† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant No. 11475223), the National Magnetic Confinement Fusion Science Program of China (Grant No. 2015GB101003), and the JSPS-NRF-NSFC A3 Foresight Program in the field of Plasma Physics (Grant Nos. 11261140328 and 2012K2A2A6000443).
The structure of the sheath in the presence of energetic particles is investigated in the multi-fluid framework. Based on the orbital motion limited (OML) theory, the dust grain charging inside the sheath of plasma containing energetic particles is examined for the carbon wall, and then the effect of the energetic particles on the stationary dust particle inside the sheath is discussed through the trapping potential energy. It is found that with the increase of energetic ion concentration or energy, the size of dust staying in levitation equilibrium decreases and the levitating position is much closer to the wall. In the case of deuterium ions as energetic ions, the bigger dust particle can be trapped by the sheath than in the case of hydrogen ions as energetic ions. When the energetic electron component is present, the levitating position of dust particle in the sheath depends strongly on the energetic electron. The levitating dust particle is closer to the wall as the energetic electron energy or concentration is increased. In addition, with the increase of temperature of thermal background ion, the size of dust particle trapped by the sheath decreases and the levitating positions of dust particles with the same size radius inside the sheath move toward the wall. Our results can be helpful in investigating the property of the sheath where the energetic particle component is present.
The dust particle is an important and serious problem in a large variety of plasma applications including plasma processing and fusion research. These particles mainly result from plasma surface interaction process. The plasma-surface interaction may become strong when an energetic particle component is present. For instance, in tokamak discharge, during radiofrequency wave heating or neutral beam injection (NBI), the energetic particles appear in the device edge region.[1–5] It is found that hot spots would be a main source for the dust production during long pulse discharge with lower hybrid current drive (LHCD),[6] and dust rate increases with increasing the power of NBI.[7] As is well known, the presence of energetic particles can change the sheath structure.[8–13] A natural question is whether energetic particles can influence dust particles in the sheath. Combining with other diagnostic tools such as laser scattering and fast cameras, the investigation of the dust particles inside the sheath is a way to study the lifetime of the wall material because a lot of dust particles imply the strong plasma-wall interaction.[14]
On the other hand, the presence of a dust particle inside the sheath can provide a useful guide in measuring the sheath characteristics.[15–18] By using fine dust probes, sheath characteristics can be investigated in considerable detail, though indirectly, due to the fact that the charge on the dust particle depends on the local plasma condition. The dynamics of a dust particle in such a surrounding can provide information about the electron and ion fluxes, and also about the sheath potential and electric field. The present investigation is motivated by such a goal for the sheath of plasma containing energetic particles and thus, will be undertaken in a study of the dust behavior in the levitated equilibrium inside the sheath after ascertaining the comparative role of energetic particles in the sheath characteristics.
The dust particle in a plasma sheath has become a topic of many studies in fusion, astrophysical and laboratory plasmas. Dust particles in the sheath can be considered to be isolated if the dust components are sparse.[14,15,19,20] However, they can modify the structure of the plasma-sheath region if its density is relatively high.[17,18,21,22] In the case where the energetic electrons exist in the sheath, both an isolated grain and a dense cloud of dust particles in a sheath with plasma wall bias have been studied.[17,18,20] In a floating sheath, the wall potential is determined by the zero total current at the wall, and it depends strongly on the energetic electron and the secondary electron emission (SEE). Furthermore, it can demonstrate a sharp transition at a certain energetic electron flux.[8–11] In addition, most of the researches mentioned above have been carried out on the assumption that ion temperature is far lower than electron temperature so that the effects of the ion directional velocity on the dust charging and dynamics are omitted.[15,17–20] In some plasmas such as fusion plasma, ion temperature is usually comparable to or even higher than electron temperature and their directional velocity may be very large if the sheath of plasma contains energetic ions. As a result, the charging of the dust particle depends on the ion directional velocity.[23]
In order to investigate the dust particle in an electrostatic sheath of plasma containing energetic ions or energetic electrons, we focus on the effects of the energetic particles on the dust charging and levitation for carbon (C) wall. In addition, we also discuss the variations of the dust charging and levitation for the different temperatures of thermal background ion. The rest of this paper is organized as follows. In Section
To investigate the levitating dust particles in a steady-state sheath of plasma containing energetic particles, we present a 1D, multi-fluid unmagnetized electrostatic sheath model, whose constituents are thermal background electrons and single charge ions, the energetic particles including ions or electrons, and the emitted electrons from the wall. In the model, the impurity sputtering from the wall surface and collision are not taken into account. Moreover, we assume that the impurity density is much lower than that of background ions and thus the impurity does not disturb the plasma.[14,24,25] The dust particle is assumed to be isolated and then cannot modify the sheath. For the thermal background and energetic ions, the continuity and momentum equations are
The electrons from the bulk plasma region far away from the sheath are assumed to be Maxwellian. After they enter into the sheath, their 1D velocity distribution function approaching the wall can be described by a truncated Maxwellian distribution function,
Then, a profile of secondary electron density is
Next, we normalize Eqs. (
For a collisionless sheath, the floating potential can be obtained directly from the current balance condition. When the energetic ion component is present, an estimate of the energetic ions affecting significantly on the sheath indicates that their concentration αi is greater than
With the given values of αi, αe, βi,j, βe,j, and βT, after obtaining the sheath floating potential, we can calculate the sheath boundary conditions at the sheath edge including the background and energetic ion velocity and background ion density, and then obtain the profiles of sheath potential and plasma parameters inside the sheath based on Eqs. (
To investigate the effect of the background ion temperature on the sheath of plasma containing energetic ions or energetic electrons, we plot and compare the profiles of the sheath potential, background electron and ion density, and energetic particle density in the sheath in Fig.
The dust particle begins to be charged by the collection of plasma and electron emission, when it meets the plasma. Usually, for a micron-sized charged grain, the dust charging time is much smaller than the characteristic time of the dust motion,[19,23] and then the equilibrium dust charge is related to the dust surface potential
In Eq. (
The thermal and energetic electron collection currents are
Before presenting the numerical results of dust charging, it is worthwhile examining the applicability of the OML theory. The OML theory can be used to calculate dust charging currents if the condition Rd << λDe is satisfied.[17,19] This implies that the dust particle radius Rd should be less than Debye length λDe; otherwise, there is no space for the dust particle to move from infinity to the grain surface, which is used for calculating the cross sections. However, in recent work,[32] it has been shown that there is very good agreement between OML theory and simulations until at least Rd/λDe = 5. With the above sheath model parameters, we have λDe ≈ 7.0 × 10−5 m. On the other hand, as mentioned in Section
Making use of Eqs. (
According to Fig.
The effect of the background ion temperature on normalized dust surface potential is also shown in Fig.
The dynamics of dust in a plasma sheath has been investigated recently.[15,19,20] The dust can levitate in the sheath if it satisfies the force balance. Under this condition, the dust particle is stationary in the sheath and then the fine dust probe can provide the information about the sheath potential and sheath electric field, and about the plasma flux. Here, any moving dust particle is not considered since it is difficult to provide the sheath information.
To discuss the levitating of a dust particle in the sheath, we consider the total resultant force acting on the dust particle, which consists of gravitational force, electric force, and ion drag force, and is expressed as follows:
The ion drag force is normally made up of the contribution to the drag by the ions that are directly collected by the dust grain (denoted as Fid,coll) and the scattering part due to the Coulomb interaction between the dust grains and the ions (not collected by the grain) orbiting in the dust grain sheath (denoted as Fid,orb). For a negatively charged grain, the drag collection is given by[14]
![]() | Fig. 7. (color online) (a) Forces and (b) trapping potential energy profiles as a function of distance from the sheath to the wall. |
First, we study the effect of the energetic ions on the stationary dust particle inside the sheath. Figure
Next, we compare the levitating positions of dust particle inside the sheath in two cases of hydrogen ion and deuterium ion as energetic ions. From Fig.
Now, we investigate the effect of the energetic electron on the stationary dust particle inside the sheath. The levitating positions of the dust particles with different radii are shown in Fig.
Finally, we investigate the effects of the background thermal ion temperature on the levitating position of dust inside the sheath of plasma containing energetic particles. We show the plots of the levitation position of dust particle versus dust radius for energetic ion with different ratios of thermal ion temperature to electron temperature in Fig.
Using numerical simulations of the multi-fluid equations, we study the dynamics of a plasma sheath in the presence of energetic ions or energetic electrons. It is found that the sheath potential depends on both energetic particle concentration and energy. By comparing two cases of the hydrogen ion and deuterium ion as the energetic ion component, the results show that the profiles of the sheath potential and plasma variables including thermal background densities of ion and electron are almost the same in the sheath, except that hydrogen energetic ion density is larger. The pronounced effect of the energetic electron is present for the carbon wall even for a small fraction of energetic electron concentration or energy. In addition, the influence of the ratio of the thermal background ion temperature to electron temperature on the sheath structure is also examined.
Based on the OML theory, a dust grain charging in the sheath of plasma containing energetic particles is investigated. The dust surface potential decreases with increasing energetic ion concentration or energy, while it increases with increasing thermal background ion temperature. In the case of a hydrogen ion as the energetic ion component, the dust acquires less negative charges than in the case of a deuterium ion as the energetic ion component. When the energetic electron component is present, it changes the dust surface potential. A large negative dust surface potential near the wall is observed when the energetic electron concentration or energy exceeds a certain value. The reason is that there are almost no thermal electrons near the wall and the energetic electron from the plasma region plays an important part in the charging process of dust. Even if the energetic ion or electron component is present, the dust surface potential decreases with the increase of thermal background ion temperature.
By analyzing the trapping potential energy, we can obtain the stable point of dust whose total resultant force is zero in the sheath. When the energetic ion concentration or energy increases, the radius of dust particle trapped by the sheath decreases and the levitating position is much closer to the wall. The difference in the levitating position of the dust particle inside the sheath between the cases of hydrogen ion and deuterium ion as energetic ions decreases, and the bigger dust particle can be trapped by the sheath in the case of deuterium ion as energetic ions. By investigating the effect of the energetic electron on the stationary dust particle inside the sheath, it is found that the levitating positions of the dust particles with different sizes depend strongly on the energetic electron concentration and energy. Once the energetic electron dominates the sheath, the large radius dust particle can levitate inside the sheath and variation of their levitating position is small for Rd < 20 μm because of the deep well of the trapping potential energy near the wall is formed due to the large electric field force. With the increase of energetic electron concentration or energy, the levitating position of the dust particle is away from the wall. As the thermal background ion temperature increases, for the cases of sheath plasma containing energetic ions and energetic electrons, respectively, the levitating position of the dust particle inside the sheath moves toward the wall for the same size dust particles, and the larger size dust particle cannot stay in levitation equilibrium for the higher background ion temperature.
Our results presented in this work are of importance for plasma diagnostic when the energetic particle component is present in plasma since the plasma-wall interaction can be studied in considerable detail by using fine dust probes, which will provide, though indirectly, a useful guide in measuring the sheath characteristics because the charge on the grain is a function of the local plasma condition. Finally, note that our sheath model and calculation of dust charging rely on several assumptions for the sake of simplicity. For example, magnetic field is not taken into account. For a magnetized sheath, the sheath structure depends strongly on the magnitude and direction of the magnetic field,[33] and the dust particle charging process is modified by the magnetic field.[34] In future work, we will investigate dust levitating in the magnetized sheath when the energetic electron component is present.
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