# Simulated temperature of a tungsten spot facing large plasma heat loads

J. Moritz, S. Heuraux, M. Lesur, E. Gravier, F. Brochard, L. Marot, P. Hiret

► To cite this version:

J. Moritz, S. Heuraux, M. Lesur, E. Gravier, F. Brochard, et al. Simulated temperature of a tungsten spot facing large plasma heat loads. Nuclear Materials and Energy, 2024, pp.101753. 10.1016/j.nme.2024.101753. hal-04729195

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Submitted on 9 Oct 2024

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# Simulated temperature of a tungsten spot facing large plasma heat loads*

J. Moritz<sup>a,*,1</sup>, S. Heuraux<sup>a</sup>, M. Lesur<sup>a</sup>, E. Gravier<sup>a</sup>, F. Brochard<sup>a</sup>, L. Marot<sup>b</sup> and P. Hiren<sup>b</sup>

## ARTICLE INFO

### Keywords:

Thermionic current

Space-charge-limited regime

Tungsten surface

PIC simulations

## ABSTRACT

In fusion devices like ITER, plasma-wall interactions are a significant concern due to the high heat fluxes, often tens of MW/m², impacting the first wall. These intense heat fluxes can lead to the formation of hot spots on components facing the plasma, such as tungsten, used in divertor plates and antennas. This results in material erosion and plasma core contamination. Our study investigates the thermal behavior of tungsten surfaces under these conditions using fluid modeling and Particle-In-Cell (PIC) simulations. We examine the effects of thermionic electron emission on the sheath potential and heat transmission. The simulations reveal that thermionic emission can decrease the sheath voltage, increasing the surface temperature due to enhanced heat flux due to electrons. Additionally, we explore how the ratio between the spot size (S) and the surrounding surface length (L<sub>y</sub>) influences the surface temperature. We find that a higher L<sub>y</sub>/S ratio allows the surface to reach higher temperatures before the system enters the space-charge-limited regime, where thermionic current is maximized and considerably larger than the case where the entire surface is emissive (L<sub>y</sub> = S).

## 1. Introduction

The use of tungsten (W) as a plasma facing material in future fusion devices such as ITER has raised the issue of plasma core pollution with high-Z metal which can eventually lead to radiative disruption. There are several sources of contamination of the plasma core with W triggered by plasma-wall interactions, including sputtering, droplet ejection following the local melting of elements facing the plasma such as the W mono-blocks that make up the divertor, or unipolar arcs. While sheath rectification close to ion cyclotron range of frequency antennas for instance can lead to physical sputtering of W [1], misaligned mono-blocks in the divertor can experience temperatures up to the W melting temperature due to heat flux of tens of MWm⁻² [2, 3] and induce droplet ejection. These melting events are associated with strong thermionic emission and a net current passing through the sample and returning to the vessel [4]. Several authors have then investigated the role of sheaths near hot emissive W surfaces during ELM phases [5, 6, 7] or use successfully numerical tools such as the MEMOS-U code to investigate the macroscopic melt motion driven by the intense volumetric $\vec{J} \times \vec{B}$ force showing the large implication of thermionic emission [8, 9]. The latter is also involved in unipolar arcs arising between the metallic surface and the plasma [10]. These arcs can drive a considerable amount of current of several amperes and are fed by the surface material which is released from it instead of the neutral gas or main plasma. They leave the surface with many micron range craters where the material can be splashed away as droplets [11, 12].

Thermionic emission of electrons is then particularly relevant in fusion devices and is well known as altering the energy transmission through the sheath [13]. The decrease in sheath potential can increase the heat flux to the wall, raising the surface temperature T<sub>s</sub> and the thermionic current J<sub>s</sub> according to the Richardson-Dushman equation:

$$J_s = AT_s^2 \exp\left(-\frac{B_w}{k_b T_s}\right), \quad (1)$$

where A is the Dushman's constant and $B_w$ the electron work function. For tungsten [14], $A = 60 \times 10^4$ Am⁻²K⁻² and $B_w = 4.55$ eV.

The accumulation of negative charges near the wall, due to thermionic emission, is what causes the decrease in sheath potential $\phi_s$ and the increase in heat flux on the surface by the plasma electrons. This leads to an increase in T<sub>s</sub> and thus in J<sub>s</sub> until equilibrium is eventually reached [15]. This feedback loop is also of great importance when applied to dust in tokamaks because it leads to an enhancement of the dust heating by the plasma and its vaporization [16]. The reduction of the sheath potential cannot continue indefinitely until it vanishes though. At a critical and maximum value of the current $J_s = J_s^*$, the electric field on the surface E<sub>s</sub>, which is generally negative to accelerate ions and repel electrons, eventually cancels out. This marks the onset of the space-charge-limited (SCL) regime where the sheath potential is at its minimum $\phi_s^*$ and the thermionic current is regulated.

In this study, we investigate the behavior of a single spot on a tungsten surface when exposed to a hot plasma with varying temperature and density. In the first part of the paper, we present a fluid model to evaluate the spot temperature in a 1D approximation. This model accounts for the circulation of electronic current between the spot and

*

<figure><img src="image_2.png" /></figure>

jerome.moritz@univ-lorraine.fr (J. Moritz)

<figure><img src="image_3.png" /></figure>

(J. Moritz)

ORCID(s):

<footnote><sup>a</sup>Institut Jean Lamour, CNRS - Université de Lorraine, 2 allée André Guinier, Nancy, 54011, France</footnote>

<footnote><sup>b</sup>Department of Physics, University of Basel, Klingelbergstrasse 82, Basel, CH-4056, Switzerland</footnote>
<figure><img src="image_4.png" /><figcaption>**Figure 1:** Sketch of the studied plasma bounded by a grounded conductive wall on its left side. The sheath potential profile $\phi(s, y)$ is indicative of what is expected in 2D, with a slightly lower sheath potential at the vertical of the spot than in the rest of the plasma. The profile of $\phi_f$ depicted in the figure is indicative of what is expected for the sheath potential in the absence of thermionic emission.</figcaption></figure>

the surrounding surface like in the unipolar arc picture [17]. In the second part, 2D Particle-In-Cell (PIC) simulations corroborate the trends deduced from the fluid model and highlight the importance of the ratio between the spot size and the surrounding surface on the maximum thermionic current.

## 2. Simulated system and fluid model

The simulated system is a hydrogen plasma bounded on one side by a conducting wall that is grounded. On the other side, at $x = L_x$, the electric field is assumed zero. The tungsten surface contains a spot of a fixed size $S = 20 \ \mu m$ [18], whose thermal conductivity is reduced compared to the surrounding surface of length $L_y$, as shown in figure 1. The spot area could be associated for instance with a point defect induced by He irradiation [19], or it could correspond to a porous redeposit, or even be a dust particle with poor thermal contact with the surface, although in the latter case, it would be preferable to model it through a thermal contact conductance [20]. The spot thermal conductivity $\kappa$ in our calculations varies between 2.5 and 160 Wm$^{-1}$K$^{-1}$, but we will mainly focus on a single specific case where $\kappa = 15$ Wm$^{-1}$K$^{-1}$, which is about 10 times smaller than pristine tungsten.

As a first attempt, we assume that the peripheral surface around the spot has the thermal conductivity of bulk tungsten and remains at temperature $T_0$ throughout the simulations, which is also the temperature of the cool side of the $l = 1$ cm thick wall as shown in figure 1. We arbitrary set this temperature to 300 K, but another operating temperature could be eventually chosen. As a consequence, there is no heat conduction in the y direction nor temperature or

thermal conductivity gradients. There is also no magnetic field included in our model, which limits its applicability, as in tokamak plasmas, the magnetic field lines are grazing the surface, and prompt redeposition can completely change the sheath properties with respect to thermionic emission. The latter approximations need to be improved in future studies.

The objective of the present simple model and simulations is to self-consistently calculate the spot temperature $T_s$ as a function of hydrogen plasma parameters (temperature and density), the spot's thermal conductivity, and the $S/L_y$ ratio. The heat flux from the plasma to the surface $Q_p$ depends on the density at the sheath entrance $n_s$, the ionic $T_i$ and electronic $T_e$ temperatures, as well as the recombination energy for hydrogen $E_i$ and the work function $B_{w}$. It writes [15]:

$$Q_p = \frac{J_i}{e} (2k_bT_i + e\phi_s + E_i - B_{w}) + \frac{J_{es}}{e} (2k_bT_e + B_{w}), \quad (2)$$

where $J_i = en_s c_s$ is the ion current and $J_{es} = \frac{en_s c_e}{4} \exp(-\frac{e\phi_s}{k_b T_e})$ is the electron one; $c_s$ is the ion acoustic velocity and $c_e = \sqrt{\frac{8k_b T_e}{\pi m}}$ where $m$ is the electron mass and $\phi_s$ is the electric potential at the sheath entrance (which is positive with respect to the grounded left wall).

The heat flux $Q_c$ from the surface in our model is attributed to conduction through the wall, radiation, and the heat flux carried by thermionic electrons penetrating the plasma. It writes:

$$Q_c = \frac{\kappa}{l} (T_s - T_0) + \epsilon\sigma T_s^4 + \frac{J_s(T_s)}{e} (B_{w} + 2k_b T_s), \quad (3)$$

where $\epsilon$ is the emissivity of the surface and $\sigma$ is the Stefan-Boltzmann constant.

To determine the potential at the sheath entrance $\phi_s$, which affects both heat fluxes, it is necessary to balance the charges at the wall so that the total current is zero and quasi-neutrality is maintained. According to the notations in figure 1, we can write that:

$$L_y J_i + S J_s = S J_{es} + (L_y - S) J_e. \quad (4)$$

Assuming the electron flux at the spot is very close to that reaching the surrounding surface, $J_{es} \approx J_e$, to reduce the system's dimensionality, we have $J_i + S/L_y \times J_s \approx J_e$. This allows us to calculate $\phi_s$ as (see reference [21]):

$$e\phi_s = e\phi_f - k_b T_e \log \left( 1 + \frac{S}{L_y} \frac{J_s}{en_s c_s} \right), \quad (5)$$

where $\phi_f$ is the well-known floating wall potential given by $-\frac{k_b T_e}{2e} \log(2\pi \frac{m}{M}(1 + \frac{T_i}{T_e}))$; $M$ is the considered ion mass (hydrogen). In the following we will assume that $n_s = \frac{1}{2}n_0$, with $n_0$ the plasma density. Note that the ambipolarity condition given by equation 4 does not allow for the direct use of this model in the case of thermionic current emitted at the mono-blocks during macroscopic melting events where
<figure><img src="image_5.png" /><figcaption>**Figure 2:** a) Heat flux from the plasma $Q_p$ for different $L_y/S$ ratios and for a plasma density $n_0 = 0.2 \times 10^{19} \text{ m}^{-3}$. $Q_c$ is the heat flux from the surface. b) $Q_p$ and $Q_c$ for a plasma density $n_0 = 0.32 \times 10^{19} \text{ m}^{-3}$. c) Spot temperature $T_s$ with respect to the plasma density $n_0$ for different $L_y/S$ ratios. In all simulations, $k_b T_i = k_b T_e = 35 \text{ eV}$.</figcaption></figure>

there should be a net current integrated over $L_y$ (if $L_y$ is the sample size).

The surface temperature $T_s$ for given plasma conditions can thus be calculated by finding the intersection of the $Q_p(T_s)$ and $Q_c(T_s)$ curves: by fixing $T_s$, we can calculate the current $J_s$ and so the sheath potential $\phi_s$ using expression 5, leading to the determination of the two heat fluxes through equations (2) and (3). An additional condition is necessary in calculating $T_s$ because the sheath potential cannot decrease to zero as explained earlier and one needs to evaluate the electric field at the surface $E_s$ and determine the critical temperature at which it cancels in order to get $J_s^*$. Note that the present case can only be rigorously treated in two dimensions, accounting for potential variations around the spot in both x and y directions. However, as we did above to calculate currents at the surface to derive equation (5), we can rely on a one dimensional case to determine the critical couple ($J_s^*$, $\phi_s^*$) at which $E_s(J_s^*, \phi_s^*) = 0$. Therefore as we did in our previous study [21], we used here the expression of $E_s$ derived by Hobbs and Wesson in their model [13]. We thus neglect the motion of electrons in the y-direction and assume that they accumulate in front of the spot as in a one-dimensional case.

<figure><img src="image_6.png" /><figcaption>**Figure 3:** a) Variation of the spot temperature $T_s$ for attached-like divertor conditions with respect to the plasma density $n_0$ calculated thanks to PIC simulations. b) Spot temperature determined by fluid modeling as presented in section 2.</figcaption></figure>

In figure 2a are depicted the variations of the heat fluxes $Q_p$ and $Q_c$ with respect to $T_s$ calculated with equations (2) and (3) for a hydrogen plasma with $k_b T_i = k_b T_e = 35 \text{ eV}$, $n_0 = 0.2 \times 10^{19} \text{ m}^{-3}$ and $\kappa = 15 \text{ Wm}^{-1}\text{K}^{-1}$. The general trend of these flux curves, particularly of $Q_p$, can be explained by the relative weight of the thermionic current in equation (5), which gives the sheath potential. This weight is monitored by the ratio $L_y/S$: the larger it is, the closer $\phi_s$ is to $\phi_f$, and the heat flux deposited by the electrons on the surface decreases. Similarly, a larger current $J_s$ can be emitted by the spot before reaching the SCL regime, where $Q_p$ no longer varies with $T_s$ ($\phi_s$ remaining equal to $\phi_s^*$ over this temperature range). The intersection of the characteristics $Q_p = Q_c$ further shows that for this specific density, there is a bifurcation in the cases $L_y/S = 1$ and $L_y/S = 2$. This type of bifurcation has already been identified numerically and experimentally by several authors [15, 22]. Although three distinct solutions are found (two stable and one metastable), the system will tend to transition abruptly between the two stable solutions, meaning it will shift from a regime of low thermo-emission to a highly emissive regime and reciprocally when plasma density is slightly changed.

When the plasma density increases as in figure 2b, the heat flux curves $Q_p$ are shifted upward. Where a bifurcation appeared, for example, for $L_y/S = 2$ and $n_0 = 0.2 \times$
$10^{18}$ m⁻³, the intersection $Q_p = Q_c$ shows here that the system has transitioned to the high-temperature regime, corresponding to a region of strong thermionic emission. More surprisingly, the bifurcation disappears for high ratios $L_y/S > 8$: the slope of the $Q_c(T)$ curve becomes steeper than that of $Q_p(T)$, and only one intersection point is found in the region of interest, i.e., in the range 2500-3000 K. The variations of $T_s$ with plasma density extracted from these types of characteristics are shown in figure 2c. Generally, it can be noted that:

* In the regime where the spot emits little or no current $J_s$ ($T_s < 2500$ K), the impact of the $Ly/S$ ratio is marginal, and the spot temperature varies similarly with $n_0$. It is indeed the plasma density that drives the behavior of the spot because other parameters ($T_i$, $T_e$, or $\phi_s$) are unchanged.

* The spot temperature at which the bifurcation occurs when traversing the curve from ascending or descending densities increases with the $L_y/S$ ratio because the latter reduces the weight of $J_s$ in expression 5. Finally, for the same plasma density, in the SCL regime, the spot temperature is lower when the $L_y/S$ ratio increases, giving a larger area for the return of the plasma electron current to the surface.

### 3. PIC simulations

PIC simulations were used to verify if the 1D approximation from the previous section remains valid for determining the surface temperature of an isolated spot. The simulated system is similar to that presented in figure 1. At each iteration, the heat flux from the plasma deposited on the spot is calculated, and $T_s$ is determined by equating this flux with $Q_c$ given by equation (3). This allows for the injection of a specific number of thermionic electrons at the same iteration using the Richardson formula [equation (1)]. This procedure is repeated as long as necessary to reach a steady state. For more details on our numerical model, interested readers may consult reference [21].

We specifically studied two plasma conditions corresponding to attached or detached divertor plasma with typical temperatures of $k_b T_i = k_b T_e = 35$ eV and $k_b T_i = k_b T_e = 5$ eV, respectively. The plasma density $n_0$ was varied in both cases over the range $10^{18} - 10^{20}$ m⁻³ to calculate the evolution of the spot temperature and eventually reach the SCL regime. Its maximum value for attached divertor conditions is $1.2 \times 10^{19}$ m⁻³ and $16 \times 10^{19}$ m⁻³ for detached ones. The thermal conductivity of the spot was kept constant to $15$ Wm⁻¹K⁻¹. Note that to facilitate scaling with plasma density in the PIC simulations and maintain a constant number of cells in the x direction, the size $L_x$ is fixed at $50\lambda_d$. This is small compared to what is usually used in PIC simulations, but in the absence of magnetic field and with a thin sheath, is large enough to capture the physics of the system. When the entire surface emits thermionic electrons, i.e., $L_y = S$, the simulations are conducted using a 1D PIC

<figure><img src="image_7.png" /><figcaption>**Figure 4:** a) Variation of the spot temperature $T_s$ for detached-like divertor conditions with respect to the plasma density $n_0$ calculated thanks to PIC simulations. b) Spot temperature determined by fluid modeling as presented in section 2.</figcaption></figure>

code as in our previous study [21] and corresponding results labeled $L_y = S$ in the following. When current feedback is possible through the area surrounding the spot with $L_y > S$ (with a fixed $S = 20$ µm), then $L_y$ is set to $50\lambda_d$ for the same reasons as previously mentioned for $L_x$ and the use of a 2D PIC code is mandatory. The latter results are labeled $L_y > S$ in the text and in the figures.

In figure 3 is displayed the temperature variation of the spot with respect to the plasma density for attached-like divertor conditions obtained with the PIC code (a) and compared to that calculated with the fluid approach detailed in section 2, (b). The same comparison for a detached-like divertor plasma is depicted in figure 4.

In both simulated plasma conditions and for both approaches (PIC vs fluid), the same trend is observed, namely the disappearance of the bifurcation (the rapid jump of $T_s$ at a critical density) when $L_y > S$, with a smooth variation of $T_s$ vs $n_0$ until the SCL is reached. Then a change in the slope marks the entry into the latter and the spot temperature increases more rapidly. A typical 2D potential profile corresponding to this regime is shown in figure 5 with a virtual cathode at the spot vicinity regulating the thermionic emission. Note that as the Debye length decreases with $n_0$, it turns out $L_y/S > 30$ and 3 in both attached and detached cases (see figure 6). Furthermore, the spot temperature is always higher when $L_y/S = 1$ in the SCL regime but is equivalent to the case $L_y > S$ for lower densities, as
<figure><img src="image_8.png" /><figcaption>**Figure 5:** a) Zoom of the 2D density plot of the plasma potential normalized to $T_e$ in the detached-plasma case for $n_0 = 15 \times 10^{19} \text{ m}^{-3}$. b) Corresponding variation of the normalized potential in the y direction for different abscissas exhibiting the virtual cathode at the spot location.</figcaption></figure>

predicted by the fluid modeling. Although PIC results do not exhibit the typical "S" curve of the fluid approach and the associated hysteresis, they are in relative good agreement. We evidenced such hysteresis in our previous PIC study by managing the time delay at which thermionic electrons were injected into the system [21] (after or before the sheath building), but did not reproduce this approach here. Finally, the fluid modeling does not allow for precise determination of the entry into the SCL regime when $L_y/S > 1$, which is certainly due to the 1D approximation made for the calculation of the sheath potential.

Concerning the thermionic current penetrating into the plasma, its amplitude is directly correlated to the ratio $L_y/S$ as depicted in figure 6. When $L_y/S = 1$, $J_s$ increases from almost zero to its maximal value $J_s^*$, which is 2.9 and $3.5en_0c_s$ for the attached and detached divertor case respectively, at the bifurcation threshold. When ambipolarity is preserved due to the return of the electron current around the spot, in the region of size $L_y-S$, as explained by equation (4), it is possible to significantly increase the current $J_s^*$, i.e., the surface temperature before the apparition of the SCL regime. For a typical attached divertor plasma, $J_s/n_0c_s$ saturates at 12.9 around 3420 K and at 6 at a temperature of 3700 K in the case of a detached divertor plasma; both values are quite larger to what is evidenced for $L_y/S = 1$.

<figure><img src="image_9.png" /><figcaption>**Figure 6:** Variation of the thermionic current density $J_s$ for a) attached-like and b) detached-like divertor plasma conditions and for the case $L_y = S$ and $L_y = 50\lambda_d > S$. The ratio $L_y/S$ plotted in yellow is indicative of the surface available for the return of the plasma electron current in order to achieve quasi-neutrality.</figcaption></figure>

## 4. Conclusion

In this paper, we have investigated the thermal behavior of a localized spot over a tungsten surface under the thermal load of a hot plasma whose density was varied. Our calculations show that if the entire surface emits electrons ($L_y = S$), then a bifurcation regime appears for a critical heat flux (which itself depends on the plasma temperature and its density). During this bifurcation, the sheath potential decreases abruptly to its minimum and the space-charge-limited regime appears: the surface temperature increases rapidly with a maximum current. Now, if we consider a thermally isolated spot on a surface with reduced conductivity, the bifurcation regime disappears ($L_y > S$). The surface temperature then increases continuously with the plasma density until it reaches the limited regime, and then continues to increase with a different slope. The maximum current density emitted by the spot in this case is much higher because there is a possibility for the electron current to circulate between the surface surrounding the spot and the plasma, as in the unipolar arc model.
# 5. Acknowledgments

This work was supported by the French National Research Agency (Agence Nationale de la Recherche) under project SHEAR ANR-19-CE30-033-01. It has been also carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No 101052200 — EUROfusion). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.

# 6.

## CRediT authorship contribution statement

**J. Moritz:** Conceptualization, Writing, Software, Formal analyses, Resources. **S. Heuraux:** Formal analyses, Writing and editing. **M. Lesur:** Writing and editing, Software. **E. Gravier:** Formal analyses, Writing and editing. **F. Brochard:** Writing and editing, Resources. **L. Marot:** Writing and editing, Resources. **P. Hiret:** Writing and editing, Formal analyses.

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