Migration of tungsten dust in tokamaks: role of dust–wall collisions

When plastic deformation is involved, continuum mechanics approaches do not yield closed form solutions for the stress fields. As a result, a plethora of theoretical models is available in the literature, each employing different approximations for the description and the onset of the plastic phase [20–26]. The MIGRAINe code implements the Thornton and Ning (T&N) model [14] for the normal impact of elastic–perfectly plastic adhesive spheres owing to its physical transparency, the incorporation of adhesion that is important for micrometre-sized grains and the simple analytical expression for the normal restitution coefficient.

The main assumptions of the T&N model are that adhesion-induced plastic deformation is negligible, while dissipation by plastic deformation and dissipation by interface adhesion are additive quantities. Consequently, the normal impact of an elastic–perfectly plastic adhesive sphere can be decomposed into the impact of an adhesive elastic sphere characterized by a restitution coefficient e_⊥,adh and the impact of an elastic–perfectly plastic non-adhesive sphere characterized by a restitution coefficient e_⊥,pl with the total restitution coefficient given by $(1-e_{\perp}^2)=(1-e_{\rm \perp,adh}^2)+(1-e_{\rm \perp,pl}^2)$.

Elastic adhesion is modelled according to the Johnson–Kendall–Roberts (JKR) theory [27]. Adhesion forces are only considered inside the area of contact, the introduction of surface energy increases the equilibrium contact radius and the contact pressure follows a reduced modified Hertzian profile owing to negative adhesive loads. In the analysis of the energetics of the process the adhesive sticking velocity naturally arises, defined by

$$\begin{equation} v_{\rm s}^{\rm adh}=\frac{\sqrt{3}}{2}\pi^{1/3}\sqrt{\frac{1+6\times2^{2/3}}{5}} \left[\frac{\Gamma^5}{\rho_{\rm d}^3E^{*2}R_{\rm d}^5}\right]^{1/6}. \end{equation} \tag{ 1 }$$

This critical velocity expresses the fact that for $v_{\perp}^{i}\leq v_{\rm s}^{\rm adh}$ the dust impact energy is so low that adhesion forces are strong enough to make the grain stick to the wall, whereas for $v_{\perp}^{i}>v_{\rm s}^{\rm adh}$ the collision is inelastic even when ignoring plastic deformation owing to the irreversible work $\frac{1}{2}m_{\rm d}(v_{\rm s}^{\rm adh})^2$ of the adhesion forces. In the above $\frac{1}{E^*}=\frac{1-\nu_1^2}{E_1}+\frac{1-\nu_2^2}{E_2}$ with ν_i the Poisson ratios and E_i Young's moduli of the colliding bodies, ρ_d the mass density of the dust grain, and Γ the total surface energy per unit area given by $\Gamma=2\sqrt{\gamma_1\gamma_2}$ with γ_i the surface energy per unit area of each colliding body.

Elastic–perfectly plastic impacts are treated by assuming that the loading stage consists of an elastic phase during which the contact pressure distribution has a Hertzian profile followed by a plastic phase during which the contact pressure distribution has a Hertzian profile truncated at the limiting contact pressure p_y. The unloading stage is assumed to be elastic but with a curvature 1/R_p, where R_p > R_d, due to the permanent deformation of the contact surface. Notice that p_y is strongly related to the yield strength σ_y of the material, i.e. the stress at which the onset of plastic deformation occurs. Typically, p_y = 1.6σ_y or p_y = 2.8σ_y is assumed in the literature [28–30], but even higher values have been reported [31]. This ambiguity stems from the existence of an elasto-plastic phase for central contact pressures in the range {1.6σ_y , 2.8σ_y} [12, 13]. This phase lies in the transition region from fully elastic to fully plastic flow and is not accounted for in the T&N model. In the analysis of the energetics of the process the yield velocity v_y naturally arises, defined by

$$\begin{equation} v_{\rm y}=\frac{\pi^2}{2\sqrt{10}}\left[\frac{p_{\rm y}^5} {\rho_{\rm d}E^{*4}}\right]^{1/2}. \end{equation} \tag{ 2 }$$

This critical velocity expresses the fact that for $v_{\perp}^{i}>v_{\rm y}$ the dust impact energy is high enough to cause plastic deformation and irreversible plastic work, whereas for $v_{\perp}^{i}\leq v_{\rm y}$ the collision is totally elastic when neglecting adhesion.

By superposition one ends up with an expression for the general case of elastic–perfectly plastic adhesive spheres. The final formula will strongly depend on the ratio between the adhesive sticking velocity and the yield velocity. Let us first define the sticking velocity $v_{\rm s}^{\rm th}$ as the velocity below which the total normal restitution coefficient is zero, e_⊥ = 0. When $v_{\rm y}<v_{\rm s}^{\rm adh}$, plastic work affects sticking and consequently the sticking velocity is generally larger than the adhesive sticking velocity $v_{\rm s}^{\rm th}>v_{\rm s}^{\rm adh}$. In contrast, when $v_{\rm y}>v_{\rm s}^{\rm adh}$, plastic work does not affect sticking and $v_{\rm s}^{\rm th}=v_{\rm s}^{\rm adh}$.

It is more practical to distinguish between these two regimes by defining the dust threshold radius R_thres, i.e. the radius for which the adhesive sticking velocity and the yield velocity are equal. Combining (1) and (2) we end up with

$$\begin{equation} R_{\rm thres}=\frac{6^{3/5}}{\pi^2}\,\left(1+6\times2^{2/3}\right)^{3/5}\, \frac{\Gamma E^{*2}}{p_{\rm y}^3}. \end{equation} \tag{ 3 }$$

For metallic dust impinging on a metal surface typical values are R_thres ∼ 1–50 µm. Therefore, both regimes are of interest for tokamak applications.

(I) The R_d ⩾ R_thres regime is characterized by $v_{\rm s}^{\rm th}=v_{\rm s}^{\rm adh}$. It has been treated by Thornton and Ning [14]:

$$\begin{eqnarray} \fl &e_{\perp}(v_{\perp}^{i},R_{\rm d})=0,\quad \underline{v_{\perp}^{i}\leq v_{\rm s}^{\rm adh}}\nonumber\\ \fl &e_{\perp}(v_{\perp}^{i},R_{\rm d})=\left[1-\left(\frac{v_{\rm s}^{\rm adh}} {v_{\perp}^{i}}\right)^2\right]^{\frac{1}{2}},\quad \underline{v_{\rm s}^{\rm adh}\leq v_{\perp}^{i}\leq v_{\rm y}}\\ \fl &e_{\perp}^2(v_{\perp}^{i},R_{\rm d})=\frac{6\sqrt{3}}{5}\left[1-\frac{1}{6} \left(\frac{v_{\rm y}}{v_{\perp}^{i}}\right)^2\right] \left[\frac{v_{\rm y}}{v_{\perp}^{i}}\right]^{1/2}\nonumber\\ \fl &\times\left[\frac{v_{\rm y}}{v_{\perp}^{i}}+\frac{2}{\sqrt{5}} \sqrt{6-\left(\frac{v_{\rm y}}{v_{\perp}^{i}}\right)^2}\right]^{-\frac{1}{2}} -\left(\frac{v_{\rm s}^{\rm adh}}{v_{\perp}^{i}}\right)^2,\quad \underline{v_{\perp}^{i}\geq v_{\rm y}}.\nonumber \end{eqnarray} \tag{ 4 }$$

(II) The R_d < R_thres regime is characterized by $v_{\rm s}^{\rm th}>v_{\rm s}^{\rm adh}$. Extending the T&N model we have

$$\begin{eqnarray} \fl &e_{\perp}(v_{\perp}^{i},R_{\rm d})=0,\quad \underline{v_{\perp}^{i}\leq v_{\rm s}^{\rm th}}\nonumber\\ \fl &e_{\perp}^2(v_{\perp}^{i},R_{\rm d})=\frac{6\sqrt{3}}{5} \left[1-\frac{1}{6}\left(\frac{v_{\rm y}}{v_{\perp}^{i}}\right)^2 \right]\left[\frac{v_{\rm y}}{v_{\perp}^{i}}\right]^{1/2}\\ \fl &\times\left[\frac{v_{\rm y}}{v_{\perp}^{i}}+\frac{2}{\sqrt{5}} \sqrt{6-\left(\frac{v_{\rm y}}{v_{\perp}^{i}}\right)^2}\right]^{-\frac{1}{2}} -\left(\frac{v_{\rm s}^{\rm adh}}{v_{\perp}^{i}}\right)^2,\quad \underline{v_{\perp}^{i}\geq v_{\rm s}^{\rm th}}\nonumber \end{eqnarray} \tag{ 5 }$$

with the sticking velocity $v_{\rm s}^{\rm th}$ given by $v_{\rm s}^{\rm th}=v_{\rm y}/w_0$, where w₀ is the unique real solution of the equation $\frac{6\sqrt{3}}{5}(1-\frac{1}{6}w^2)=(\frac{v_{\rm s}^{\rm adh}}{v_{\rm y}})^2w^{3/2}(w+\frac{2}{\sqrt{5}}\sqrt{6-w^2})^{1/2}$ in the interval 0 < w₀ ⩽ 1. The ratio of the sticking velocity to the adhesive sticking velocity for various values of the ratio $v_{\rm y}/v_{\rm s}^{\rm adh}$ is plotted in figure 4.

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Tangential interactions are more complicated. The contact area changes during the impact leading to a continuous redistribution of stresses; the tangential displacements and deformations of the impacting bodies may vary throughout the instantaneous contact area leading to scenarios where one part is sliding while simultaneously another part is sticking, but the tangential compliance also has a strong dependence on the loading history [32–34].

A useful approximation would be to neglect elastic displacements and plastic deformations by employing rigid body theory in the treatment of the tangential component of the impact. Within such an analysis, there are two possible regimes, rolling or sliding. Typically, the rolling regime is realized for near normal angles and the sliding regime for more oblique angles [35]. In what follows, we shall assume that we are always in the sliding regime, where a simple analytical expression for e_|| can be found.

The tangential restitution coefficient can be expressed as a function of the normal restitution coefficient with the aid of the impulse ratio $f=\frac{|P_{||}|}{|P_{\perp}|} =\frac{|\int_0^{\tau_{\rm imp}}F_{||}(t)\,\rmd t|}{|\int_0^{\tau_{\rm imp}}F_{\perp}(t)\,\rmd t|}$, where τ_imp is the impact duration and F_|| , F_⊥ are the tangential and normal components of the total contact forces, respectively [35–38]. Combined with Newton's second law this yields e_|| = 1 − f(1 + e_⊥) cot θ_i. Generally, the impulse ratio is different from the Coulomb friction coefficient μ. However, in the sliding regime f = μ and overall $e_{||}(v_{\perp}^{i},v_{||}^{i},R_{\rm d})=1-\mu (v_{\perp}^{i}/v_{||}^{i})\{1+e_{\perp}(v_{\perp}^{i},R_{\rm d})\}$.

This simplified analysis is indicative of the coupling between the normal and tangential directions during the impact. As aforementioned, the T&N model treats purely normal impacts, thus one would anticipate that for oblique impacts $e_{\perp}(v_{\perp}^{i},v_{||}^{i},R_{\rm d})\neq e_{\perp}(v_{\perp}^{i},R_{\rm d})$. However, finite element simulations of oblique elastic and elastic–plastic impacts [37] have revealed that (i) within the rolling regime $e_{\perp}(v_{\perp}^{i},v_{||}^{i},R_{\rm d})$ remains constant and equal to $e_{\perp}(v_{\perp}^{i},R_{\rm d})$, (ii) within the sliding regime $e_{\perp}(v_{\perp}^{i},v_{||}^{i},R_{\rm d})$ remains constant with a value slightly lower than $e_{\perp}(v_{\perp}^{i},R_{\rm d})$. Hence, in this work we shall assume that $e_{\perp}(v_{\perp}^{i},v_{||}^{i},R_{\rm d})=e_{\perp}(v_{\perp}^{i},R_{\rm d})$ for all impact angles.

It is important to note that during oblique impacts, since the tangential impulse is non-zero, the grain will inevitably acquire a rebound angular velocity even if it impinges without an initial spin [37, 38]. Since rotational dust dynamics are not treated in the present version of the MIGRAINe code, it is more self-consistent to assume a frictionless contact, μ = 0. In this case the rebounding grain will not spin and

$$\begin{equation} e_{||}(v_{\perp}^{i},v_{||}^{i},R_{\rm d})=1. \end{equation} \tag{ 6 }$$

Engineered surfaces are not smooth on the micrometre scale, unless designed for dedicated applications. Micrometre-scale roughness of the wall refers to length scales larger than the contact radius of the impacts and will therefore not affect the micro-physics of dust–wall collisions as defined by the sticking velocity, the yield velocity and the friction coefficient. However, it will have a major effect on the impact geometry by determining the impact angle.

Micrometre-scale roughness is modelled here by assuming that the wall is dispersed with random triangular corrugations. The geometry of the problem is illustrated in figure 5(a). At the impact point, let $\hat{{\bit n}}_{\rm sm}$ be the unit vector normal to the smooth surface and $\hat{{\bit t}}_{\rm sm}$ be the unit vector tangential to the smooth surface in the impact plane. We define $\hat{{\bit n}}_{\rm rough}$ to be the unit vector normal to the rough surface, $\hat{{\bit n}}_{\rm rough}=\cos{\varphi}\hat{{\bit n}}_{\rm sm}+\sin{\varphi}\cos{\chi}\hat{{\bit t}}_{\rm sm}+\sin{\varphi}\sin{\chi}\hat{{\bit n}}_{\rm sm}\times\hat{{\bit t}}_{\rm sm}$, where both the polar angle φ and the azimuthal angle χ are random variables. It is further assumed that there is no preferred direction in the smooth surface plane and hence χ follows a uniform probability distribution in the interval [0, 2π].

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In the normal plane to the smooth surface, defined by the random choice of azimuthal angle χ, as illustrated in figure 5(b), we assume the presence of a right triangle of base with constant length d and height with random length h following a zero-mean Gaussian probability distribution of standard deviation σ_surf. In this case, the rough surface lies on the hypotenuse and tan φ will follow a zero-mean Gaussian distribution of standard deviation σ_surf/d. Since only the absolute value of φ is of interest, it is more formal to say that tan φ follows a half-normal distribution of standard deviation $\sigma_{\rm surf}\sqrt{1-2/\pi}/d$. Overall,

$$\begin{equation*} \chi\sim \mathcal{U}\left[0,2\pi\right], \end{equation*}$$

$$\begin{equation} \tan{\varphi}\sim \mathcal{N}\left[0,\sigma_{\rm surf}^2/d^2\right]. \end{equation} \tag{ 7 }$$

Clearly, this treatment of micrometre-scale roughness is equivalent to randomizing the wall normal vector.

Real surfaces are never smooth on the nano-scale. Nano-scale roughness cannot modify the impact geometry but can play an important role in adhesion, since, depending on the stiffness of the material, it can lead to a significant reduction of the contact area between the colliding bodies and consequently a strong reduction in the adhesive sticking velocity [39]. It has long been realized that roughness hardly affects the adhesion of materials with very low elastic moduli (e.g. for rubber E ∼ 0.01 GPa), while it can reduce the adhesion of stiff materials by orders of magnitude (e.g. for tungsten E ∼ 400 GPa and for INCONEL E ∼ 200 GPa) [39].

Mathematical descriptions of nano-roughness typically model the rough surface as an average smooth surface plane dispersed with a random number of hemispherical asperities of the same radius r_asp (≪R_d) [40–43]. The centre of these asperities is offset from the smooth plane in a probabilistic manner, such that the asperity heights follow a Gaussian zero-mean distribution with a standard deviation σ_asp [41]. Within this framework, the application of the JKR theory of elastic adhesion [41] has demonstrated that the effect of roughness on adhesion can be expressed by the dimensionless parameter $\Theta=E^{*}\sigma_{\rm asp}^{3/2}r_{\rm asp}^{1/2} /(r_{\rm asp}\Gamma)$. This adhesion parameter represents the statistically averaged competition between the compressive forces exerted by the higher asperities that tend to detach the colliding bodies and the adhesive forces exerted by the lower asperities that tend to stick the colliding bodies. For Θ ≫ 1 (large σ_asp) adhesion is significantly inhibited and the adhesive sticking velocity $V_{\rm s}^{\rm adh}$ is significantly reduced even by orders of magnitude, whereas for Θ≪1 (small σ_asp) $V_{\rm s}^{\rm adh}$ approaches its theoretical value for smooth surfaces, given by (1).

In the relevant scenario of tungsten dust grains impinging on an INCONEL wall, the E^* modulus is large enough to guarantee that Θ ≫ 1 even for relatively low roughness. In this light, nano-scale roughness is accounted for in the MIGRAINe code by assuming that the actual adhesive sticking velocity is a random variable following a uniform probability distribution in the interval between zero and its theoretical value for smooth surfaces, i.e.

$$\begin{equation} V_{\rm s}^{\rm adh}\sim\ \mathcal{U}\left[0\,,v_{\rm s}^{\rm adh}\right]. \end{equation} \tag{ 8 }$$

Table 1 presents the numerical values of the material properties that are necessary for the calculation of the normal restitution coefficient for W dust impinging on an INCONEL wall. As already mentioned, there is no consensus in the literature for the value of the limiting contact pressure p_y. In what follows, we shall assume p_y = 2.8σ_y as our reference value, but we shall also explore the sensitivity of our results to p_y.

In figure 6(a) the dependence of the normal restitution coefficient on the normal impact velocity for different dust sizes is shown. For p_y = 2.8σ_y, R_d < R_thres ∼ 10 µm and therefore we are in regime II, $v_{\rm s}^{\rm th}>v_{\rm s}^{\rm adh}$. By definition, the origin of the curves is located at the sticking velocity. The sticking velocity is nearly inversely proportional to the dust radius, whereas the yield velocity is independent of the dust size. For normal velocities much larger than the yield velocity, the curves collapse on each other, since the main mechanism of energy loss is plastic work, whose effect on the restitution coefficient is independent of size. On the other hand, for normal velocities smaller than or of the order of the yield velocity, the curves strongly diverge, since energy dissipation is mainly due to adhesive work, which strongly depends on size.

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In figure 6(b) the dependence of the normal restitution coefficient on the normal impact velocity for a 1 µm radius particle and for different values of the limiting contact pressure is shown. According to (2) the yield velocity scales as p_y^5/2, recalling that v_y is the threshold velocity at which plastic work occurs, the higher the value of p_y the less energy is lost in the impact. For p_y = 1.6σ_y, 2.8σ_y and 4σ_y the threshold radii are R_thres = 50 µm, 10 µm and 3 µm, respectively. Therefore, we are still in regime II, where $v_{\rm s}^{\rm th}>v_{\rm s}^{\rm adh}$, this is demonstrated by the inverse dependence of the sticking velocity on the limiting pressure.

Finally, figure 7 shows the normal restitution coefficient for various values of the adhesive sticking velocity lower than the theoretical limit, as required by nano-roughness. As anticipated, e_⊥ tends to unity as $V_{\rm s}^{\rm adh}$ approaches zero, since due to the absence of adhesive work the collision will be nearly elastic in the low velocity limit. Notice also that the probabilistic reduction of $V_{\rm s}^{\rm adh}$ according to (8) is responsible for a transition from the $v_{\rm y}<V_{\rm s}^{\rm adh}$ to the $v_{\rm y}>V_{\rm s}^{\rm adh}$ regime.

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Due to the lack of relevant data on dust–surface collisions, there are no means of calibrating the theoretical restitution coefficient curves against experimental evidence. To our knowledge, the only empirical study of dust impacts for tokamak-relevant materials was reported in [44], where micrometric particles were accelerated by means of a modified pellet injection system and their sticking velocity $v_{\rm s}^{\rm th}$ was determined. In that work, abnormally high values for the sticking velocity in excess of 40 m s⁻¹ were presented in the case of W grains of size R_d < 8 µm impacting stainless steel (SS) targets (note that the mechanical properties of SS and INCONEL are very close). Such values are in clear contradiction with all theoretical predictions as well as with experimental data on micrometre particle impacts. Additional experiments have revealed that this overestimation likely occurs due to the agglomeration of smaller and larger grains in the dust mixture inside the sabot, nullifying the correlation between the sabot speed and the actual impact velocity of the smaller grains.

Due to the wide spread in size of the injected dust, trajectories of grains with various radii, all starting from the injection point with a downward velocity of 1.3 m s⁻¹, are studied. To gain an insight on the overall dynamics we first consider the case of an ideal smooth wall, i.e. without introducing the effect of roughness.

The trajectories for different initial R_d are shown in figure 8. The motion is mainly determined by the toroidal ion drag and the centrifugal effect due to the curvature of the ion flow—as discussed in the previous studies [9, 10]. The overall 1/R_d scaling of the acceleration due to the ion drag force induces an initial selection process on the dust size; particles with initial radii above some critical value R_c reach the hot plasma and vaporize, while smaller particles acquire enough speed to collide with the wall. This is detailed in the inset of figure 8, where R_c = 1.7 µm. The value of R_c depends on the ion flow and the details of the heat balance. For example, flow velocity values at the injection point of 10 ± 5 km s⁻¹ yield R_c = 1.7 ± 0.5 µm. Grains with initial sizes R_d < R_c are able to migrate and are further separated by the functional dependence of the restitution coefficient: since the sticking velocity is a decreasing function of size, larger grains can bounce off several times and continue to hover in the SOL, while smaller ones are more likely to stick.

The consistency with the experimentally observed size selection of the collected grains indicates that the main processes are accounted for. On the other hand, clearly the main experimental evidence—dust reaching the collector—is not reproduced. The particles do not travel further than 45° toroidally, neither do they reach the equatorial plane, where the collectors are located (figure 8). This is due to the following trend of deterministic collisions; the normal component of the velocity is always reduced, resulting in more grazing impact angles, which drive the trajectories towards the sticking condition. Although some modifications to the profiles, such as, e.g., a steeper T_i decay near the wall, can improve the transport and widen the size selection, sticking will still occur well before the collector position. The same is valid for variations of the yield velocity (limiting contact pressure p_y); although increasing from p_y = 1.6σ_y to p_y = 4σ_y does imply more elastic rebounds (see figure 6(b)), the advance in poloidal transport between these two cases is within ∼10°.

We, thus, proceed to a more realistic scenario—collisions with a rough wall. As discussed in sections 3.3 and 3.4, micrometre and nano-scale irregularities can be modelled by randomizing the wall normal and the adhesive sticking velocity, respectively. Below we show results for the case of initial R_d = 1 µm, not only because dust grains of this size are representative of the initial R_d < R_c population which is able to migrate, but also because within this population, they have the highest radius upon their first collision (nearly the same as their initial radius) and hence are the most likely to be transported far.

When only the randomization of the adhesive sticking velocity is applied, the results are quite similar to those of figure 8 since the particles keep on bouncing at increasingly grazing angles, which eventually leads them to stick rather close to their first impact point.

The results with randomization of the surface normal differ drastically, as seen from figure 9 (for σ_surf/d = 0.5). Such a randomization permits for the tangential part of the rebound velocity, essentially conserved in the impact, to have a component directed away from the wall, which allows the particles to spend a sufficient time in regions of stronger ion flow, and hence enhances dust transport. In fact, as observed from the figure, in this case a fraction of the injected dust is able to reach the collector position. Values of σ_surf/d larger than 0.5 do not significantly improve the distance of migration as they tend to correspond to a uniform distribution of the surface normal, allowing more particles to bounce backwards or directly towards the core.

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Finally, figure 10 presents the results with both randomizations taken into account. The number of particles reaching the collector does not differ significantly from that in figure 9 (drastically enhanced statistics would be necessary to draw a firm qualitative statement). The poloidal projection in figure 10, however, shows that more particles were vaporized by the hot plasma, nearly 80% as compared with ∼50% in figure 9. In fact, as the reduction of the adhesive sticking velocity further increases the restitution coefficient (see figure 7), particles that rebound towards the hot plasma will tend to penetrate deeper and will be more likely to vaporize. Therefore, the effect of the second randomization on the distance of migration is twofold; a higher restitution coefficient inhibits sticking, but, when coupled with the randomization of the surface normal, it can also enhance the probability of vaporization.

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Below we follow the enumeration of the experimental results as given in section 2.

The collection of dust by poloidally exposed samples and not by toroidal ones—experimental results (i) and (ii)—can be explained as follows: dust reaches the Si collectors with typical velocities of several m s⁻¹ poloidally and ∼100 m s⁻¹ toroidally. It impacts poloidally exposed samples at grazing angles and is efficiently trapped as a result of the collector geometry described in section 2. On the other hand, it impacts nearly normally on toroidally exposed samples, and bounces off owing to the low critical velocity values for W dust impinging on a Si surface (whose order of magnitude is similar to those in figure 6). Such an interpretation is supported by results of the previous campaign where C dust was injected and collected by toroidally exposed aerogel samples [18]. Aerogel is a highly porous material and micrometre dust with a velocity of 100 m s⁻¹ is efficiently captured, in contrast to similar impacts on any solid collector [4].

The selectivity of the size of migrating dust—experimental result (iii)—is caused by the loss of dust particles due to vaporization (upper threshold) and sticking to the wall (lower threshold). The particles with an initial radius below R_c lose a fraction of their mass on their way to the wall, causing them to experience their first collision with a radius not above 1 µm. Therefore, our simulations predict that no particles of larger radius can be collected. As mentioned above, the exact values of R_c and of the maximum size of the particles reaching the wall depend on the details of the plasma profiles, hence the agreement with the experimental values is rather reasonable. The lower limit in the experimentally observed sizes can be explained by the fact that small particles stick very efficiently and are unlikely to undergo more than a few collisions with the wall, and thus reach the collector.