# Real-time helium bubble growth in tungsten by in-situ GISAXS

L Corso, S Curiotto, F Cheynis, P Muller, E Bernard, M Cabie, C Martin, F Leroy

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L Corso, S Curiotto, F Cheynis, P Muller, E Bernard, et al. Real-time helium bubble growth in tungsten by in-situ GISAXS. Nuclear Materials and Energy, 2025, 10.1016/j.nme.2025.101894. hal-04792519v2

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# Real-time helium bubble growth in tungsten by in-situ GISAXS☆

L. Corso<sup>a,b,*</sup>, S. Curiotto<sup>b,c</sup>, F. Cheynis<sup>b,c</sup>, P. Muller<sup>b</sup>, E. Bernard<sup>a</sup>, M. Cabie<sup>c</sup>, C. Martin<sup>d</sup>, F. Leroy<sup>b,d</sup>

## ARTICLE INFO

Keywords:

Tungsten

Helium bubbles

GISAXS

Growth kinetics

Coalescence

Diffusion coefficient

## ABSTRACT

Nuclear fusion, as a carbon-free energy source, is actively pursued by the international community. However, the irradiation by helium (He) of the walls facing the plasma can lead to important damages at the surface and deeper in the bulk, affecting the properties and life span of the materials. To address the effects of He irradiation, we have studied in real time, during He bombardment at 1273 K, the nucleation and growth of He bubbles nearby the surface of tungsten by grazing incidence small angle X-ray scattering (GISAXS). We could disentangle the surface and bubble contributions on the measured GISAXS patterns by adjusting the X-ray incident angle below and above the critical angle of total external reflection. The occurrence of tilted diffuse scattering rods proves the presence of facetted bubbles buried inside the tungsten matrix. The time evolution of the X-ray diffuse scattering has been used to evaluate the growth kinetics. A close comparison with analytical modelling of the growth process suggests a growth kinetics dominated by the migration-coalescence of bubbles during which the bubble motion is hindered by the nucleation of ledges at the bubble facet surface.

## Introduction

In fusion reactors, the most stressed part of the first physical barrier facing the hot deuterium-tritium plasma is the divertor. In recent reactors like ITER [1], WEST [2], ASDEX-upgrade [3] or EAST [4], tungsten (W) has been chosen as material for the divertor. In ITER, the divertor will have to support extreme heat loads up to 20 MW/m² during short plasma instabilities and intense particles bombardment [5] composed of deuterium, tritium and helium (He) as the product of the fusion reaction. Despite the low energy of most of He irradiating the divertor (< 100 eV [6]), well below the displacement damage threshold of W by He (~500 eV [7,8]), He bombardment affects significantly W. In particular He nanobubbles are formed close to the surface [9–13] and affect the physical properties of W [14–18]. A major concern regarding the nuclear safety is the tritium retention increase due to the presence of these He-bubbles [19–21]. In addition, bubbles could initiate the formation of the nanostructured W “fuzz” [22–25] which is dramatic for the component erosion and life-time.

The formation of the bubbles is expected to follow the subsequent

steps: the early stage of the formation of bubbles consists in the so called “trap-mutation” mechanism described in Ref. [25]. Briefly, the low solubility of He in W is the driving force for the formation of He clusters in interstitial positions. Once the local pressure is high enough, a W atom is expelled from its crystallographic position as self-interstitial atom (SIA). The vacancy thus generated will act as a trap for the He cluster, and the vacancy-He cluster complex that is formed is called bubble.

Since the 60s such gas bubbles in metals have been studied and two possible mechanisms have been proposed for their growth: migration-coalescence (MC) and Ostwald ripening (OR). Barnes and Mazey have observed the migration and coalescence of bubbles in copper by transmission electron microscopy (TEM) in presence of a temperature gradient [26]. The migration of a bubble requires the displacement of the surrounding metal atoms. Three different elementary processes have been considered to address this motion [27]:

i) The surface diffusion process by which metal atoms diffuse on the bubble surface.

<footnote><sup>a</sup> CEA, IRFM, F-13108 Saint Paul-lez-Durance, France</footnote>

<footnote><sup>b</sup> Aix Marseille Univ, CNRS, CINAM, Marseille, France</footnote>

<footnote><sup>c</sup> Aix Marseille Univ, CNRS, Centrale Med, FSCM, Marseille, France</footnote>

<footnote><sup>d</sup> Aix Marseille Univ, CNRS, PIIM, Marseille, France</footnote>

<footnote>* This article is part of a special issue entitled: 'PSI-26' published in Nuclear Materials and Energy.</footnote>

<footnote>* Corresponding author.
E-mail address: loic.corso@cea.fr (L. Corso).</footnote>
ii) The volume diffusion process, where atoms diffuse through the metal matrix around the bubble.

iii) The vapor diffusion process that consists in the sublimation of atoms at one side of the bubble and the condensation at the opposite side.

However, as observed by Willertz and Shewmon [28] in gold, the motion of faceted bubbles is much slower than predicted by those processes. In this case an additional diffusion process should be considered: iv) the ledge nucleation limited diffusion [28,29] where the limiting step is the nucleation of an ad-vacancy nucleus on the bubble facet. Goodhew and Tyler [30] have studied the growth of faceted He bubbles in vanadium, niobium and niobium-zirconium alloys between 1123 K and 1723 K and concluded that bubbles growth is limited by the ledge nucleation process. Let us note that for a power gain Q = 10 during deuterium--tritium operation in ITER, the surface temperature of the component close to strike point on the vertical target of the divertor is expected in the range 900 K to more than 2000 K depending on the neutral pressure [6]. Therefore, a range of high temperatures is expected at the ITER divertor, including temperatures above 1073 K, at which bubbles in W are expected to develop facets [31–34]. As it has been shown in previous works [31,35], the equilibrium shape of He bubbles in W is the truncated rhombic dodecahedron composed of {100} and {110} facets.

The second proposed mechanism for bubble growth is Ostwald ripening (OR). It consists in a flux of He atoms from small bubbles towards larger bubbles. The driving force arises from the reduction of the free energy of the gas in larger bubbles.

Both mechanisms have been observed in steels [36,37] and temperature is an important parameter as a transition between MC at low temperature and OR at high temperature has been reported [38].

In W, only coalescence has been observed experimentally at 1273 K by in situ TEM [39]. In this regard, kinetic Monte Carlo (KMC) simulations considering the MC mechanism, have shown that, between 1003 K and 1433 K, bubbles migrate, burst at the surface and leave holes [40]. In addition, Molecular dynamic simulations have shown that single He atoms can escape from He-clusters only when the temperature reaches 2000 K [41] or 2100 K [42] suggesting that the OR mechanism is suppressed below this temperature. However, Teo et al. [43] have found for polycrystalline W exposed to 20 eV He plasma at 573 K and 1073 K that OR seems to be the mechanism underlying the bubbles growth during post-implantation annealing up to 998 K since no bubble motion has been seen by TEM.

In this work, we have investigated the growth mechanism of He bubbles in W. Our approach aims to be as close as possible to a “model system” in order to provide experimental data that could be implemented in future simulations of bubbles growth. In this context, W single crystals have been studied in ultra-high vacuum (UHV) environment to avoid contamination. He bombardments have been performed using ion beams to control the incident ion energy (2 keV). An energy of 2 keV is important to precisely control the He-ion energy and to maximize the flux. While most He ions implanted in fusion reactors are expected to have an energy lower than 100 eV, some of He ions and neutrons will escape the confinement with high energy inducing displacement damage in the W microstructure [44]. An energy of 2 keV can be considered as a proxy for low energy He plasma and simultaneous displacement damage. In-operando grazing incidence small angle X-ray scattering (GISAXS) measurements have been recorded in real time during He irradiation to measure the size evolution of the bubbles. In this way, bubbles are not altered during the measurement, as it could be the case for thin lamellae studied by TEM. In addition, statistical information over a large area (mm²) are obtained [45]. The chosen implantation temperature is 1273 K and corresponds to the foreseen temperature of ITER surface components during operation.

## Experiments

### Samples

Two high purity (> 99.999 %) W single-crystalline samples have been used in this study. Samples were purchased from MaTeck GmbH. The surfaces are oriented along {100} and {110} crystallographic planes with a precision below 0.1°. Samples are disk-shaped, with 9 mm diameter and 1 mm thickness. The surfaces are polished and the final roughness is below 0.01 µm. The samples are prepared under UHV to remove the remaining atomic contamination at the surface and to decrease the surface roughness until atomic steps and terraces are visible by low energy electron microscopy (LEEM). The procedure is described in [46]. It consists in cycles of annealing at 1273 K under a O₂ partial pressure (10⁻⁶ mbar) followed by a high temperature flash up to 2273 K to remove the remaining tungsten oxide layer. The level of carbon and oxygen contamination is checked to be below the detection limit of Auger electron spectroscopy (AES). Finally, a 5 nm layer of gold is deposited on the surface to prevent recontamination during transportation to the European synchrotron radiation facility (ESRF) for He bombardment and in-operando GISAXS studies.

## Experimental set-up

Samples are introduced in the INS2 setup of the BM32 beamline at ESRF [45]. Before He implantation, the gold layer is removed by heating the sample at 1373 K. The desorption of gold is monitored by grazing incidence X-ray diffraction (GIXD). The in-plane orientation of the W samples is obtained by GIXD. For each sample, the He implantation has lasted around 11 h. The sample temperature is maintained at 1273 K. He ions energy is 2 keV, to maximise the flux which is estimated at 2.4 × 10¹⁷ m⁻² s⁻¹ from the sample current and the cumulated fluence is close to 1.0 × 10²² m⁻². GISAXS patterns are recorded in real time all along the implantation process. A detailed description of the GISAXS principle can be found in Ref. [45] and the GISAXS geometry is presented in Fig. 1. The main parameters are the incident and the scattered X-ray wave vectors (respectively $k_i$ and $k_f$), with $|k_i| = |k_f| = k_0 = \frac{2\pi}{\lambda}$. $\alpha_i$ and $\alpha_f$ are the incident and out-of-plane scattering angles, $2\theta_f$ is the in-plane scattering angle and $\omega$ is the azimuthal angle. The GISAXS signal results from the variations of the electronic density. It means that chemical or morphological alterations are detected. Here, the GISAXS signal arises from the local drop of the electronic density inside the He-bubbles with respect to the W-matrix. The X-ray energy used for GISAXS/GIXD measurements is 10 keV (wavelength $\lambda = 0.124$ nm). The refractive index with X-ray is $n = 1 - \delta - i\beta$ with $\delta = 2.79 \times 10^{-5}$ and $\beta = 1.778 \times 10^{-6}$ [47] and the critical angle for total external reflexion is about 0.4°. The X-ray penetration depth is tuned to probe different thicknesses in W by changing the incident angle between 0.2° and 0.6°. At $\alpha_i = 0.6°$, the probed depth is 40 nm, enough to analyse the bubble region. The detector-sample distance is 802 mm for W(100) and 1246 mm for W(110). The detector is a charge-coupled device (CDD) camera composed of two modules. The black horizontal zone visible on the

<figure><img src="image_3.png" /><figcaption>Fig. 1. GISAXS geometry.</figcaption></figure>
GISAXS patterns (Figs. 1–4) is the gap between those two modules.

An additional {110} oriented sample similar to the previous one, has been He implanted in the same conditions (i.e. ion energy 2 keV, temperature 1273 K and fluence $1.0 \times 10^{22} m^{-2}$). After He implantation the sample temperature has been maintained during 11 h at 1273 K. A thin lamella for TEM observation was extracted from this sample by focused ion beam (FIB) with a FEI Helios 600 Nanolab Dual Beam. To prevent the degradation of the bubble zone, a platinum (Pt) layer has been deposited on the surface of the cutting area. The Pt deposition for the TEM lamella preparation consists of two steps: i) a thin Pt layer of a few hundred nanometers thick is deposited with electron beam assistance. During this step, the electron beam current is 1.4 nA and the beam voltage is 3 kV. ii) a thick Pt layer of approximately 3 µm is then deposited on top of the first layer by ion beam deposition. For this step, 30 keV gallium ions are used and the ion beam current is 0.28 nA. The final thickness of the lamella is estimated at 70 nm by the FIB images. TEM analysis was performed with a Jeol JEM-2100F apparatus.

## Results

### Reference before implantation

Fig. 2 shows the GISAXS patterns recorded for W(100) and W(110) before implantation at $\alpha_i = 0.6^\circ$. A thin vertical rod termed “specular rod” is clearly visible on all GISAXS patterns. The specular rod is due to the X-ray diffuse scattering from the sample surface. The very small width of the specular rod indicates that reflection occurs on extended terraces with low morphological roughness at the surface.

### Bubbles nucleation during the early stages of implantation

The maximum energy transferred by a He atom to a W atom during the first elastic collision is given by $E_{max} = 4 \frac{m_{He}m_{W}}{(m_{He}+m_{W})^{2}}E_{i}$ [48]. Here, $m_{He}$ and $m_{W}$ are the atomic masses of He and W atoms and $E_{i}$ is the incident ion energy. In the case of 2 keV incident ion energy, the maximum energy transferred to a W atom is 166.6 eV well above the displacement threshold of tungsten $E_{d} = 90$ eV [7]. Hence, implantation is expected to generate crystallographic damages especially in the first nanometers. In addition, the bursting of bubbles at the sample surface [23,31,49–51] could significantly affect the surface morphology and thus the GISAXS pattern. To discriminate the surface contribution raised from bubble bursting and damaged surface under 2 keV irradiation and the contribution of buried bubbles on GISAXS patterns, the incident angle $\alpha_i$ has been varied between 0.2° and 0.6°. Below the critical angle ($\alpha_c = 0.4^\circ$) an evanescent wave penetrates slightly into the sample volume. The penetration depth is given by $D = [-2k_{0}\sqrt{\alpha_{i}^{2} - \alpha_{c}^{2} - 2i\beta}]^{-1}$ [45]. Therefore, at 0.2° incident angle, the penetration depth is $D = 1.50nm$ and GISAXS signal is extremely sensitive to the surface morphology while mostly insensitive to the buried He bubbles.

<figure><img src="image_4.png" /><figcaption>Fig. 2. GISAXS patterns before He implantation of W(100) a), W(110) b), with incident angle $\alpha_i = 0.6^\circ$. The thin vertical rod is due to the X-ray reflection on the flat sample surfaces.</figcaption></figure>

Fig. 3 shows the GISAXS patterns recorded during the first stage of implantation for all samples at two distinct incident angles, above (0.6°) and below (0.2°) the critical angle. Above the critical angle (Fig. 3a and b), an off-specular diffuse scattering signal is now visible. This signal arises from the sub-surface region since such signal is absent at an incident angle below the critical angle (Fig. 3c and d). The diffuse scattering signal is rather similar for all samples and isotropic, indicating the presence of spherical bubbles buried below the sample surface. Indeed, in the early stage of He implantation, small He-vacancies complexes (bubbles) nucleate with a shape close to a sphere. Similar GISAXS patterns of spherical bubbles have been observed by Thompson and coworkers after He implantation at 973 K and $10^{26} m^{-2}$ fluence [52].

### Bubble growth and faceting

Fig. 4 shows the time evolution of the GISAXS patterns during He implantation for the W(100) and W(110) surfaces and implanted doses corresponding approximately to 20 % ($2.0 \times 10^{21} m^{-2}$), 50 % ($5.0 \times 10^{21} m^{-2}$) and 100 % ($1.0 \times 10^{22} m^{-2}$) of the final implanted dose. For both samples, the same behaviour is observed: the specular rod becomes broader. This behaviour can be assigned to an increase of the surface roughness during He implantation. This arises presumably from surface damages caused by the 2 keV He ions and bubbles bursting at the surface. In addition, a lateral rod highlighted by a red ellipse in Fig. 4c and f is more and more pronounced with implantation. This signal is due to faceted objects, the lateral rod results from the preferential X-ray diffuse scattering along the crystal truncation rod coming from the faceted bubble surface. The angle between the lateral rod and the surface normal is relative to the facet orientation. For W(100), it extends at 45° with respect to the surface normal. For W(110) the X-ray diffuse scattering rods make an angle of 60° with respect to the surface normal. In both cases, these lateral rods can be assigned to the presence of {110} facets. As shown by Corso et al. [35] similar X-ray scattering rods have been measured by GISAXS using other incident azimuths angles ($\omega$) and revealing also the presence of {100} facets at the bubbles surface.

### Discussion

The average size $d_{110}$ of the {110} facet can be estimated by:

$$d_{110} = \frac{2\pi}{\Delta q} \qquad (1)$$

where $\Delta q$ is the full width at half maximum (FWHM) of the diffuse scattering rod.

The experimental uncertainty on GISAXS measurements lies on the fit quality of the intensity profile along the rod width. When the facet size is small (< 2 nm), the rod is broad and the ratio between the scattered intensity and the background is not favourable. In this case, the maximum relative uncertainty on the facet size is 40 %. When facets are larger, (> 2 nm) the relative uncertainty decreases rapidly and tends to 5 %.

Assuming that the bubbles have reached a quasi-equilibrium shape at 1273 K [35], the bubble radius $R$ (defined here as the half distance between two opposite {100} facets) is given by

$$R = \frac{d_{100}}{2} + \frac{\sqrt{2}}{2}d_{110} \qquad (2)$$

where $d_{hkl}$ is the size of the hkl facet.

Furthermore, $d_{100}$ can be directly estimated from the {110} facet size and surface energy anisotropy ratio ($\gamma_{100}/\gamma_{110}$) derived in [35] as $\frac{\gamma_{100}}{\gamma_{110}} = \frac{d_{100}+\sqrt{2}d_{110}}{\sqrt{2}d_{100}+d_{110}}$, this leads to the following relation:

$$d_{100} = d_{110} \frac{\frac{\gamma_{100}}{\gamma_{110}} - \sqrt{2}}{1 - \frac{\gamma_{100}}{\gamma_{110}} \sqrt{2}} \qquad (3)$$
<figure><img src="image_5.png" /><figcaption>Fig. 3. GISAXS patterns recorded at the first stage of the 2 keV He implantation at 1273 K. Two incident angles have been used, for a) and b) above the critical angle to probe the bubble region and for c) and d) below the critical angle to probe only the extreme surface. The significant difference is due to the rapid formation of bubbles below the surface.</figcaption></figure>

Finally, injecting Eq.(3) in Eq.(2) the radius is given by:

$$R = \frac{d_{110}}{2} \left( \sqrt{2} + \frac{\frac{\gamma_{100}}{\gamma_{110}} - \sqrt{2}}{1 - \frac{\gamma_{100}}{\gamma_{110}} \sqrt{2}} \right) \quad (4)$$

Thus, the variations of {110} facet size and bubble radius are correlated by a simple linear relation. Let us note that it has been estimated by Corso et al. [35] that $\gamma_{100}/\gamma_{110} = 1.03$.

Fig. 5 represents the bubble radius evolution measured by in-operando GISAXS considering the two crystallographic orientations of the surface. As seen in Fig. 5a, bubble growth occurs in three steps and is identical for both crystallographic orientations. First, the bubble radius is below the detection limit (<0.5 nm) for a so-called incubation time. Then, once the fluence reaches approximately $5.80 \times 10^{20} \text{ m}^{-2}$ the bubble radius increases drastically up to around 4 nm beyond which the growth rate slows down significantly.

We propose two possible explanations for the observed incubation time. The first possible scenario has been proposed by Trinkaus et al. [53]. The incubation phase corresponds to a regime during which bubbles are not stable and spontaneously form and dissociate. Then, since the He concentration in the W matrix continuously increases during He implantation, a critical bubble size is finally reached, above which the bubbles are stable and can grow. In our experiments, the fluence needed to overpass the incubation phase is around $5.80 \times 10^{20} \text{ m}^{-2}$ and the critical bubble radius would be between 0 and $1.2 \pm 0.3 \text{ nm}$. The second probable scenario has been proposed by Curiotto et al. and is based on KMC simulations [54]. The diffusion of facetted voids of

different sizes in a bcc solid has been simulated. It has been shown that the diffusion coefficient first increases with the void size, reaches a maximum and then decreases to follow classical scaling power laws for diffusion of 3D objects. In our experiment, if the MC of bubble is the governing coarsening mechanism, the motion of small bubbles is initially hampered due to a low diffusion coefficient, until their size increases sufficiently for the diffusion coefficient to reach its maximum strongly favouring the coalescence phenomenon. This moment might correspond to the incubation time. Then, since the bubble size increases the diffusion coefficient decreases continuously limiting drastically the number of coalescence events.

To determine whether OR or MC is governing the coarsening mechanism, the radius evolution of the bubbles has been compared to theoretical predictions according to the long-time regime of void growth (see Fig. 5).

The expected radius evolution by OR mechanism has been derived by Greenwood [55] and is $R(t) \propto t^{1/2}$.

For the MC mechanism, Goodhew and Tyler [30] have derived the time dependence of the mean bubble radius assuming that the volume of bubbles is conserved after coalescence. They have found the following radius variation:

$$\frac{dR(t)}{dt} = \frac{8\pi c D_b}{3R(t)} \quad (5)$$

with $D_b$ the bubble diffusion coefficient that depends on the transport process (given in Table 1) and c the volume fraction of bubbles in the sample (assumed to be constant).

Continuous implantation of He ions into the W sample will naturally
<figure><img src="image_6.png" /><figcaption>Fig. 4. GISAXS patterns during 2 keV He implantation at 1273 K with incident angle αᵢ = 0.6°. The evolution of the W(100) is shown in the first line from a) to c) and W(110) in the second line from d) to f). The first column corresponds to 20 % of maximum dose, the second one to 50 % and the third one to 100 % of the total implanted dose. The lateral rod (red ellipse) is due to the faceting of the bubbles by {110} facets. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

increase the volume fraction “c” in the sample. Meanwhile, c is reduced by different He desorption processes such as bubble bursting, or diffusion of He atoms and clusters to the surface with subsequent desorption. All these processes release He atoms from the W crystal. Considering that the rate of He desorption is proportional to the volume fraction of He c inside the sample, the total rate change of c with time is hence:

$$ \frac{dc(t)}{dt} = F - \frac{c(t)}{\tau} \qquad (6) $$

where F is the creation term (i.e. the incoming flux) and τ related to the characteristic time needed for He desorption events to occur.

The solution of Eq. (6) is:

$$ c(t) = F\tau(1 - \exp(-t/\tau)) \qquad (7) $$

For $t < τ$: $c(t) ≈ Ft$, i.e. in the early stages of implantation, before bubble bursting, the He volume fraction increases linearly with time. This behaviour has been observed by Rivera et al. [56] using Kinetic Monte Carlo (KMC) simulations in conditions similar to those of our study (3 keV, flux of $2 \times 10^{17} m^{-2} s^{-1}$ and temperature of 1300 K to be compared with our conditions: 2 keV, $2.4 \times 10^{17} m^{-2} s^{-1}$ and 1273 K) and considering immobile He bubbles.

For $t \gg τ$: $c(t) ≈ Fτ$, i.e. at long time the He volume fraction is constant due to a steady state (saturation) that is reached between He continuously implanted and He desorption by bubble bursting. This behaviour has been reported in ref. [44] for nickel, steel alloys and niobium. In W, the He saturation has been experimentally observed in refs. [57,58]. Precisely, the total He retention starts to saturate at a fluence about $1.0 \times 10^{21} m^{-2}$ and it reaches its maximum value ($\sim 1.5 \times 10^{21} m^{-2}$) at the fluence of $1.0 \times 10^{22} m^{-2}$ [57].

Thus, in our study the volume fraction “c” evolves from a linear dependence with time at the early stage (fluence $< 1.0 \times 10^{21} m^{-2}$) to a

constant value at the late stage (fluence $\sim 1.0 \times 10^{22} m^{-2}$). Nevertheless, in order to get an initial insight into the growth mechanism of bubbles, we assume that a steady state is established between implanted and desorbed He, i.e. c is constant.

After integration of Eq. (5), scaling power laws can be deduced for the bubble radius: $R(t) = A t^{\frac{1}{n}}$ with $n = 5$ for vapour and volume transports and $n = 6$ for surface transport. The prefactor A contains all the constants and is different for each transport regime.

For the ledge nucleation regime, the time dependence of the radius is hence $R(t) = \frac{\ln(Bkt)}{k}$ where $B = \frac{8\pi^2 cD_s}{18a_i}$ and $k = \frac{\pi\epsilon}{2k_B T}$ are constants.

The time scale in the expressions has been shifted to consider the incubation time τ such that $t \to t - \tau$. In other words, the experimental points have been fitted by power law functions expressed by $R(t) = A(t - \tau)^{\frac{1}{n}}$ (A and τ are fitted parameters) except for the ledge nucleation limited regime case where $R(t) = \frac{\ln(Bk(t-\tau)+1)}{k}$. In such a way, when $t = \tau$ the radius is $R(t = \tau) = 0$ nm.

The best fit is obtained for bubble growth with a limiting elementary mechanism of bubble motion by the ledge nucleation mechanism at the bubble facets (black curve). OR (green lines) clearly does not overlap the experimental data and other MC mechanisms are not as satisfactory (blue and red curves) as the ledge nucleation-based process. Let us note that the models of radius evolution are based on the assumption of a constant bubble volume fraction c. This implies that the time scale for the increase of c is much longer than the growth rate of He bubbles and we can assume a weak effect of the concentration increase. We therefore provide values for the fit parameters, particularly the ledge energy, which should be considered cautiously as a first estimation. However, we stress that the qualitative interpretation of the growth mechanism would not be impacted by considering “c” as time-dependent.

The fitted parameters for the ledge nucleation regime are given in
<figure><img src="image_7.png" /><figcaption>**Fig. 5.** The experimental bubble radius evolution in W(100) (blue dots) and W(110) (red triangles) during 2 keV He implantation at 1273 K by in-situ GISAXS analysis. The full line curves represent the fit corresponding the different growing mechanisms. In green Ostwald ripening, in blue is the non-distinguishable case of migration-coalescence by vapour or volume diffusion, in red the migration-coalescence limited by surface diffusion and in black the ledge nucleation migration-coalescence regime. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>
<caption>**Table 1**</caption>

Bubble diffusion coefficient regarding the elementary processes of the bubble motion from [30].

<table><thead><tr><th>Elementary processes</th><th>Bubble diffusion coefficient: D<sub>b</sub></th><th></th></tr></thead><tbody><tr><td>Vapor transport</td><td>D<sub>b,vap</sub> = <sup>D<sub>g</sub>PΩ<sup>2</sup>q</sup>&frasl;<sub>k<sub>B</sub>TR(t)<sup>3</sup></sub></td><td>D<sub>g</sub>: diffusion coefficient of metal atom in the bubble gas<br>P: vapor pressure of the metal<br>Ω: the atomic volume<br>q: the deviation of the bubble gas from ideal gas<br>T: temperature<br>k<sub>B</sub>: the Boltzmann constant</td></tr><tr><td>Volume transport</td><td>D<sub>b,vol</sub> = <sup>D<sub>v</sub>Ω</sup>&frasl;<sub>R(t)<sup>3</sup></sub></td><td>D<sub>v</sub>: diffusion coefficient of vacancy in the metal matrix<br>Ω: the atomic volume</td></tr><tr><td>Surface transport</td><td>D<sub>b,surf</sub> = <sup>4D<sub>s</sub>Ω<sup>2</sup></sup>&frasl;<sub>a<sub>s</sub><sup>2</sup>R(t)<sup>4</sup></sub></td><td>D<sub>s</sub>: surface coefficient diffusion the metal atom<br>a<sub>s</sub>: jump distance of the moving atom<br>ε: ledge energy formation</td></tr><tr><td>Ledge nucleation regime</td><td>D<sub>b,ledge</sub> = <sup>πD<sub>s</sub>R(t)</sup>&frasl;<sub>6α<sub>s</sub></sub> exp(<sup>-πR(t)ε</sup>&frasl;<sub>2k<sub>B</sub>T</sub>)</td><td></td></tr></tbody></table>

**Table 2.** Assuming that the diffusivity of W atoms at the bubbles inner surface does not depend on the initial surface orientation (i.e. B constant), we can estimate the ledge energy $ε$ to about $2.63 \pm 0.04 \times 10^{-11}$ J m$^{-1}$.

For comparison the ledge energy $ε$ has also been reported for the W surface in vacuum and lies in between $2.18 \times 10^{-10}$ and $4.95 \times 10^{-10}$ J m$^{-1}$ at 0 K under vacuum [59]. The deduced value of the ledge energy $ε$ from GISAXS analysis is thus an order of magnitude smaller. This difference could result from the influence of the sample temperature (1273 K) and He atoms trapped inside the bubbles that may reduce the ledge nucleation energy barrier. Indeed, it has been shown [41] that W atoms could be expelled from their crystallographic positions in pressurized He bubbles. This type of mechanism clearly indicates that the mobility of W atoms may be strongly enhanced at the surface of He bubbles and favour the nucleation of ledges on facets by reducing the energy barrier. A growth mechanism based on coalescence is supported by our TEM observations. Coalescence of bubbles was clearly observed several times as shown in Fig. 6 (see red arrows).

At last, the extraction of the bubbles radius as function of time by the migration coalescence mechanism allows us to estimate the evolution of the bubble diffusion coefficient over time if the volume fraction of bubbles is known. Indeed, rewriting the expression of $D_{b,ledge}$ with the expressions of B and k, the bubble radius dependence of the diffusion coefficient $D_{b,ledge}$ is given by:

$$D_{b,ledge} = \frac{3}{8\pi c} BR(t) \exp(-kR(t)) \quad (8)$$

where $R(t)$ is experimentally measured and B and k are deduced by the previous fit.

Since c can be estimated by TEM analysis it should be possible to quantify the bubbles’ diffusion coefficient. In order to do so, TEM analysis has been performed on an extra W(110) sample. The sample has been implanted in the same conditions then maintained at 1273 K during 11 h after implantation allowing bubbles to grow and to be easily

visible by TEM. The parameter c is expected to stay constant during the whole plateau as the bubble growth is described by the constant volume model. The bubble radius measured by GISAXS at the end of the temperature plateau is $10.3 \pm 0.5$ nm. The number of bubbles in the volume represented by the red dashed rectangle in Fig. 7 was counted manually. The area is limited in depth at 40 nm that corresponds to the maximum penetration of X-ray at $α_i = 0.6°$.

This number is probably under-estimated and experimental errors could result in missing small bubbles, due to bubble overlap and possibly truncated bubble at the lamella surfaces. The estimated bubble density is $1.04 \times 10^{23}$ m$^{-3}$. The volume fraction c expressed as the product of the bubble volume (R$^3$) and density [30] is $c = 0.11$. It means around 11 % of the volume is filled with bubbles.

Fig. 8 shows the evolution of $D_b$ as a function of He dose. The sharp decrease of $D_b$ from $10^{-2}$ to $10^{-4}$ nm$^2$ s$^{-1}$ with the implanted dose is clear evidence that the bubble diffusion coefficient drops: it is dominated by its exponential decay.

We can estimate that $D_b$ decreases by two orders of magnitude during implantation, indicating that the bubbles are much less mobile as they grow. At some point, the mobility of bubbles is too low and coalescence is less frequent resulting in a very slow evolution of the bubble size.

## Conclusion

The growth of He bubbles in W single crystals has been studied by in-operando GISAXS measurements, i.e. in real time during the exposure to 2 keV He ions at 1273 K. The evaluation by GISAXS of the mean bubble radius offers a statistical relevant vision on the bubble growth process.

<figure><img src="image_8.png" /><figcaption>**Fig. 6.** TEM micrograph of W(110) implanted with 2 keV He ion at 1273 K and after post-implantation temperature plateau at 1273 K during 11 h. Two bubbles marked by the red arrows are coalescing. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<caption>**Table 2**</caption>

Fit parameters for the different mechanisms.

<table><thead><tr><th rowspan="2">Sample</th><th>Ostwald ripening</th><th colspan="4">Migration-coalescence</th></tr><tr><th>A (nm s<sup>-1/2</sup>)</th><th>Vap/vol transport<br>A (nm s<sup>-1/5</sup>)</th><th>Surface transport<br>A (nm s<sup>-1/6</sup>)</th><th>Ledge nucleation regime<br>B (nm s<sup>-1</sup>)</th><th>ε (J m<sup>-1</sup>) × 10<sup>-11</sup></th></tr></thead><tbody><tr><td>W(100)</td><td>0.03</td><td>0.61</td><td>0.84</td><td>0.69</td><td>2.67</td></tr><tr><td>W(110)</td><td>0.03</td><td>0.62</td><td>0.87</td><td>0.69</td><td>2.59</td></tr></tbody></table>
<figure><img src="image_9.png" /><figcaption>Fig. 7. TEM micrograph of W(110) implanted with 2 keV He ion at 1273 K and after post-implantation temperature plateau during 11 h. The bubble density has been estimated in the area delimited by the red dashed rectangle. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 8. Bubble diffusion coefficient evolution during the bubble growth by migration-coalescence limited by ledge nucleation in W(100) (blue) and W (110) (red). Dot representations of the diffusion coefficient (blue circle, red triangle) are deduced from the experimental measurements of the bubble radius and full lines are deduced from the bubble radius prediction by the best fit. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

Ostwald ripening and migration-coalescence mechanism were considered to fit the evolution of the bubble radius as function of time. We attribute the rapid change of bubble growth to a mechanism of migration-coalescence of the bubbles that is limited by the nucleation of a new ledge on the bubble surface. This result is consistent with the presence of facets at the bubbles surface, exposing {110} and {100} crystallographic planes. This process makes the bubble diffusion coefficient decrease rapidly with bubble size, limiting collision frequency and coalescence. We believe this experimental work will provide valuable data for future modelling studies on the bubble behaviour. Finally, we aim at decreasing the implantation energy to below the sputtering threshold energy of W by He to get closer to tokamaks relevant conditions.

### CRediT authorship contribution statement

**L. Corso:** Writing – review & editing, Writing – original draft, Methodology, Investigation, Formal analysis, Conceptualization.
**S. Curiotto:** Writing – review & editing, Investigation, Formal analysis.
**F. Cheynis:** Writing – review & editing.
**P. Muller:** Writing – review & editing.
**E. Bernard** : **M. Cabie:** Writing – review & editing, Investigation.
**C. Martin:** Writing – review & editing, Resources.
**F. Leroy:** Writing – review & editing, Supervision, Resources, Investigation, Formal analysis.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgments

This work has been carried out within the framework of the EURO-fusion 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. This work has been also carried out within the framework of the French Federation for Magnetic Fusion Studies (FR-FCM). In particular, we are grateful to L. Martinelli from BM32 staff (ESRF) for his help during the GISAXS measurements and to D. Chaudanson and A. Altié for their help in TEM characterizations. We gratefully acknowledge the helpful discussions with Dr. T. Swinburne, Pr. J. Mougenot and Dr. M-F Barthe.

### Data availability

Data will be made available on request.

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