2.1 Dust sizes

Dust-size distributions in fusion experiments are conveniently measured by collecting samples from the experiments. We show an example from a carbon-arc discharge here and analyse the distribution using an optical microscope and ImageJ (Fiji distribution) in figure 1. Previous Raman measurements indicate similarities between dust produced in a carbon arc and in the National Spherical Torus eXperiment (NSTX) (Raitses et al. Reference Raitses, Skinner, Jiang and Duffy2008).

Figure 1.

(Left) Dust of different sizes and irregular shapes collected from an arc discharge using graphite as the cathode and a mixture of hydrogen and argon. (Right) Another graphite electrode nearby shows redeposition of carbon.

Figure 2.

Dust-size distribution from a carbon-arc discharge. The number of dust particles increases rapidly as the size decreases, consistent with in situ measurements (Voinier et al. Reference Voinier, Skinner and Roquemore2005).

Two models are used to describe the size distribution in figure 2, a power-law model and an exponential-law model,

(2.1) $$\begin{eqnarray}n_{d}=n_{0}(r/r_{0})^{-{\it\alpha}}\end{eqnarray}$$

and

(2.2) $$\begin{eqnarray}n_{d}=n_{0}\exp [-(r/r_{0})^{{\it\beta}}].\end{eqnarray}$$

For the stainless steel (SS) collector, the two models gives slightly different fits to the data. The fitting power in (2.1) is ${\it\alpha}=1.89$ . In the exponential model (2.2), ${\it\beta}=0.331$ . For the C-redeposit sample, the fitting power is ${\it\alpha}=1.63$ . The exponential model gives a better fit with ${\it\beta}=0.564$ .

2.2 Dust velocities

For dust with a small initial velocity ( ${\sim}$ a few $\text{m}~\text{s}^{-1}$ ), it can be accelerated by ion-drag forces in fusion plasmas at the edge as well as inside the edge pedestal. Due to ablation, the dust motion can be quite complicated (Pigarov et al. Reference Pigarov, Krasheninnikov, Soboleva and Rognlien2005; Wang et al. Reference Wang, Skinner, Delzanno, Krasheninnikov, Lapenta, Pigarov, Shukla, Smirnov, Ticos and West2008; Bacharis et al. Reference Bacharis, Coppins and Allen2010; Krasheninnikov et al. Reference Krasheninnikov, Smirnov and Rudakov2011; Ratynskaia et al. Reference Ratynskaia, Vignitchouk, Tolias, Bykov, Bergsker, Litnovsky, Den Harder and Lazzaro2013). We shall only estimate the magnitude of velocities for MPI development. When the dust velocity ( $\boldsymbol{u}_{d}$ ) is small compared with plasma flow ( $\boldsymbol{u}_{i}$ ), the force on the dust is approximately (Baines et al. Reference Baines, Williams and Asebiomo1965; Ticos et al. Reference Ticos, Wang, Delzanno and Lapenta2006a )

(2.3) $$\begin{eqnarray}\boldsymbol{F}=2{\rm\pi}r_{d}^{2}kT\mathop{\sum }_{i}n_{i}\frac{\boldsymbol{s}_{i}^{3}}{s_{i}}.\end{eqnarray}$$

The sum is for different ion species. The normalized relative velocity $\boldsymbol{s}_{i}=(\boldsymbol{u}_{i}-\boldsymbol{u}_{d})/v_{i}$ is the ion flow ( $\boldsymbol{u}_{i}$ ) relative to the dust motion, with the normalization $v_{i}=\sqrt{2kT/m_{i}}$ . It is interesting to estimate how fast dust can be accelerated by ion-drag force in fusion plasmas. For a constant ion drag and initial dust velocity $\boldsymbol{u}_{d}\sim 0$ , (2.3) gives

(2.4) $$\begin{eqnarray}s(t)=\frac{1}{\displaystyle \frac{t}{{\it\tau}_{0}}+\frac{1}{s_{0}}},\end{eqnarray}$$

where the characteristic time ${\it\tau}_{0}$ is given by

(2.5) $$\begin{eqnarray}{\it\tau}_{0}=\frac{m_{d}v_{i}}{2{\rm\pi}r_{d}^{2}kTn}.\end{eqnarray}$$

As expected, for the same dust size, higher density W particles take longer to be accelerated to the same speed than lower density particles of Be or C. ${\it\tau}_{0}$ for fusion-relevant conditions is illustrated in figure 3 for carbon dust, which is in the hundreds of ms to tens of seconds range. Extrapolations to other materials and plasma conditions are possible using (2.5). When a dust grain moves for a distance $L$ in the fusion plasma, it will reach a velocity $u_{d}$ given by

(2.6) $$\begin{eqnarray}u_{d}\sim \sqrt{\frac{2Lv_{i}}{{\it\tau}_{0}}},\end{eqnarray}$$

assuming the plasma flow is of the same order as the thermal velocity $v_{i}$ . An example is shown in figure 4. Tens of $\text{m}~\text{s}^{-1}$ to hundreds of $\text{m}~\text{s}^{-1}$ is even possible depending on the dust mass and size.

Figure 3.

Entrain time ${\it\tau}_{0}$ as a function of deuterium plasma density and temperature for carbon dust of $1~{\rm\mu}\text{m}$ in radius. Dust granules are unlikely to approach plasma flow velocities in the $\text{km}~\text{s}^{-1}$ range.

Figure 4.

Expected dust velocity as a function of acceleration distance due to ion-drag force – carbon dust ( $10~{\rm\mu}\text{m}$ in radius). The deuterium plasma density is $10^{19}\,\text{m}^{-3}$ .

3.1 Ablation model

The ablation equation is

(3.1) $$\begin{eqnarray}\frac{1}{M_{0}}\frac{\text{d}m_{d}}{\text{d}t}=-\frac{A_{d}{\it\Gamma}^{\infty }f_{s}}{E_{0}},\end{eqnarray}$$

which for a spherical micropellet may be rewritten as

(3.2) $$\begin{eqnarray}\frac{\text{d}r_{d}}{\text{d}t}=-f_{s}c_{a}.\end{eqnarray}$$

$m_{d}$ , $A_{d}$ , ${\it\rho}_{d}$ and $r_{d}$ are the dust or micropellet mass, surface area, mass density and radius, respectively. $M_{0}$ is the mass of individual atoms/molecules. $E_{0}$ , the evaporation/sublimation energy per atom/molecule, is listed in table 2 for several materials. We introduce a characteristic ablation speed $c_{a}$ that satisfies

(3.3) $$\begin{eqnarray}c_{a}\equiv \frac{M_{0}}{{\it\rho}_{0}}\frac{{\it\Gamma}^{\infty }}{E_{0}}.\end{eqnarray}$$

The heat flux at far away from the dust ${\it\Gamma}^{\infty }=\sum _{j}{\it\Gamma}_{j}^{\infty }$ is the sum of the thermal fluxes of electrons and ions. ${\it\Gamma}_{j}^{\infty }=n_{j}\bar{v}_{j}(2kT_{j})/4$ and $\bar{v}_{j}=\sqrt{8kT_{j}/{\rm\pi}m_{j}}$ with corresponding mass ( $m_{j}$ ), temperature ( $T_{j}$ ) and density ( $n_{j}$ ) for ions and electrons, respectively. $f_{s}$ is the heat flux shielding factor. If $0<f_{s}<1$ , the heat flux is reduced due to ablation cloud shielding and dust charging and other secondary effects. If $f_{s}>1$ , the heat flux is enhanced due to dust–plasma interactions.

Table 2.

Material properties for ablation models.

a Liquid density or solid density when sublimates.

b Based on evaporation/sublimation energy. For carbon, the value of $589~\text{kJ}~\text{mol}^{-1}$ comes from Doehaerd et al. (Reference Doehaerd, Goldfinger and Waelbroeck1952).

c Per atom.

d Sublimation temperature.

3.2 Pre-ablation phase

As recognized in Kuteev et al. (Reference Kuteev, Sergeev and Tsendin1984), there is a pre-ablation phase for most non-cryogenic materials before evaporation and sublimation begin. We compiled a material property table based on Wikipedia and NIST condensed phase thermochemistry data (NIST 2016), shown in table 2. The pre-ablation energy per atom ( $E_{1}$ ) is the mean energy that each atom will gain before evaporation or sublimation begins. Another way to estimate $E_{1}$ is to subtract the evaporation/sublimation energy ( $E_{0}$ ) from the surface bonding energy of the atoms to the solid.

In the pre-ablation phase, $\text{d}m_{d}/\text{d}t\sim 0$ . We may calculate the ability of dust and micropellets to penetrate through edge plasma and cross the separatrix as a function of size and velocity without losing any mass. We assume a constant plasma density of $10^{19}~\text{m}^{-3}$ and temperature profiles $T_{e}(z)=T_{i}(z)=T_{a}+(T_{b}-T_{a})z/z_{0}$ , where $T_{a}=10$ eV and $T_{b}=200$ eV. $z=0$ corresponds to the wall position. $z_{0}=10$ cm is the width of the SOL.

The electron and ion heating rates, ${\it\Gamma}_{e}$ and ${\it\Gamma}_{i}$ , can be related to the electron and ion currents ( $I_{e}$ and $I_{i}$ ) for Maxwellian distributions of electrons and ions,

(3.4a,b ) $$\begin{eqnarray}{\it\Gamma}_{e}=-\frac{I_{e}}{e}2kT_{e},\quad {\it\Gamma}_{i}=\frac{I_{i}}{Z_{i}e}\frac{2kT_{i}-Z_{i}e{\it\phi}_{d}}{1-Z_{i}e{\it\phi}_{d}/kT_{i}},\end{eqnarray}$$

with $-I_{e}=I_{i}$ , the vanishing current condition. The following assumptions are made: the sticking coefficients of electrons and ions on the dust are one, independent of their energies. The thermionic and secondary electron emissions, as well as radiative heating and cooling, are negligible. The vanishing current condition is used to find ${\it\phi}_{d}$ self-consistently through

(3.5) $$\begin{eqnarray}\bar{v}_{e}\exp \frac{e{\it\phi}_{d}}{kT_{e}}=\bar{v}_{i}\left(1-\frac{Z_{i}e{\it\phi}_{d}}{kT_{i}}\right).\end{eqnarray}$$

For $T_{i}=T_{e}=T$ and $Z_{i}=1$ (hydrogen and deuterium plasmas), the ratio $e{\it\phi}_{d}/kT$ is independent of $T$ and has the value of –2.504 and –2.776 for hydrogen and deuterium plasma, respectively.

For an SOL of width $L_{0}$ , the dust penetration without evaporation gives the minimum dust velocity required before ablation starts,

(3.6) $$\begin{eqnarray}u_{d}^{min}=\int _{0}^{L_{0}}\,\text{d}z\frac{{\it\Gamma}_{e}+{\it\Gamma}_{i}}{N_{d}E_{1}},\end{eqnarray}$$

where the integration is along the trajectory of the dust. $N_{d}$ is the initial number of atoms/molecules contained in the dust and $E_{1}$ is given in table 2. $u_{d}^{min}$ for several materials are shown in figure 5.

Figure 5.

Minimum dust velocity required to penetrate a ‘model’ edge plasma with a constant density of $10^{19}~\text{m}^{-3}$ and a linear variation of temperature from 10 to 200 eV. The same pre-ablation model can be applied to other realistic edge-plasma scenarios.

3.3 The shielding factor $f_{s}$ and surface neutral density $n_{surf}$

In the ‘small’ or orbital motion limit (OML), the shielding factor $f_{s}$ for Maxwellian distributions ( $T_{i}=T_{e}=T$ ) can also be calculated as

(3.7) $$\begin{eqnarray}f_{s}=\frac{{\it\Gamma}_{e}+{\it\Gamma}_{i}}{{\it\Gamma}^{\infty }}=\frac{\bar{v}_{e}}{\bar{v}_{e}+\bar{v}_{i}}\exp \left(\frac{e{\it\phi}_{d}}{kT}\right)+\frac{\bar{v}_{i}}{\bar{v}_{e}+\bar{v}_{i}}\left(1-\frac{e{\it\phi}_{d}}{2kT}\right).\end{eqnarray}$$

The temperature dependence cancels out for $T_{i}=T_{e}=T$ , which yields $f_{s}=0.134$ and 0.102 for hydrogen and deuterium plasmas and the self-consistent $e{\it\phi}_{d}$ as given above. $f_{s}$ as a function of $e{\it\phi}_{d}$ is shown in figure 6.

Figure 6.

Shielding factor ( $f_{s}$ ) as a function of dust potential ${\it\phi}_{d}$ (normalized to plasma temperature $T$ ).

For sufficiently large dust, $f_{s}<1$ due to the shielding of the ablation cloud, which consists of neutral atoms and so-called ‘secondary’ plasma. Ionization of the neutral atoms gives rise to the secondary plasma, to distinguish it from the surrounding plasma or the ‘primary’ plasma. The shielding of the cloud is not effective when the condition

(3.8) $$\begin{eqnarray}\int _{r_{d}}^{\infty }\,\text{d}rn_{0}(r,r_{d})M_{0}\leqslant {\it\rho}_{0}R_{e}\end{eqnarray}$$

is met. The left-hand side is the areal density of the neutral-atom cloud. $R_{e}$ on the right-hand side is the electron range in the cloud corresponding to a density ${\it\rho}_{0}$ . The product ${\it\rho}_{0}R_{e}$ is a material property. The neutral density $n_{0}(r,r_{d})$ is given by

(3.9) $$\begin{eqnarray}n_{0}(r,r_{d})=n_{surf}(r_{d}^{\prime })\exp \left(-\frac{r-r_{d}^{\prime }}{{\it\Lambda}_{e}}\right),\end{eqnarray}$$

where ${\it\Lambda}_{e}=u_{0}/(n_{e}\langle {\it\sigma}_{i0}v_{e}\rangle )$ is the neutral ionization mean free path, $n_{e}$ the electron density and $\langle {\it\sigma}_{i}v_{e}\rangle$ the average ionization rate coefficient. $u_{0}$ is the speed of the neutrals leaving the dust surface and it remains constant until they collide with another atom or heavy ion. Electrons are too light to deflect the neutrals. Here we introduce a new variable $n_{surf}(r_{d})$ , which is the surface neutral density when the dust is at radius $r_{d}$ . $r_{d}^{\prime }$ is related to the instantaneous radius $r_{d}$ through a time delay ${\it\delta}t$ as

(3.10) $$\begin{eqnarray}{\it\delta}t=\frac{r-r_{d}^{\prime }}{u_{0}}=\int _{r_{d}}^{r_{d}^{\prime }}\,\text{d}s\frac{1}{f_{s}c_{a}}\end{eqnarray}$$

based on the ablation (3.2) and (3.3) for $c_{a}$ . Assuming that $f_{s}$ remains constant for the period when the dust or micropellet shrinks from a size $r_{d}^{\prime }$ to $r_{d}$ , solution of equation (3.10) for $r_{d}^{\prime }$ gives

(3.11) $$\begin{eqnarray}r_{d}^{\prime }=\frac{f_{s}c_{a}r+u_{0}r_{d}}{u_{0}+f_{s}c_{a}},\quad r\geqslant r_{d}.\end{eqnarray}$$

From neutral particle flux conservation, the surface density $n_{surf}(r_{d})$ is given by

(3.12) $$\begin{eqnarray}n_{surf}(r_{d})=\frac{{\it\Gamma}^{\infty }f_{s}}{E_{0}u_{0}}=\frac{{\it\rho}_{0}}{M_{0}}\frac{f_{s}c_{a}}{u_{0}}.\end{eqnarray}$$

The neutral atoms expand thermally at the evaporation temperature with a radial velocity of up to $u_{0}=\sqrt{2kT_{0}/3M_{0}}$ . Here the factor 3 takes into account that the total thermal energy $kT_{0}$ from evaporation is equally shared among the three degrees of freedom. The transition radius as a function of $n_{surf}(r_{d})$ has been calculated for Li and C in figure 7.

Figure 7.

Transition dust radius as a function of the surface neutral density ( $n_{surf}$ ) for Li and C spheres. $N_{0}=2.69\times 10^{19}\,\text{m}^{-3}$ (the ideal gas density at STP). The plasma is assumed to be at 100 eV with a density of $10^{19}~\text{m}^{-3}$ .

5.1 Granule dropper experiments in NSTX

A compact piezoelectrically actuated granule dropper has enabled several successful injections of both low- $Z$ (Li) and high- $Z$ (W) dust into tokamaks (Mansfield et al. Reference Mansfield, Roquemore, Carroll, Sun, Hu, Zhang, Liang, Gong, Li and Guo2013). By varying the amplitude of the piezoelectric oscillation at a resonance around 2 kHz, the dropper provided a regulated mass injection which was rapidly aerosolized as it impacted the edge plasma. Commercial spherical Li powder with a $44~{\rm\mu}\text{m}$ average diameter was stabilized against reaction with air by a 30 nm thick mantle of $\text{Li}_{2}\text{CO}_{3}$ . When dust particles were gravitationally accelerated, their velocities were around 3 $\text{m}~\text{s}^{-1}$ . When used in conjunction with a mechanical propeller, granule velocities of up to tens of $\text{m}~\text{s}^{-1}$ have been achieved. In long-pulse tokamaks, the dust dropper has shown the ability to reliably provide real-time Li injection rates between 1 and $120~\text{mg}~\text{s}^{-1}$ for periods of up to 30 s (Hu et al. Reference Hu, Ren, Sun, Zuo, Yang, Li, Mansfield, Zakharov and Ruzic2014).

The three-dimensional motion of lithium dust in NSTX was tracked using two toroidally separated fast visible cameras (Nichols et al. Reference Nichols, Roquemore, Davis, Mansfield, Skinner, Feibush, Boeglin, Patel, Abolafia and Hartzfeld2011). Between 10 and 250 mg of Li dust were dropped into H-mode discharges with neutral beam heating power of 2–6 MW, plasma current ( $I_{p}$ ) 0.9 MA and discharge duration 1.0–1.3 s. While a statistically representative sample could not be obtained, it was clear that most ( ${\sim}90\,\%$ ) dust particles drifted perpendicular to field lines, while a smaller ( ${\sim}10\,\%$ ) population accelerated along field lines and an even smaller ( ${<}1\,\%$ ) population changed direction mid-flight. Even at the highest injection rates, no Li dust particles were observed to cross the separatrix. Additionally, no discharges were lost to disruptions due to the injection of Li dust, and in some cases the introduction of Li dust improved confinement.

In the case of W injection, an approximate rate of $3~\text{mg}~\text{s}^{-1}$ was achieved for the duration of a 700 ms NSTX discharge (Clementson et al. Reference Clementson, Beiersdorfer, Roquemore, Skinner, Mansfield, Hartzfeld and Lepson2010). In that case, the W particles took the form of irregularly shaped crystals with $5~{\rm\mu}\text{m}$ average diameter and maximum dimension of $10~{\rm\mu}\text{m}$ . W dust trajectories were also obtained, but they are not directly comparable to Li trajectories because NSTX was operating with a reversed toroidal field at the time. Significant W radiation was observed in the core during the dust injection, and W dust appeared to have crossed the separatrix, but nevertheless the discharge did not disrupt. Furthermore, during the subsequent discharges with no W injection, there was no spectroscopic evidence of W residue in the core, indicating that injected W particles had been pumped away or deposited on a plasma-facing component (PFC) surface.

5.2 MPI technologies

In addition to gas injection and pellet injectors based on high-pressure gas acceleration, several mass-injection technologies are shown in figure 10 in the velocity versus radius plot. The velocity scales with radius as $u_{d}\propto r_{d}^{-1}$ because of the fact that the forces of acceleration are proportional to $r_{d}^{2}$ while the mass scales with radius as $m_{d}\propto r_{d}^{3}$ . In comparison with fuelling pellets moving at several hundred $\text{m}~\text{s}^{-1}$ , higher speeds and smaller masses can be advantageous for many applications (Plöckl et al. Reference Plöckl, Lang, Jehl, Prechtl and Sotier2011). Blower guns, centrifuge launchers (IPP 0000) and rail guns are some other possibilities for larger mass. Supersonic molecular beam injectors are examples for smaller mass (Xiao et al. Reference Xiao, Diamond, Kim, Yao, Yoon, Ding, Hahn, Kim, Xu and Chen2012).

Figure 10.

A comparison of different mass-injection technologies. Along the lowest slanted line from the top to the bottom are an electrostatic dust accelerator (Shu et al. Reference Shu, Collette, Drake, Grün, Horányi, Kempf, Mocker, Munsat, Northway and Srama2012), a plasma-drag accelerator (Ticos et al. Reference Ticos, Wang, Dorf and Wurden2006b ) and a mechanical lithium propeller (Mansfield et al. Reference Mansfield, Roquemore, Carroll, Sun, Hu, Zhang, Liang, Gong, Li and Guo2013). Along the slanted line above is a two-stage gas gun (Physics Applications Inc. 0000). No technology exists today along the top slanted line.

5.3 Materials of interest

For PMI research, it is useful to examine dust transport of carbon, Be, W and compounds of these first wall/divertor materials. For ELM pacing and disruption mitigation, additional low- $Z$ (Li and LiD for example) and high- $Z$ materials should be examined. Material selection is also constrained by availability, safety (fire hazard and health hazard) and the method of injection.

Recent induced ELM pacing experiments with conventional cryogenic pellets in DIII-D as well as room-temperature lithium granule injection in EAST (Wu Reference Wu2007) have motivated further ELM pacing studies (Baylor et al. Reference Baylor, Commaux, Jernigan, Brooks, Combs, Evans, Fenstermacher, Isler, Lasnier and Meitner2013; Mansfield et al. Reference Mansfield, Roquemore, Carroll, Sun, Hu, Zhang, Liang, Gong, Li and Guo2013). In NSTX-U (U for ‘upgrade’ that just finished recently), for example, both boron carbide splinter powder and vitreous carbon microspheres have undergone laboratory testing in preparation for the upcoming run campaign. Both granule types were impurity assayed by Evans Analytical Group (EAG) utilizing glow-discharge mass spectroscopy. While the carbon microspheres were found to be of high purity (99.9 %+) the boron carbide splinter powders were found to contain fractional percentages of the impurities Si and Fe at the 0.57 % and 0.15 % levels, respectively. The carbon and boron carbide granules have been grouped into three average sizes of 300, 600 and $900~{\rm\mu}\text{m}$ . It is anticipated that 50–100 Hz injection of these microgranules into the edge of NSTX-U plasmas will accelerate the frequency of ELMs and thereby reduce the peak heat load to the divertor.

5.4 Possible new experiments

Extensive laboratory experiments will be needed to develop MPI technologies for various applications. Major fusion experiments such as NSTX-U, DIII-D, JET, W7-X (Beidler et al. Reference Beidler, Grieger, Herrnegger, Harmeyer, Kisslinger, Lotz, Maassberg, Merkel, Nuhrenberg and Rau1990), EAST and others can be complemented by smaller scale laboratory experiments, which allow better access for systematic examination of MPI physics and technology. Several facilities in the USA, shown in table 3, are used as illustrations here. The Hybrid Illinois Device for Research and Applications (HIDRA) facility (Andruczyk et al. Reference Andruczyk, Ruzic, Allain and Curreli2015) allows dust transport and ablation research. The Magnetized Dusty Plasma eXperiment (MDPX) (Thomas et al. Reference Thomas, Konopka, Artis, Lynch, Leblanc, Adams, Merlino and Rosenberg2015) permits dust charging at elevated temperatures and dust generation. The Material Plasma Exposure eXperiment (MPEX) (Rapp et al. Reference Rapp, Biewer, Canik, Caughman, Goulding, Hillis, Lore and Owen2013) allows examination of dust dynamics and erosions in divertor-like conditions.

Table 3.

Possible facilities for new micropellet injection experiments and development.