1. Introduction
The interaction between the edge of a fusion plasma and the plasma-facing components (PFCs) is one of the most important issues in fusion (Sharpe, Petti & Bartels Reference Sharpe, Petti and Bartels2002; Grisolia et al. Reference Grisolia, Rosanvallon, Sharpe and Winter2009; Ratynskaia et al. Reference Ratynskaia, Thorén, Tolias, Pitts, Krieger, Vignitchouk and Iglesias2020; Ratynskaia, Bortolon & Krasheninnikov Reference Ratynskaia, Bortolon and Krasheninnikov2022a). The melting of the surface layer on the PFCs and delamination of the co-deposits are considered to be the main reasons for the generation of dust particles. These particles lead to substantial deleterious effects on the discharge performance, including disruption and radiation loss (Liu et al. Reference Liu, Chen, Mao, Sun and Duan2013).
In recent years, studies of dust particles in fusion devices have been carried out in many aspects, including the generation and re-deposition of dust particles (Tolias et al. Reference Tolias, Ratynskaia, De Angeli, De Temmerman, Ripamonti, Riva, Bykov, Shalpegin, Vignitchouk and Brochard2016; Ratynskaia et al. Reference Ratynskaia, Tolias, Bykov, Rudakov, De Angeli, Vignitchouk, Ripamonti, Riva, Bardin and van der Meiden2016, Reference Ratynskaia, Tolias, De Angeli, Rohde, Herrmann, Ripamonti, Riva, Thorén, Vignitchouk and Sieglin2018), the interaction between dust particles and plasma (Laux, Balden & Siemroth Reference Laux, Balden and Siemroth2014; Autricque et al. Reference Autricque, Khrapak, Couëdel, Fedorczak, Arnas, Layet and Grisolia2018; Block & Melzer Reference Block and Melzer2019), particle injections and transports (Delzanno & Tang Reference Delzanno and Tang2014; Lunsford et al. Reference Lunsford, Sun, Maingi, Hu, Mansfield, Xu, Zuo, Diallo, Osborne and Tritz2018) and the characteristics of dust particles (Arnas et al. Reference Arnas, Irby, Celli, De Temmerman, Addab, Couëdel, Grisolia, Lin, Martin and Pardanaud2017; Pardanaud et al. Reference Pardanaud, Dellasega, Passoni, Martin, Roubin, Addab, Arnas, Couëdel, Minissale and Salomon2020; Pan et al. Reference Pan, Ding, Peng, Yan, Zhu and Chen2022). The applications of fast cameras provide an important method for particle and plasma diagnosis (Ratynskaia et al. Reference Ratynskaia, Castaldo, Bergsåker and Rudakov2011; Shalpegin et al. Reference Shalpegin, Brochard, Ratynskaia, Tolias, De Angeli, Vignitchouk, Bykov, Bardin, Bystrov and Morgan2015). It is possible to obtain the particle trajectories from the wall materials to the core plasma.
During their transport, dust particles sample a wide range of plasma conditions (Winter Reference Winter1999; Fortov et al. Reference Fortov, Ivlev, Khrapak, Khrapak and Morfill2005; Morfill & Ivlev Reference Morfill and Ivlev2009). As one of the most interesting phenomena, fragmentation of dust particles has been observed in tokamaks (Holgate et al. Reference Holgate, Simons, Andrew and Stavrou2019). Since the dust fragments are usually small in size and have low heat capacity, their penetration depth from the scrape-off layer (SOL) is correspondingly small. As a result, fragmentation of dust particles leads to lower $Z$
impurities and less Bremsstrahlung in the core and can mitigate the disruptions triggered by localized plasma cooling to some extent. Therefore, understanding the mechanism of fragmentation and the dynamics of fragments is of great importance to control the hazards caused by these dust impurities.
In fusion, the plasma temperature reaches ${\sim }1$
keV in the core, but drops dramatically in the SOL. Nevertheless, it still exceeds the melting temperature of materials of the PFCs such as tungsten and molybdenum. The dust particles appear as droplets and may undergo an electrostatic breakup into several fragments (Duft et al. Reference Duft, Lebius, Huber, Guet and Leisner2002; Koromyslov, Grigor'Ev & Rybakova Reference Koromyslov, Grigor'Ev and Rybakova2002; Achtzehn et al. Reference Achtzehn, Müller, Duft and Leisner2005). The instability condition of a charged particle in vacuum or neutral gas was derived by Rayleigh (Reference Rayleigh1882) and reads $Q^2>16 {\rm \pi}\gamma a^3$
, where $Q$
is the particle charge, $\gamma$
is the liquid surface tension and $a$
is the particle radius. Recent studies show that the particles are more likely to be destroyed due to electrostatic breakup than to slower processes such as evaporation in fusion devices (Coppins Reference Coppins2010; Holgate & Coppins Reference Holgate and Coppins2018). Rotation is another main cause of fragmentation (Krasheninnikov et al. Reference Krasheninnikov, Tomita, Smirnov and Janev2004; Liao & Hill Reference Liao and Hill2017; Holgate et al. Reference Holgate, Simons, Andrew and Stavrou2019). The particles in plasma rotate with high angular velocities due to various mechanisms. In a low-temperature plasma, the angular velocity can reach up to $10^5\ {\rm rad}\ {\rm s}^{-1}$
(Hutchinson Reference Hutchinson2004), while in a high-temperature plasma, it can reach up to $10^9\ {\rm rad}\ {\rm s}^{-1}$
due to Lorentz force (Krasheninnikov Reference Krasheninnikov2006).
In this article, we report a direct observation of the spontaneous fragmentation of dust particles in the Experimental Advanced Superconducting Tokamak (EAST) using a high-speed camera. Equation-of-motion simulations were performed and the results were consistent with the experimental observations. Finally, the criterion for the fragmentation is briefly discussed.
2. Experimental observations
The EAST is a fully superconducting tokamak device, which aims at implementing long-pulse high-confinement plasma operations (Keilhacker et al. Reference Keilhacker, Becker, Bernhardi, Eberhagen, ElShaer, FuBmann, Gehre, Gernhardt and Gierke1984; Li et al. Reference Li, Guo, Wan, Gong, Liang, Xu, Gan, Hu, Wang and Wang2013). The major and minor radii are $1.7$
and $0.45$
m, respectively. The upper divertor has an actively water-cooled tungsten–copper monoblock structure while the lower divertor is made of graphite. The first wall materials are of molybdenum (Wang et al. Reference Wang, Guo, Xu, Liu, Gan, Wang, Gong, Liang, Zou and Hu2014; Wan et al. Reference Wan, Li, Guo, Liang, Xu, Wang, Gong and Garofalo2015). In order to study various phenomena in the plasma, a high-speed CCD camera (Phantom V710) has been installed in $K$
port of the EAST with a fixed focal length and aperture (Jia et al. Reference Jia, Yang, Zhong, Zeng, Lu and Shu2015; Chen et al. Reference Chen, Zhong, Yang, Li, Liang, Zhang and Hu2019). The optical system views the tangential direction of the tokamak and the field of view is designed to be $-29^{\circ }$
to $29^{\circ }$
in vertical orientation and $-29^{\circ }$
to $18^{\circ }$
in horizontal orientation. The data acquisition speed and spatial resolution of the whole system are up to $50\,000$
frames per second (fps) and $6$
mm per pixel, respectively.
Multiple fragmentation events were recorded in the discharge by the high-speed camera. As a typical example observed in discharge shot 46618, a single dust particle travelled with a high velocity in the SOL and broke into two pieces. The two fragments obtained a transverse velocity and further travelled apart, as shown in figure 1. The dust particles appear as bright spots in the dark background in the video with a recording rate of $5000$
fps. The recorded two-dimensional projected motions were tracked by the simple particle identification and tracking (SPIT) algorithm.
Figure 1. A typical fragmentation event of a dust particle recorded in discharge shot 46618 in the EAST. The fragmentation occurred in the region marked by the dotted rectangle in (a). The recording rate is $5000$
fps. (b–f) The particle positions at five consecutive times and (g) the full trajectories of the particles before and after the fragmentation. The white crosses represent the results of the particle tracking by the SPIT algorithm.
The evolution of the plasma parameters in this discharge is shown in figure 2. The neutral gas pressure is about $3.5$
Pa. The toroidal field strength $B_{\rm t}$
is about $1.85$
T. The current increased in the first second and reached a steady state until the disruption at $t\sim 2.2$
s. The average electron density in the core increased relatively fast and reached a plateau of $1.3\times 10^{19}\ {\rm m}^{-3}$
for the rest of the discharge. The electron temperature was measured by electron cyclotron emission (ECE) and is about $3$
keV in the core. Note that the electron temperature and density in the plasma edge (SOL) are much lower than those in the core. Besides, the evolution of the emission of tungsten and molybdenum, corresponding to the divertor and wall materials in the EAST, are shown in figure 2(e, f).
Figure 2. The evolution of the plasma parameters including current (a), electron temperature (b) and density (d) in the core, gas pressure (c) and emission of molybdenum (e) and tungsten ( f) in discharge shot 46618.
More fragmentation events observed in different discharges in the EAST are shown in figure 3. The dust particles travelled in different directions and with various velocities. Some trajectories, such as shown in figure 3(a,d), are similar to that shown in figure 1, despite the different initial velocities of the particles and locations of the observations.Footnote 1 Others obtained larger transverse velocity than the longitudinal one that the daughter particles indeed accelerated after fragmentation, as shown in figure 3(b,e). In some events, one of the daughter particles further broke into two pieces, as shown in figure 3(c, f). Note that one trajectory of the particle after the fragmentation event (figure 3c-ii) was too short to follow and thus was neglected in the analysis. These motions of particles were associated with not only the size but also the local plasma conditions.
Figure 3. A few examples of fragmentation events observed in the EAST. The white plus symbols represent the positions of the dust particles tracked by the SPIT algorithm. Single fragmentation events are shown in (a,b,d,e). Multiple fragmentation events are shown in (c, f), highlighted by the yellow dotted rectangles.
The normalized mass loss $L=(m_0-m_1-m_2)/m_0$
during the fragmentation and the mass ratio $R=m_1/m_2$
of two daughter particles for the events shown in figures 1 and 3 can be estimated based on momentum conservation, where $m_0$
is the mass of the initial dust particle and $m_{1,2}$
are the mass of the daughter particles. The results are listed in table 1. The errors are caused by the uncertainties of the high-speed camera. It is assumed to be one-quarter of a pixel, which may be exaggerated. Except for the event shown in figure 3(d), the estimated mass ratio is close to unity, indicating that two daughter particles have similar size. Note that the negative mass loss may be a measurement error, caused by the low temporal resolution of the recording. (Actually, $L$
is expected to be positive due to jetting associated with droplet breakup and small fragments which may be invisible to the camera.)
3. Numerical simulations
In order to study the dynamics of the fragments after breakup, a simple equation-of-motion simulation was applied to fit the trajectories (Block & Melzer Reference Block and Melzer2019). Since the mass ratio of the daughter particles is close to unity, we assumed that they were spheres and had the same radius $a_1 = a_2$
. We further assumed that those particles were made of tungsten. Two dominant forces taken into account are the electric repulsion (modelled by the screened Coulomb or Yukawa interaction potential Lampe & Joyce (Reference Lampe and Joyce2015)) and the damping force due to friction against neutral gas background (Epstein drag Epstein (Reference Epstein1924); Baines et al. (Reference Baines, Williams, Asebiomo and Agacy1965)). The equation of motion of the daughter particles then reads
where $\boldsymbol{F}_{{\rm Yuk}} = (Q_1 Q_2 /r^2) (1+ r/\lambda_{{\rm D}})\exp (-r/\lambda_{{\rm D}})$
is the Yukawa repulsive force, $r$
is the interparticle distance, $\lambda_{{\rm D}}$
is the Debye length and $\boldsymbol{F}_{{\rm damp}}$
is the damping force.
As an example, here we simulated the trajectories of the fragmentation event in figure 1. The temperature and density are set as $T_{\rm e}=T_{\rm i}=200$
eV and $n_{\rm e}=n_{\rm i}=10^{13}\ {\rm cm}^{-3}$
, respectively, resulting in a Debye length $\lambda _{\rm D} \approx 24\ \mathrm {\mu }{\rm m}$
. Before the fragmentation, the dust particle was assumed to rotate with an angular velocity $\omega =10^7\ {\rm rad}\ {\rm s}^{-1}$
and the initial velocity was $150\ {\rm m}\ {\rm s}^{-1}$
. The charge of the particle was calculated with the orbital motion limited theory (Allen Reference Allen1992). The electron and ion collection and the thermal emission were taken into account. The charge of the particle $Q$
can be described as
where $I_{\rm e} = \sqrt {8{\rm \pi} } a^2 n_{\rm e} v_{\rm Te}\exp (-z)$
is the electron current, $I_{\rm i} = \sqrt {8{\rm \pi} } n_{\rm i} a^2 v_{\rm Ti} (1+z)$
is the ion current, $v_{\rm Te}=\sqrt {T_{\rm e}/m_{\rm e}}$
is the electron thermal velocity, $v_{\rm Ti}=\sqrt {T_{\rm i}/m_{\rm i}}$
is the ion thermal velocity, $I_{\rm th} = 4{\rm \pi} a^2 A_{\rm eff}T_{\rm s}^2 \exp (-W_{\rm f}/k_{\rm B} T_{\rm s})$
is the thermal emission, $A_{\rm eff} = 60\ {\rm A}\ {\rm cm}^{-2}\ {\rm K}^{-2}$
is the effective Richardson constant, $T_{\rm s}$
is the temperature of the dust surface and $W_{\rm f} = 4.55$
eV is the work function of tungsten (Delzanno, Lapenta & Rosenberg Reference Delzanno, Lapenta and Rosenberg2004; Komm et al. Reference Komm, Ratynskaia, Tolias and Podolnik2020; Tolias et al. Reference Tolias, Komm, Ratynskaia and Podolnik2020). The reduced particle charge $z$
can be calculated for the spherical particle as $z=|Q|e/aT_{\rm e}$
. Here it is assumed that the charge is negative. In the simulations, six surface temperatures above the tungsten melting temperature were selected and calculated. For high surface temperature, the dust charge is positive. The dust can recapture the thermionic electrons, resulting in a factor $(1-zT_{\rm e}/T_{\rm s})\exp {(zT_{\rm e}/T_{\rm s})}$
in the thermal emission currents (Vignitchouk, Tolias & Ratynskaia Reference Vignitchouk, Tolias and Ratynskaia2014; Autricque et al. Reference Autricque, Fedorczak, Khrapak, Couëdel, Klumov, Arnas, Ning, Layet and Grisolia2017). Aslo z-containing factors in the expressions for ion and electron currents should be interchanged. But since z is very close to zero in this case (figure 4d), this does not result in any appreciable effect. Here, slight magnetization of electron collection and thin sheath effects on ion collection were neglected in our simulations (Vignitchouk, Ratynskaia & Tolias Reference Vignitchouk, Ratynskaia and Tolias2017).
Figure 4. Equation-of-motion simulations of dust particles after fragmentation. (a–c) The electron and ion currents and the thermal emission, where colours represent different surface temperature $T_{\rm s}$
. (d) The reduced charges $z$
of the particles. (e) The interaction force and damping force exerted on the daughter particle in the initial stage, highlighted by the dotted rectangle in ( f) where the trajectories of relatively large particles in experiment (red plus) and simulations (blue and green lines) are shown. For different surface temperature of the dust particles, the trajectories are indistinguishable in the simulations.
The initial charge of each daughter particle was set as half the value of the particle charge before fragmentation. As soon as the fragmentation occurred, the two daughter particles acquired additional velocity due to the rotation and moved in opposite directions with an initial relative velocity ${\sim }36\ {\rm m}\ {\rm s}^{-1}$
with respect to the centre of mass.Footnote 2 The two daughter particles were recharged by the electron and ion current and the thermal emission in the plasma during the transport. The electron and ion current and the thermal emission also evolved as the particle was recharged, as shown in figure 4. By varying the angle $\alpha$
between particle velocity direction and fragmentation direction, we fit the particle trajectories in the simulations to the experimental observations. As a result of the fit, an angle $\alpha \sim 57^\circ$
was obtained and the radius of the dust particle before fragmentation was ${\sim }10\ \mathrm {\mu }{\rm m}$
. It turns out that the particle trajectories are indistinguishable for different surface temperature, as shown in figure 4( f). The acquired initial velocities of the daughter particles from the rotation play an essential role in their trajectories after fragmentation.
4. Discussion
The essential physics behind the instability of a charged liquid particle is relatively simple. The inward restoring force per unit area associated with the surface tension is $F_{\rm st}=2\gamma /a$
(Landau & Lifshitz Reference Landau and Lifshitz1980). If electrical charge is present on the surface of such a sphere, an outward force (‘negative pressure’) is exerted on the surface (Landau & Lifshitz Reference Landau and Lifshitz1984). The magnitude of this force is $F_{\rm el}=\frac {1}{2}\sigma E = Q^2/(8{\rm \pi} a^4)$
, where $\sigma =Q/4{\rm \pi} a^2$
is the charge surface density and $E=Q/a^2$
is the electric field. If $F_{\rm el}>F_{\rm st}$
, the outward electric force cannot be compensated by the inward surface tension force and the droplet becomes unstable. This results in the well-known Rayleigh instability criterion $Q^2>16{\rm \pi} a^3\gamma$
(Rayleigh Reference Rayleigh1882). If we take the rotational effect into account, an additional centrifugal force $F_{\rm ce} = a^2 \rho \omega ^2 /4$
points outwards, where $\rho$
is the mass density of the particle. Then the instability condition becomes $F_{\rm el} + F_{\rm ce} > F_{\rm st}$
. For the plasma parameters assumed in § 3, the instability condition is fulfilled and the fragmentation of the dust particle happens, which agrees with our experimental results. The underlying mechanism leading to the high angular velocities has not been identified yet and is the subject of much speculation (Ratynskaia, Vignitchouk & Tolias Reference Ratynskaia, Vignitchouk and Tolias2022b).
5. Conclusion
To summarize, we have described the spontaneous fragmentation of dust particles in magnetic confinement fusion in this article. Multiple fragmentation events were observed in the EAST and the trajectories were recorded using a high-speed camera. The size ratio of the daughter particles has been estimated based on momentum conservation. Equation-of-motion simulations have been performed to study the particle dynamics after fragmentation. The simulation results are consistent with the experimental observations. Our results show that the surface temperature of the dust particles plays a significant role in the charging and the particle interaction. However, the trajectories of the fragments mainly depend on their acquired initial velocities after fragmentation.
In future, we plan to conduct a statistical study of dust disintegration systematically and install an infrared camera to the EAST to measure the temperatures directly in the experiments.
Acknowledgements
The authors thank the referees for the valuable comments to improve the quality of this article.
Editor Edward Thomas, Jr. thanks the referees for their advice in evaluating this article.
Funding
This work was supported by the National Natural Science Foundation of China (grant numbers $12035003$
and $11975073$
); the National Key R&D Program of China (grant number $2017$
YFE$0301104$
); and Fundamental Research Funds for the Central Universities (grant number $2232023$
G-$10$
).
Declaration of interests
The authors report no conflict of interest.
Data availability statement
The data that support the findings of this study are available from the corresponding author upon reasonable request.
