# Tungsten surface temperature calculated by means of PIC simulations: bifurcation and space charge limited regime

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

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J. Moritz, S. Heuraux, M. Lesur, E Gravier, F Brochard, et al., Tungsten surface temperature calculated by means of PIC simulations: bifurcation and space charge limited regime. 19th International Conference on Plasma-Facing Materials and Components for Fusion Applications, May 2023, Bonn, Germany. hal-04145346

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Submitted on 29 Jun 2023

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# Tungsten surface temperature calculated by means of PIC simulations: bifurcation and space charge limited regime

J. Moritz*), S. Heuraux a), M. Lesur a), E. Gravier a), F. Brochard a), N. Lemoine a), P. Hiret b), L. Marot b)

a) Institut Jean Lamour UMR 7198 CNRS - Université de Lorraine 2 allée André Guinier F-54011 Nancy Cedex, France.

b) Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056, Basel, Switzerland.

## Context of the Simulations

- 1d/3V PIC simulations ($L=100L_d$)

- Tungsten surface with $\epsilon = 0.25$

$$B_w = 4.55 \text{ eV}$$

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

- Heat flux balance at the wall at each iteration

$$Q_p = Q_c$$

With [1] $Q_c = J_s(2k_bT_s + B_w) + \epsilon \sigma T_s^4 + \frac{K}{t}(T_s - T_0)$

→ gives $T_s$ and then $J_s$ to be injected + loop until the steady state ($T_0 = 300$ K and wall thickness $t=1$ cm).

### - 2 Simulation scenarios:

S1: The routine calculating $T_s$ and $J_s$ is run at the beginning of the simulations. During the establishment of the sheath, the surface receives a very large heat flux due to the electrons → high $T_s$. Then the wall eventually cools down to the steady state.

S2: The routine calculating $T_s$ and $J_s$ is launched after 10 plasma periods (waiting time) for the sheath to be established. The cathode can then be cold before the steady state is reached contrary to scenario 1.

## Sheath building & Space potential + plasma density

<figure><img src="image_3.png" /><figcaption>Plasma potential vs. W thermal conductivity</figcaption></figure>

<figure><img src="image_4.png" /></figure>

- During the sheath building, the heat flux due to electrons is huge → High $T_s$ and strong $J_s$ → Virtual cathode on several plasma periods (but can be temporary and cancels if further heat flux not large enough)

<caption>Ion and electron density profiles</caption>

- The reduction of $K$ makes the surface temperature increase with a larger thermionic emission → sheath shrinking and double layer occurs = virtual cathode

## Phase space and Velocity distribution

<figure><img src="image_5.png" /><figcaption>Averaged ion velocity</figcaption></figure>

<figure><img src="image_6.png" /></figure>

- Electrons accelerated through the sheath or reflected onto the virtual cathode when $J_s > J_s^*$

- Ions can be trapped into the potential dip [2,3]

<figure><img src="image_7.png" /><figcaption>Scenario S1 vs. S2: Heat Flux</figcaption></figure>

<figure><img src="image_8.png" /></figure>

- The final state of the cathode depends on its initial one in the expected bifurcation regime (similar behavior in experiments [6])

- The critical temperature ranges between 2500-3000K for the space charge limited regime to be reached

## Parametric studies & hysteresis

<figure><img src="image_9.png" /></figure>

## Heat Flux & Fluid Approach and bifurcation

<figure><img src="image_10.png" /><figcaption>PIC Heat flux from the surface</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Temperature bifurcation S curve</figcaption></figure>

- The heat flux on the surface depends on the sheath potential.

- In the strong emission regime, the current is limited + reflected electrons increase the heat flux

- The heat flux carried away from the surface → mostly conduction for all the range of $K$ investigated

$$Q_c = J_s(2k_bT_s + B_w) + \epsilon \sigma T_s^4 + \frac{K}{t}(T_s - T_0)$$

$$Q_p = n_s c_s (2k_b T_i + E_i - B_w + e \phi_s) + \frac{n_s c_e}{4} \exp\left(-\frac{e \phi_s}{k_b T_e} (2k_b T_e + B_w)\right)$$

$e \phi_s = e \phi_f - k_b T_e \log\left(1 + \frac{J_s}{en_s c_s}\right)$ until the electric field at the surface cancels [4,5] then $J_s = J_s^* \quad \phi_s = \phi_s^*$

- There is a regime of the thermal conductivity where a bifurcation occurs → the final temperature of the surface depends on its history = hysteresis phenomenon

- The history in the PIC simulation is related to the scenario run → coming from a high or a low temperature before reaching the steady state

## References:

[1] E. Hantzschke, in Handbook of Vacuum Arc Science and Technology, William Andrew Publishing, Park Ridge, NJ, 1996 pp. 151 - 208

[2] Campanell and Umansky, Phys. Rev. Lett. 116, 085003 (2016)

[3] T. Intrator et al. J. Appl. Phys. 64, 2927 (1988)

[4] Hobbs and Wesson, Plasma Physics 9, 85 (1967)

[5] Tokar, Nedospasov, and Yarochkin, Nuclear Fusion 32, 15 (1992)

[6] Ye, Takamura and Ohno, J. Nucl. Mater. 241, 1243 (1997)

## 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 was 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)