Electrode durability and sheared-flow-stabilized Z-pinch fusion energy

The sheared-flow-stabilized (SFS) Z-pinch¹ is one of the fusion systems currently under development for commercial applications in the nascent private fusion industry.² While the plasma confinement performance of SFS Z-pinches as measured by the Lawson criterion³ is steadily improved on devices like the Fusion Z-pinch Experiment (FuZE)⁴ and FuZE-Q,⁵ Zap Energy is also looking ahead to the engineering design of power plants based on this technology.⁶ One of the key factors in potential SFS Z-pinch plant reliability and maintenance intervals, and, therefore, economics, is the durability of the electrodes that sustain the pinch current.

Z-pinch fusion relies on the self-compression that occurs in a plasma column due to the passage of a strong unidirectional current and has a long history in controlled thermonuclear fusion research. Early promising results on the ZETA device in the late 1950s briefly made Z pinches the front runner in fusion energy research.⁷ However, progress in Z-pinch performance was short-lived due to the rampant plasma instabilities endemic to a basic Z pinch. By the 1960s, research on Z pinches as systems for fusion energy was largely abandoned in favor of other approaches. This situation began to change in the 1990s with the proposal to stabilize Z-pinch plasmas via sheared flow.⁸ 

Zap Energy's work commercializing the sheared-flow-stabilized (SFS) Z-pinch concept grew out of earlier development efforts at the University of Washington with LLNL collaborators. FuZE employs high power-handling electrodes, flexible gas injection, and independently switched capacitor bank modules to tailor the discharge current and gas distribution to establish stabilizing sheared flow and pinch current.⁴ Recent experiments corroborate expected deuterium–deuterium (DD) fusion neutron production rates sustained for ∼10 μs, which is thousands of times the magnetohydrodynamic (MHD) instability growth time typical of an unstable Z pinch with similar parameters.⁹ The measured neutron spectra are also consistent with thermonuclear production.¹⁰ Experimental campaigns are underway on FuZE-Q,⁵ the successor device following FuZE, aimed at increasing the pinch current, duration, and DD neutron production.

The SFS Z-pinch approach to fusion energy aims to improve the economic viability of fusion power by creating a system that can use the deuterium–tritium (DT) fuel cycle as advantageously as possible.⁶ SFS Z-pinch plasmas planned for prototypical power plants have lengths of ∼0.5 m, radii of ∼0.15 mm, and densities ∼ 10²⁶ m⁻³. Compared with typical magnetic and inertial confinement approaches, the SFS Z pinch is intrinsically small in physical size, low in technical complexity, and high in power density. Table I lists approximate plasma parameters at each step between the FuZE and a nominal SFS Z-pinch power plant core.^6,11 In all cases, the plasma column length is assumed to remain 0.5 m long.

The current conceptual core design uses a vertically oriented plasma operated in pulsed mode, Fig. 1. The upper electrode is the cathode, also referred to as the inner electrode. The cathode tip is called the nose cone, which is composed of graphite in FuZE and FuZE-Q. The coaxial tube around the upper electrode and the sides of the flowing liquid metal wall that transiently supports the SFS Z-pinch current during its initial formation are also referred to as the outer electrode. The liquid metal pool at the bottom of the first wall serves as the anode during pinch sustainment. The tank of liquid metal surrounding the pinch region serves the functions of heat exchange and tritium breeding medium and forms the main body of the conceptual core. The liquid is pumped from the sides of the tank up and over a weir wall to form a central cavity surrounded by flowing liquid metal. The SFS Z-pinch formation system sits on top of the tank and repeatedly fires pulses of plasma which assemble and generate fusion reactions within the liquid metal lined cavity. Each plasma pulse will produce a nominal yield of 19 MJ of fusion energy as described in Table I. Nominal thermal power is 200 MW, but variation of the pulse rate allows for controlled output power and load following. This core design has the promise to elegantly solve many of the engineering challenges associated with the DT fuel cycle in other fusion technology approaches.⁶ In particular, the extensive use of liquid metal walls and blanket will protect most of the system from radiation damage. The plasma cathode is an exception. The cathode carries and shapes the ∼1 MA currents necessary to form and briefly sustain each SFS Z-pinch plasma pulse.⁴ This function requires a solid cathode structure which is in direct contact with the plasma and, therefore, directly exposed to high heat and particle flux, and high neutron flux emanating from the fusion plasma.

The SFS Z-pinch electrodes are exposed to intense fluxes of heat, plasma particles, and fusion neutrons. While the anode is a continuously renewed pool of liquid metal, the heat and plasma fluxes result in cathode erosion that could limit the operational life of a basic solid SFS Z-pinch cathode to roughly a day in a power plant, see Sec. II. This is far shorter than the lifetime limit expected from neutron damage to the cathode structure which early calculations suggest is on the scale of months. Therefore, the priority in extending the life of the SFS Z-pinch cathode is mitigating the erosion damage, which is discussed in Sec. III. The degree to which cathode and anode materials contaminate the SFS Z-pinch plasma and the impact that contamination has on plasma performance are active areas of study but are not the focus of this work.

The simulations described in Sec. II place the peak instantaneous heat flux on the SFS Z-pinch cathode at ∼500 MW/cm² during the plasma discharge. However, the pinch plasma is only on for ∼200 μs during the ∼100 ms between pulses, equivalent to a duty cycle of 0.2%. The average power impinging on the cathode is, therefore, ∼1 MW/cm². To place these numbers in the broader context of fusion devices, we note that ∼1 MW/cm² (∼10 GW/m²) is substantially larger than the peak outer divertor perpendicular surface heat flux of ∼350 MW/m² projected for a compact, reactor-class tokamak.¹² An SFS Z-pinch cathode has a small (∼10 cm²) area to protect and no magnetic field outside the plasma whereas the tokomak divertor area is >1 m² and located inside a high magnetic field toroidal magnet. In addition, the conceptual SFS Z-pinch core design, Fig. 1, has a coaxial geometry with a cylindrical blanket and straightforward access to the cathode system from above. Thus, there are many options for SFS Z-pinch electrode cooling and damage mitigation, discussed in Sec. III, that either do not apply to tokamak divertors or would be much more difficult to implement in a divertor scenario.

Electrode erosion is a complex process involving several interacting mechanisms. It is convenient to conceptually categorize these mechanisms into thermal effects and plasma effects. Thermal effects include melting and sublimation of the electrode material due to strong heating regardless of the source of the heating (e.g., resistive, radiative, and particle impact). Thermal effects are typically most acute while the electrode is in contact with plasma, though they do not require the presence of plasma. For example, in a pulsed arc device where the electrodes become incandescently hot, they can continue to sublimate between plasma pulses. The presence of plasma in contact with electrode surfaces causes erosion effects specifically associated with impact of energetic charged particles on the solid surface (e.g., physical and chemical sputtering). Considerations of sputtering alone can overestimate electrode erosion by neglecting the redeposition of ejected material back onto the surface. Given the complexity of the situation, it is difficult to derive correct erosion rates for a particular system from the first principles. However, many measurements of empirical erosion rates are available.¹³ 

Table II gives examples of several erosion rate values for common electrode materials under a variety of conditions. Several trends are clear. First, the ratio of eroded mass to the amount of charge passed through the electrode is roughly constant for a particular material and scale of current. Second, the erosion rate for a material tends to increase substantially with order of magnitude increases to the current passing through the electrode. Erosion rates for typical materials and currents can range from ∼10 μg/C at about 100 A to ∼100 mg/C at about 1 MA. Third, carbon, which is often used in the form of graphite, has some of the lowest erosion rates for currents up to the 100 kA range.

The ZaP-HD device tested cathode nose cones composed of solid copper sprayed with a tungsten coating 400 to 500 μm thick.²¹ The tungsten coating eroded away entirely in a circular area approximately 25 mm across over the course of thousands of plasma pulses each of which passed on the order of 10 C of charge through the electrode at 300 kA current.²¹ An order of magnitude estimate for the tungsten coating erosion rate gives ∼100 μg/C, which is comparable to the vacuum arc values cited in Table II, but much lower than the cited high-pressure arc erosion numbers. To avoid the tungsten coating erosion issue, the FuZE program switched to a graphite cathode nose cone.^4,21 The graphite cathode proved more durable and showed no pronounced damage after thousands of shots, which is consistent with the generally lower erosion rates of carbon but has also hampered efforts to quantify the erosion rate. Modeling and spectroscopic measurements are being used to investigate carbon cathode erosion in SFS Z-pinch devices.

The change in surface temperature due to heat flux incident on a surface for a time Δt is $ΔT=2q Δ t/ π b$⁠.²² Here, b is a combined constant representing the thermal properties of the material— $b= κ ρ c p$⁠, where κ is the thermal conductivity, ρ is the mass density of the material, and c_p is the specific heat capacity. Numerically integrating these temperature changes over time results in a time-dependent surface temperature.

For a representative FuZE-like WDM simulation with a peak total current of 550 kA, the result of the surface response analysis is shown in Fig. 2. An applied voltage of 10 kV is assumed. The presented values are shown as a function of the path length (S) along the nose cone of the cathode where zero is a tip and negative values are upward along the cathode body as oriented in Fig. 1. The predicted surface temperature is shown in Fig. 2(a) for the nose cone at the time of peak neutron yield rate, as a function of the path length. Near the nose cone tip, the calculated instantaneous heat flux to the surface approaches ∼0.1 GW/cm² and the modeled surface temperature exceeds 10 000 K, which is well above the sublimation temperature of graphite (T_sub = 3900 K).

An erosion model that includes sputtering and sublimation was developed for the WDM simulation. The sputtering model uses experimental values of sputtering yields for D⁺ on graphite as well as self-sputtering and includes the effect of prompt redeposition to estimate an effective physical sputtering yield (Y_eff).²³ This result and the scaled plasma flux are used to calculate a sputtering erosion flux as $Γ sput= Y eff Γ p l$⁠. Figure 2(c) shows this estimated flux. Erosion from sputtering can approach values of 10²⁴ atoms m⁻² s⁻¹ along the nose cone. Note that the prompt redeposition fraction approaches unity near the nose cone tip. This results in a rapid reduction of the erosion fluxes.

Sublimation is modeled by assuming that all the deposited energy from the fraction of Bohm flux reaching the cathode goes into heating and sublimation. Hence, this represents a worst-case scenario for material loss. To estimate the sublimated particle flux (Γ_subl), the locally deposited energy is divided by the surface binding energy (E_sb = 7.4 eV). The plume of eroded particles can also become ionized and promptly redeposit back on the surface, leading to a reduction in the erosion flux. Where the raw erosion fluxes are much larger than incident particle fluxes, they are truncated to approximate the nonlinear turn-off of erosion as the incident plasma properties are altered. The net sublimated flux is $Γ subl net= Y eff subl T surf Γ subl$⁠.

Figure 2(b) shows sublimation fluxes approaching values of 10²⁹ atoms m⁻² s⁻¹, i.e., about five orders of magnitude greater than what is predicted from physical sputtering. The calculated erosion rate of the graphite nose cone in the simulation is 861 μg/C. Equivalently, it is predicted that nearly 2 μm of the nose cone tip will be eroded over an area a few cm in diameter. These results are roughly consistent with the empirical values shown in Table II. Future work will aim to improve the physics-based modeling of erosion and continue to validate against experimental data.

The particle-in-cell (PIC) plasma simulation tool CHICAGO^24,25 has been employed as an alternative approach to estimating the erosion of the electrode surfaces during an SFS Z-pinch by combining previously demonstrated modeling techniques for dense plasma focus with neutron production^26,27 and methods for estimating electrode mass loss.²⁸ The initial simulation incorporates a coupled inductor, capacitor, resistor (LCR) circuit based on the FuZE power supply operating at ∼450 kA of total current, and a voltage plateau of ∼7 kV across the cathode to anode gap. Additionally, the simulation incorporates a resistive layer on the cathode at z = 0 cm with an electrical conductivity of 8900 S/m to emulate the carbon cathode currently used in experiments. Including this resistive layer results in higher surface temperatures leading to increased melt depths of ∼20% when compared to a perfect metal conductor. Figure 3 shows the estimated plasma density for a representative FuZE experiment. From this simulation, the energy deposition along the electrode surfaces can be calculated for Joule heating, particle energy deposition, and radiation. The estimated energy deposition rate into a 0.35 cm radius spot at z = 0 is 0.3 J/ns. This PIC simulation predicts plasma density and temperature of n ∼1–5 × 10²² m⁻³ and T ∼1.5 keV and an estimated surface heat flux at the tip of the cathode of ∼0.7 GW/cm², all of which generally corroborates the conclusions of the MHD WDM simulations.

For the FuZE experiment, the C III (C²⁺ to C¹⁺ transition) emission at 229.7 nm was observed with a telescope that has an integration spot size of 640 ± 150 μm for 40 fibers that were aligned in four rows and ten columns. The rows are 22 mm apart, and the columns are 6 mm apart. The gross eroded flux along the nose cone close to the center axis of the machine was collected during scans of the pinch time and plasma current. The experimental setup is shown in Fig. 4, and the emission intensity measured in front of the nose cone is shown in Fig. 5. Figure 5(a) shows the emission intensity before, during, and after the peak pinch current timing at a pinch current of 250 kA. Figure 5(b) shows the emission intensity as a function of plasma current at the peak pinch current time. The S/XB value can range from 1 to 1500 depending on the density and temperature in front of the nose cone. The eroded flux from this method is the minimum value of the gross eroded flux since the redeposition fraction approaches unity as the density and temperature of the plasma increase. The net erosion will be orders of magnitude lower than the gross and can be measured in the future with weight loss or witness plate experiments.

This measurement is still under development and the calculated erosion yield is lower than the experimentally derived peak gross erosion value due to a large amount of carbon light that was measured in the line integrated measurement. This led to an increase in experimentally derived sputtered flux as the collection is not solely from the electrode surface but the entire line of sight. This indicates some carbon in the plasma from sources other than the cathode nose cone. High gross erosion with high redeposition will lead to a churning of the electrode surface, which is the subject of planned electron microscopy studies.

The electrodes of an SFS Z-pinch device must transfer a large amount of charge. As described in Table I, a power plant level pinch will have a current of ∼1.5 MA lasting for ∼200 μs, which equates to 300 C of charge. A nominal 200 MW thermal SFS Z-pinch power core will fire at roughly 10 Hz.⁶ At that rate of operation, approximately 1.1 × 10⁷ C per hour will pass through the cathode. If the cathode is simple solid carbon, then we expect it will be subject to erosion similar to or greater than the very high mass losses of carbon electrodes that are seen in the operational experience of arc smelting furnaces where typical values are ∼1.7 tons per 24 h at 80 kA arc current, or ∼2.5 × 10⁻⁴ g/C.^20,30 Applying that erosion rate to a 10⁷ C per hour SFS Z-pinch electrode yields an expected mass loss of ∼3 kg per hour. A solid tungsten electrode could lose ∼600 kg per hour if SFS Z-pinch electrode erosion rates reach those seen in high current, high-pressure discharges, see Table II. In either case, such high rates of erosion would lead to impractically short maintenance cycles in an SFS Z-pinch fusion power plant without a mitigation strategy in place. Fortunately, several classes of solutions are available to reduce the cathode erosion problem to manageable levels.

One straightforward approach to dealing with electrode erosion is to start with a longer than necessary cathode and gradually feed it into the system as the tip erodes. This is the approach taken in direct current arc smelting furnaces where the electrode for a 60 MW system is a graphite rod 60 cm in diameter supported by a hydraulically operated mechanical arm that can raise and lower the electrode as required.³⁰ Given the high rate of erosion discussed above, electrode sections must be added at least once per day, which typically results in a minimum of 25 min of down time.³⁰ 

Arc smelters neither have nor need control over the geometry of the electrode tip. It remains to be seen how erosion modifies the geometry of an SFS Z-pinch electrode or the shape tolerance necessary to maintain the plasma performance. The linear geometry of the electrode and overall SFS Z-pinch power core system, as well as its small volume and mass, facilitates straightforward remote removal and replacement. Given that flexibility, mechanically fed electrodes or a very frequent removal and refurbishment schedule are potential solutions to the cathode erosion issue in an SFS Z-pinch power plant.

If continuous feed or frequent replacement of a solid electrode that is simply allowed to evaporate away at a high rate is not practical for an SFS Z-pinch power plant, then methods to cool and mitigate damage to a long-lasting solid electrode structure are necessary. One means of addressing the thermal challenges of preventing the melting or sublimation of surfaces exposed to extremely high heat fluxes is transpiration cooling. Transpiration cooling typically involves moving gas or liquid through small pores in the outer wall of the body being cooled and letting the heated gas or vaporized liquid escape. As the coolant leaves the surface it takes its entrained heat with it. Transpiration cooling has a long history in the fields of missile nose cones³¹ and hypersonic aircraft leading edges,³² where heat fluxes can exceed 1 MW/m². Similar ideas have been proposed for high-power density fusion system design. For example, the Evaporation of Lithium and Vapor Extraction (EVOLVE) concept envisioned a transpiration-cooled tungsten alloy first wall receiving a heat load of 2 MW/m² that is removed by the evaporation of liquid lithium moving through capillaries in the walls.³³ Similar approaches could be applied to the cooling of an SFS Z-pinch electrode.

While transpiration cooling can deal with the heat loads to the electrode, it does not necessarily mitigate plasma particle erosion effects if plasma is still impinging on solid electrode material between the transpiration capillaries or pores. In the limit of extremely high capillary density or high flow rate, one would expect the surface of a liquid-metal transpiration-cooled electrode to be completely covered with liquid metal. At that point, the solid substrate should be completely protected by the liquid metal layer. Indeed, the idea of using a liquid metal cathode in high-power systems goes back to at least the late 1960s in the context of electric space propulsion³⁴ and electrical switching.³⁵ This approach could also prove effective in an SFS Z-pinch power plant.

Demonstration of high gain plasma performance in the SFS Z pinch will open the pathway to implementing simple and flexible DT fusion power core designs.⁶ The electrode that carries and shapes the mega-ampere scale currents needed to assemble and briefly sustain each SFS Z-pinch plasma pulse is a vital part of that system. Any simple solid electrode will erode very quickly under expected SFS Z-pinch power plant conditions, potentially requiring daily replacement. While this seems operationally daunting for a future power plant, past industrial experience shows there are practical ways to handle frequent mechanical replacement and past research shows there are multiple engineering options with the potential to radically extend the operational lifetime of the electrode structure in the system. To that end, Zap Energy is working on several damage mitigation techniques to increase cathode longevity and reduce the maintenance frequency in future power plants.

The authors would like to acknowledge and thank Dale Welch and Dave Rose of Voss Scientific, LLC for their work under contract with Zap Energy on the setup, execution, and post analysis of the CHICAGO simulations of FuZE.

The authors have no conflicts to disclose.

Matthew Colin Thompson: Conceptualization (lead); Investigation (lead); Writing – original draft (lead); Writing – review & editing (supporting). Sean Simpson: Conceptualization (supporting); Investigation (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). Clyde Joshua Beers: Investigation (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). Jonny Dadras: Investigation (supporting); Software (equal); Visualization (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). Eric Meier: Investigation (supporting); Software (supporting); Writing – review & editing (supporting). Peter H. Stoltz: Investigation (supporting); Writing – original draft (supporting); Writing – review & editing (supporting).

The data that support the findings of this study are available within the article.