 Neutral density

Typical scale of the plume of interest for spacecraft propulsion application is in the order of several meters which is about the size of the solar panels of a communication satellite. Neutral density decreases due to expansion into vacuum. As a result, in the most part of the plume, the flow is collisionless. The most common approach to describe the neutral density in the plasma plume is a Monte Carlo simulation such as Direct Simulation Monte Carlo (DSMC).

In some case, a simple analytical model for the neutral expansion can be employed as well [223]. The profile of neutrals in the thruster plume is computed using the emission by the solid angle Ω(r,z) [223]:

(5.118)

where Γ_o is the mass flow rate of neutrals, U_a is the neutral velocity, and Ω is the solid angle [223].

The ion–neutral interactions include momentum and CEX collisions. The most important collision for plume contaminations is the CEX. The fast ions from the main beam undergo CEX collisions with neutral particles, resulting in slow-moving ions and fast-moving neutrals:

(5.119)

CEX collision rate is calculated as follows:

(5.120)

A value for σ_CEX of 5.5×10⁻¹⁹ m² has been assumed following the previous comparison with experimental data. The data for collision cross section has been analyzed by Pullins et al. [225].

5.4.1.4 Potential distribution

The PIC algorithm solves the 2D Poisson equation and iterates until the steady-state solution is reached. The sum of the primary ion density and the CEX ion density is used in the Boltzmann relation:

(5.121)

The primary ion density is calculated using the Lagrangian method.

The overall approach to calculate the primary ion density and CEX ion density is shown in Figure 5.54. Most simulations of the plasma plume employ about 100,000–500,000 particles in 300×300 PIC cells. These simulations required about 50–100 CPU time on single workstation.

5.4.3.1 Laser generated plasma expansion

The laser generated plasma plume has been described by Itina et al. [231] and by Franklin and Thareja [232]. In each case, Monte Carlo methods were employed. The focus of these studies was on use of laser ablation for thin film deposition.

In the example that will be described in the next section, a hybrid fluid-particle approach is employed for the plasma plume simulations. The heavy-particle products of the laser ablation of PVC (neutral atoms of C and H, and ions C+ and H+) are modeled as particles. Particle collisions are computed using the DSMC method [233]. Both momentum-exchange and CEX collisions are taken into account. Momentum-exchange cross sections follow the model of Dalgarno et al. [234] and the collision dynamics follows the normal DSMC procedures. The implementation of this algorithm for the unsteady plasma plume is described in Ref. [235]. The CEX processes employ the cross sections proposed by Sakabe and Izawa [236].

Acceleration of the charged particles in self-consistent electric fields is simulated using the PIC method [237]. The plasma potential, φ, is obtained from the Boltzmann relation assuming that the plasma is quasi-neutral. Recall that the ion density is known from the PIC simulations.

By further assuming the electrons are adiabatic, the electron number density, n_e, is then used in the Boltzmann relation to obtain the plasma potential:

(5.122)

where φ*,T*, and n* are reference values and γ=5/3. In the case of the μ-LPT, the reference point for the Boltzmann relation is taken as the target surface. It is assumed that the potential here is constant.

The 2D axisymmetric simulation uses a single grid for both the collision and ion motion calculations. Since charge neutrality is assumed, the PIC cells are not required to be of the order of the Debye length. Instead they are chosen to be small enough to resolve in a reasonable way the gradients in the potential. At the same time, the cells satisfy the DSMC requirement that their size be less than a mean free path. The background pressure of the order of 10⁻⁵–10⁻² Torr, typical for Laboratory conditions, is simulated. The background pressure gas is assumed to be fully composed of hydrogen atoms at a temperature of 300 K.

5.4.3.2 Example of calculation of expanding plume from microlaser plasma thruster

For simplicity, we consider here a two-component plasma consisting of carbon and hydrogen. The chlorine component of PVC is omitted due to its relatively low abundance. Two examples of a 2.5–8 W diode laser beam focused on a 25 μm radius spot were considered. The pulse duration is 3–10 ms, and the experimentally measured thrust-to-power (coupling coefficient) of about 70–200 μN/W is used in order to estimate the recoil pressure.

The particles injected into the DSMC–PIC simulations are sampled from the equilibrium velocity distribution [178]. The grid employed in the plume computation contains 300 by 300 nonuniform rectangular cells. The flow domain extends to about 3 cm in both the axial and radial directions from the center of the ablation spot. A constant time step of 2×10⁻¹⁰ s is employed that is smaller than the smallest collision and plasma timescales (the inverses of the maximum collision and plasma frequencies, respectively). A steady state is reached after 40,000 iterations and final results are obtained by averaging over a further 50,000 steps. A maximum of more than 1 million particles is employed in the simulation.

Contours of plasma densities are shown in Figure 5.59. One can see that background gas pressure strongly affects the plume expansion due to CEX and momentum transfer elastic collisions.

The distribution of plume deposition on a witness plate was obtained experimentally. The schematic of the experiment is shown in Figure 5.60. The axes in this figure are labeled in degrees, with 0° corresponding to the direction normal to the tape surface.

The distance from the laser focus to the cylinder on which the witness plate is rolled is 6 cm. The cylinder is made of metal and mounted to the vacuum chamber so that it is always in the same location. The X-axis in the images is the distance around the circumference of the cylinder. The Y-axis is the distance along the length, and the degree marks are not equally spaced. The deposit is attached to the metal cylinder as shown in Figure 5.60.

The normalized flux profiles are shown in Figures 5.61 and 5.62 for all cases (low and high power and low and high pressure, and vacuum). It is assumed that the black deposition material on the witness plate is carbon (this assumption was verified by chemical element analysis of the plume signatures using SEM) and so the simulation results in Figures 5.61 and 5.62 contain the total fluxes of carbon ions and atoms. One can see differences in the two deposition signatures due to the background pressure effect. High background pressure leads to narrow plume, while low-pressure case corresponds to large plume expansion. In turn, the strongest effect of the background pressure can be seen in the higher pressure case, while in the low-pressure case calculations in the vacuum case produces similar result as the one for low pressure. One can see that the simulations generally predict a narrowing (focusing) of the plume. These results are comparable to previous experimental and theoretical study in the similar background pressure range [238,239]. In this relatively low-pressure regime (<10⁻² Torr), the effect of ablated atom collisions with background gas (hydrogen in our case) leads to redistribution of the ablated atom velocity toward the axial direction. These predictions are confirmed by present experiments as shown in Figures 5.61 and 5.62 showing the plume narrowing.

The above example of the code validation illustrate that the hybrid PIC–DSMC approach is the appropriate tool for plasma plume simulations.

5.4.4.1 Effect of the magnetic field on the plasma plume

In this section, we describe effect of the magnetic field on the plasma potential distribution in the plume of the Hall thruster. A quasi-1D model of the three-fluid (ions, electron, and neutrals) plasma flow is considered. The model is described elsewhere [152]. The effect of the magnetic field appears in the equation for potential that can be obtained by solving the electron momentum equation for the potential:

(5.123)

where B is the magnetic field, V is the flow velocity, and β_e is the Hall parameter. The solution procedure and the approach is described in Ref. [152]. In this section, the main results of this model will be highlighted.

The axial potential distribution is shown in Figure 5.63 with the electron temperature as a parameter. One can see that the potential sharply increases initially and then saturates. For comparison, the experimental data [240] on the lines of 0° and 45° off the thruster centerline are plotted. It can be seen that, in general, the predicted results are in agreement with experiment. For instance, in the experiment, plasma potential saturation with axial distance was observed in the case of 0° as is predicted by the model.

The plasma potential distribution with magnetic field as a parameter is shown in Figure 5.64. In the case of a zero magnetic field, the plasma potential is negative and decreases with axial distance according to the Boltzmann relation. The plasma potential becomes positive when the magnetic field is finite. One can see that, in general, the plasma potential increases with increasing magnetic field.

The potential increase across the magnetic field can be understood in the following manner. The cases considered correspond to a plasma jet entering a transverse magnetic field with a high supersonic directed velocity in the condition that the magnetic field is relatively weak. This means that only electrons are magnetized, while ions are not magnetized on the spatial scale of interest. Since plasma flow across magnetic field is ambipolar, an electric field is formed that leads to the potential increase across the magnetic field.

5.4.4.2 Near-field plasma plume of PPT

The model is based on a hybrid approach involving a DSMC description of neutrals, a PIC approach for ions, and a fluid description of the electrons. In these methods, the potential distribution is usually calculated by reducing the electron momentum equation to the Boltzmann relation in the absence of a magnetic field. When the magnetic field was presented in the plasma plume domain, the electron momentum equation including the magnetic field effect should be used instead of Boltzmann relation.

5.4.4.3 Boundary conditions for the plume simulation

During the discharge, the plasma density near the propellant face is large (on the order of 10²³–10²⁴ m⁻³ [25,241]) and therefore a fluid approach can be used. The plasma layer model includes the following features (similar to the model of an electrothermal PPT [25,241]): Joule heating of the plasma, heat transfer to the Teflon surface, and Teflon ablation. Mechanisms of energy transfer from the plasma column to the Teflon wall include heat transfer by particle convection and by radiation. The Teflon ablation computation is based on a kinetic ablation model described in Chapter 1.

5.4.4.4 Plasma plume electrodynamics

As it was mentioned above, the hybrid fluid-particle approach was used for the plasma plume description in which the neutrals and ions are modeled as particles, while electrons are treated as a fluid. Elastic (momentum transfer) and nonelastic (CEX) collisions were taking into account. The particle trajectory due to collisions is calculated using the DSMC method [233]. Momentum-exchange elastic cross sections use the model of Dalgarno et al. [242], while CEX cross sections was calculated according to Sakabe and Izawa [236]. Acceleration of the charged particles is computed using the PIC method [237]. A single grid employing nonuniform, rectangular cells is used for both the DSMC and PIC steps. Since the flow is assumed to be quasi-neutral, there is no requirement to resolve the Debye length. Hence, the cells are scaled by the local mean free path. A single time step given by the reciprocal of the maximum plasma frequency is used throughout.

In the presence of a strong magnetic field, the electron density distribution deviates from that of the Boltzmann distribution [84]. In the case of a magnetic field, the electron momentum equation reads (neglecting electron inertia):

(5.124)

We have assumed quasi-neutrality therefore N_e=N_i=N. Using a definition j=eN((V_e−V_i), Eq. (5.125) is usually referred to as the generalized Ohm’s law. The electric and magnetic field distributions in the plume were calculated from the set of Maxwell’s equations (see Chapter 2). We further assume that the magnetic field has only an azimuthal component and also neglect the displacement current. The combination of Maxwell’s equations and electron momentum conservation (5.125) gives the following equation for the magnetic field:

(5.125)

where σ=e²N_e/(ν_eim_e) is the plasma conductivity which has only a weak dependence on the plasma density (in the Coulomb logarithm). Using this fact, it was assumed that plasma conductivity depends only on the electron temperature in the plasma plume. A similar approach regarding the assumption about conductivity was also employed in the plasma flow simulation in the PPT [243]. The electron temperature in the plume was assumed to be constant. The plasma conductivity is also constant (mainly due to the fact that electron temperature is assumed to be constant in the near plume [226]) and that the density gradient does not affect the magnetic field diffusion. The last assumption comes from the fact that the main density gradient is developed in the direction of magnetic field transport (as clear from the results below) and therefore does not affect magnetic field transport [32].

A scaling analysis shows that in the case of the near plume of the μ-PPT with a characteristic scale length L of about 1 mm, the magnetic Reynolds number Re_m=μσLV1 (where V is the characteristic velocity ~10⁴ m/s as shown below) and therefore the last term in Eq. (5.126) can be neglected. In addition, depending on the plasma density, the Hall effect (second term on the right-hand side of Eq. (5.125)) may be important for the magnetic field evolution. One of the first calculations of the plasma flow with Hall effect was performed by Brushlinski and Morozov (see Ref. [244] and references therein), who considered isothermal flow. The plasma densities become high at the cathode and lower at the anode. The Hall effect has a particularly noticeable influence on the magnetic field distribution. The field near the anode increases and near the cathode decreases. As a result, the current is deflected to the side and grazes the anode. Our estimations of the Hall parameter show that ωτ1 if the plasma density near the Teflon surface N>10²³ m⁻³. This is usually the case in the μ-PPT (PPT with the scale length of about few mm and low power), so the Hall effect is expected to be small and will not be considered in this analysis. Therefore Eq. (5.126) is reduced to the simple magnetic transport equation:

(5.126)

Having the magnetic field distribution, one can calculate the current density distribution from Ampere’s law:

(5.127)

The magnetic field and current distributions calculated from this model are used in PIC for ion motion calculations. The ion velocity distribution depends upon the magnetic field distribution, and the ion motion is calculated using following momentum equation:

(5.128)

The electric field in this equation was determined using the electron momentum equation and therefore the last equation reduces to the following simplified form:

(5.129)

5.4.4.5 Example of plasma plume simulations

The results are presented for a 3.6 mm (0.141″) diameter axisymmetric μ-PPT, which has a 0.9 mm diameter central electrode, 3.1 mm propellant diameter, and 0.24 mm anode wall [174]. Results presented below correspond to energy of about 2.25 J and C=0.5 μF.

Figure 5.65 shows evolution of the carbon ion (C+) component of the plasma plume during the main part of the pulse. One can see that a high-density region is developed at a few millimeters from the thruster exit plane.

This dense region exists during the entire pulse as shown in Figure 5.65, but the plasma density decreases from about 2×10²² m⁻³ at the beginning of the pulse down to 0.3×10²² m⁻³ at 5 μs. At the beginning (first 2 μs), the C+ density mainly develops a gradient in the radial direction that is a result of high directed velocity in the axial direction. Later, during the discharge pulse, the axial density gradient becomes comparable to the radial one.

The fluorine ions (F+), due to their larger mass, have different dynamics as shown in Figure 5.66. They have smaller acceleration in the axial direction even at the beginning of the pulse and therefore both axial and radial density gradients are developed. The F+ density in the plume and in the high plasma density region is larger than that of C+, because originally Teflon has composition C₂F₄ with F density twice that of C. Additionally, F ions experience less acceleration in the plume because of their higher mass that also contributes to their relative density increase.

The μ-PPT is essentially an electromagnetic accelerator as shown in the velocity phase plots (Figures 5.67 and 5.68). The phase plot of the carbon ions at 5 μs is centered at 10 km/s in the axial direction (note that the sound speed is ~5 km/s for a 4 eV plasma). Ions also experience radial expansion due to the magnetic field structure and expansion. The radial velocity in the negative direction is related to the focus formation along the axis, as shown in Figures 5.67. The fluorine ions have generally smaller axial and radial velocities comparing to carbon ions due to their higher mass.

During the entire pulse, there is a population of ions having a negative axial velocity with magnitude up to about 10 km/s (see Figures 5.67 and 5.68). This population creates the backflow contamination that occurs mainly onto the thruster itself. The carbon ions have a larger negative velocity due to their higher mobility that results in their domination in the backflux. This backflux may be mainly responsible for charring phenomena observed in the μ-PPT [173].

5.4.4.6 Comparison with experiment

In this section, the comparison of measured and predicted electron and neutral density distributions in the near-field plume for one μ-PPT design is discussed.

An experimental basis for comparison is provided using results from a two-color interferometer [245]. Electron and neutral density measurements are taken on a 6.35 mm (1/4″) diameter μ-PPT at AFRL. The interferometer uses 488 and 1152 nm wavelengths and a quadrature heterodyne technique described by Spanjers et al. [246]. The two-color technique uses the difference in phase shifts from the two wavelengths to calculate electron and neutral densities. This technique tends to ensure that both lasers are sampling the same portions of the plume [247]. Older data with less focusing is shown with typical uncertainty in Antonsen et al. [248] along with a more thorough treatment of the two-color diagnostic. The data in Figure 5.69 has a maximum uncertainty of ±1.15×10¹⁶ cm⁻³ for electron density and ±1.76×10¹⁶ cm⁻³ for neutral density at 2 µs into the discharge. For the data shown here, the beam center is located 3 mm from fuel face on the thruster centerline with a beam diameter of 6 mm. It should be noted that the discharge in the μ-PPT may be azimuthally nonuniform (nonaxisymmetric) [249]. Therefore a comparison between an axisymmetric model and experiment in this case may be questionable. However, the mentioned nonuniformity decreases with discharge energy[173] and experimental data are reproducible during many pulses. This suggests that any effect of nonuniformity may be small and our comparison is justified.

Figure 5.69 shows the experimental data coplotted with model predictions. Plasma density peaks at about 2×10²² m⁻³ and decreases by several orders of magnitude toward the pulse end. The neutral density is significantly higher and peaks at about 14×10²² m⁻³. The experimental data were taken at discharge energy of 6.0 J from a 0.417 μF capacitor. One can see that plume model predicts very well magnitude and temporal variation of both electron and neutral densities.

1. Calculate the velocity (normalized by local sound speed) at the outer edge of the Knudsen layer as a function of plasma density (10²²–10²⁴ m⁻³). Use iteration to perform analysis. Use the following conditions at the surface: n₀=10²³ m⁻³; T₀=1000 K and plasma temperature is 0.5 eV. Compare results with and without effect of magnetic field. Current is about 10³ A and characteristics size of the electrode is about 1 cm.

2. Compute the Alfven speed in a magnetic field of about 0.1 T and plasma density of about 10²⁰ m⁻³.

3. Thickness of the ionization layer in ablative pulsed plasma thrusters. Demonstrate that the thickness of the ionization layer decreases with the magnetic field strength using the analysis of plasma acceleration in pulsed plasma accelerators.

4. Compute the ablation rate of the Teflon in the case of surface temperature of about 900 K (use system of equation from Chapter 1, Section 5). Teflon surface is in contact with plasma having density of about 10²² m⁻³ and temperature of about 3 eV.

1. Estimate the frequency of near wall conductivity across the channel having width of about 2 cm. Consider xenon plasma having electron temperature of about 30 eV. (use material from Section 5.2.2)

2. Plot the dependence of the current density distribution on potential drop across the sheath. Consider the near wall conductivity. Comment on effect of the sheath. (use material from Section 5.2.2)

3. Calculate the deviation from the equipotential condition along the magnetic line in the plasma having electron temperature of about 10 eV. Consider the plasma density decrease along the magnetic field by factor of 10.

4. Compute the dependence of the sheath potential drop near the dielectric wall in the Hall thruster channel on the SEE coefficient (0–0.9). Consider Xenon plasma flow with electron temperature of about 20 eV.

5. Assume that SEE coefficient has linear dependence on the electron energy (5–20 eV). Plot the dependence of the sheath potential drop on the electron energy. Consider krypton as a propellant.

6. Anodic plasma analysis. Calculate the near anode potential drop assuming that the electron density near the anode is about 10¹⁷ m⁻³, electron temperature is about 7 eV. Discharge current is 3 A and the anode collection area is about 6 cm².

7. Consider the sheath thickness near the channel wall of the thruster with anode layer. Electron temperature can be assumed to be about 100 eV. Consider the discharge voltage of about 3 kV, bismuth as a propellant and the channel wall having the cathode potential. Plot the dependence of the maximum sheath thickness on the electron density (10¹⁶–10¹⁸ m⁻³).

1. Compute the potential drop in the near field plume of the Hall thruster. Consider a plasma flow across the 1 cm layer of a magnetic field of about 0.005 T, Hall parameter of about 10 and density decrease across the layer by factor of 2. Electron temperature is about 5 eV and ion velocity is about 14 km/s.

2. A Hall thruster 10 cm in diameter produces a Xe ion beam at 300 eV. Assume that initially ions have only axial component of velocity. Compute the effective divergence of the beam (angle) if diameter of the beam doubles in a distance of 50 cm and radial component of ion velocity.

3. Estimate the rate of particle production due to charge exchange in the near field plume of Hall thruster having outer diameter of channel of about 10 cm and the inner diameter of about 6 cm. Consider the mass flow rate of about 5 mg/s and the ionization degree of about 10 %. Assume that the ion velocity is about 16 km/s and the neutral particles velocity is about 1 km/s. Plume expansion can be neglected.

4. Compute the sputtering rate of the Kapton plate placed normal to the plasma plume. Consider conditions of the problem #2. Calculate the sputtering at following radial locations: 0, 10, 15 and 20 cm. The molecular weight of Kapton is 382 g/mol and that its mass density is 1.42×10³ kg/m³. Use the sputtering yield of Kapton as a function of ion energy and angle.