Onset of fuzz formation in plasma-facing tungsten as a surface morphological instability

Based on simulation results and theoretical analysis according to an atomistically informed continuum-scale model for the surface morphological response of tungsten plasma-facing component (PFC) in nuclear fusion devices, we demonstrate that the onset of the well-known surface nanostructure (known as “fuzz”) formation in PFC tungsten is the outcome of a stress-induced surface morphological instability. Specifically, we show that formation and growth of nanotendrils emanating from the exposed surface of PFC tungsten, a precursor to fuzz formation, is caused by a long-wavelength surface morphological instability triggered by compressive stress in the PFC near-surface layer due to over-pressurized helium (He) bubbles forming in this near-surface region through implantation of low-energy He ions from the plasma. Using linear stability theory (LST), we predict the onset of surface growth in response to low-amplitude perturbations from a planar PFC surface morphology and calculate the average spacing between nanotendrils growing from the PFC surface, in excellent agreement with self-consistent dynamical simulations of surface morphological response according to the fully nonlinear surface evolution model starting with random fluctuations from the planar surface morphology that result in nanoscale surface roughness. In addition, we examine the morphological response of the PFC surface to low-amplitude perturbations of very long wavelengths from its planar morphology and interpret fully the simulation results on the basis of a weakly nonlinear tip-splitting instability theory, which predicts a post-instability nanotendril pattern formation with nanotendril separation consistent with the LST predictions regardless of the initial surface perturbation. Finally, we compare our simulation predictions for surface nanotendril growth with experimental measurements of fuzz layer growth on the exposed surface of PFC tungsten. Our simulation results are in very good agreement with the surface growth measurements at the early stages of fuzz growth, further establishing the onset of fuzz formation in PFC tungsten as the outcome of a stress-driven PFC surface morphological instability.

Figure 1

Modeling of the onset of fuzz formation. (a) Schematic representation of the evolution of tungsten surface nanotendril growth (precursor to fuzz formation) during He plasma irradiation; the He ion fluence increases and the He bubbles formed in the near-surface region grow as time increases from ($a_1$) to ($a_4$). (b) Schematic representation of the homogenized near-surface nanobubble region of plasma-irradiated tungsten with W self-interstitial atoms migrating toward the exposed tungsten surface, at a flux $J_{\mathrm{I}}$, and the surface being subjected to a long-wavelength perturbation from the planar morphology.

Figure 2

Morphological evolution of a W(110) surface starting with a random low-amplitude perturbation from a planar morphology at an initial nanoscale surface roughness $SR=5$ nm, with the surface being exposed to a 75 eV He plasma irradiation with a He flux of $2.7\times {10}^{20}$ ions ${\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$ at $n_{\mathrm{He},s}=16\%, r_{b,s}=1$ nm, $T=1100$ K, and ${\dot{r}}_{\mathrm{burst}}=2.67\times {10}^{12}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$. [(a1)–(a4)] Top views of the tungsten surface morphology in the form of contour maps of the surface height function $h(x,y,t)$ at (a1) $t=0$, (a2) 70, (a3) 100, and (a4) 140 min after initiation of plasma exposure. [(b1)–(b4)] Cross-sectional views of tungsten surface profiles along the black horizontal solid line marked in (a1) at the time instants corresponding to [(a1)–(a4)], respectively. [(c)-(d)] 3D views of the tungsten surface morphology corresponding to the top views shown (c) in (a1) and (d) in (a4). Evolution of (e) RMS surface roughness $SR\left(t\right)$ and (f) average nanotendril height, ${\overline{h}}_{\mathrm{tendril}}\left(t\right)$.

Figure 3

Dimensionless dispersion relation $\tilde{\omega }=\tilde{\omega }\left(\tilde{k}\right)$ for the real part of the surface perturbation growth rate as a function of perturbation wave number, where $\tilde{k}\equiv \sqrt{{\tilde{k}}_x^2+{\tilde{k}}_y^2\phantom{{}^l}}$. The blue open circle and downwards pointing arrow are used to mark the maximally unstable wave number ${\tilde{k}}_{\max }\approx 1.50$.

Figure 4

Morphological evolution of a W(110) surface starting with a specific low-amplitude long-wavelength perturbation from a planar morphology, $h=h_0+{\mathrm{\Delta }}_0[exp(i2\pi x/{\lambda }_p)+exp(i2\pi y/{\lambda }_p)]$, under He incident ion energy and flux conditions identical to those in Fig. 2 at $n_{\mathrm{He},s}=16\%, r_{b,s}=1$ nm, $T=1100$ K, and ${\dot{r}}_{\mathrm{burst}}=2.67\times {10}^{12}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$, where the wavelength of the surface perturbation is ${\lambda }_p=20\mu \mathrm{m}$. [(a1)–(a4)] Top views of the tungsten surface morphology in the form of contour maps of the surface height function $h(x,y,t)$ at (a1) $t=0$, (a2) 620, (a3) 700, and (a4) 740 min after initiation of plasma exposure. [(b1)–(b4)] Cross-sectional views of tungsten surface profiles along the black horizontal solid line marked in (a1) at the time instants corresponding to [(a1)–(a4)], respectively. [(c) and (d)] 3D views of the tungsten surface morphology corresponding to the top views shown (c) in (a1) and (d) in (a4). (e) Evolution of RMS surface roughness $SR\left(t\right)$.

Figure 5

Morphological evolution of a W(110) surface starting with a specific low-amplitude long-wavelength perturbation from a planar morphology, $h=h_0+{\mathrm{\Delta }}_0[exp(i2\pi x/\left({\lambda }_p\right))+exp(i2\pi y/\left({\lambda }_p\right))]$, under He incident ion energy and flux conditions identical to those in Fig. 2 at $n_{\mathrm{He},s}=16\%, r_{b,s}=1$ nm, $T=1100$ K, and ${\dot{r}}_{\mathrm{burst}}=2.67\times {10}^{12}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$, where the wavelength of the surface perturbation is ${\lambda }_p=10\mu \mathrm{m}$. [(a1)–(a4)] Top views of the tungsten surface morphology in the form of contour maps of the surface height function $h(x,y,t)$ at (a1) $t=0$, (a2) 610, (a3) 650, and (a4) 690 min after initiation of plasma exposure. [(b1)–(b4)] Cross-sectional views of tungsten surface profiles along the black horizontal solid line marked in (a1) at the time instants corresponding to [(a1)–(a4)], respectively. [(c) and (d)] 3D views of the tungsten surface morphology corresponding to the top views shown (c) in (a1) and (d) in (a4). (e) Evolution of RMS surface roughness $SR\left(t\right)$.

Figure 6

Dependence of number of surface features in 1D surface profile, $N_f$, on the surface perturbation wavelength ${\lambda }_p$, following the surface morphological instability triggered on a PFC W(110) surface under conditions identical to those in Fig. 2 and simulation parameters identical to those in Figs. 4 and 5. The simulation results and the predictions of the weakly nonlinear tip-splitting instability theory are denoted by red open circles and blue solid line segments, respectively.

Figure 7

(a) Evolution of the RMS surface roughness, denoted as $\mathrm{\Delta }\left(t\right)$, under He plasma irradiation conditions identical to those in Fig. 2, with an initial surface roughness ${\mathrm{\Delta }}_0=5$ nm. The solid black and solid blue lines correspond to simulation results starting with specific low-amplitude perturbations from the planar surface morphology identical to those in Figs. 4 and 5, respectively. (b) Exponential surface roughness evolution during the initial stage of growth of the low-amplitude, long-wavelength surface perturbation; the open symbols and solid lines represent simulation results and optimal linear fits (in a semilogarithmic plot) to the simulation predictions, respectively. [(c) and (d)] Evolution of the RMS surface roughness, $\mathrm{\Delta }\left(t\right)$, for the two cases in (a) plotted as (c) a function of $\ln (1-t/t_f)$ and (d) a function of $\ln (1-t/t_f)$ at times $t$ near the time to failure, $t_f$; the open symbols and solid lines represent simulation results and optimal linear fits to the simulation predictions, respectively.

Figure 8

Comparison of experimental data for fuzz layer thickness as a function of He fluence in plasma-exposed tungsten with simulation results for average nanotendril height evolution shifted upwards by $h_0$ under consistent plasma exposure conditions. Experimental data and simulation predictions are denoted by red open circles and small black solid circles, respectively, while optimal fitted curves to experimental measurements and model-predicted simulation results are denoted by blue and green dashed lines, respectively. In the simulations, the He plasma irradiation conditions are identical to those in Fig. 2. The error bars in the simulation results correspond to statistical errors from five independent simulations with different initial surface configurations of random low-amplitude perturbations from the planar surface morphology.