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 1 2 where z is the distance from the sheath edge with the plasma-sheath boundary located at z = 0, φ and e are the electrostatic potential and electron charge, respectively, m_i,j is the ion mass, n_i,j is the ion number density, v_i,j is the ion fluid velocity, and p_i,j is the ion pressure of background ions and energetic ions with j = 1, 2 and is expressed as p_i,j = γ_jk_B T_i,jn_i,j with k_B being the Boltzmann constant, T_i,j the ion temperature, and γ the polytropic coefficient. Here, isothermal approximation γ = 1 is used.

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, 3 Here, subscripts e, j = e, 1 and e, j = e, 2 denote the thermal background electron and energetic electron, respectively, n_e,j is the electron number density, is the electron thermal velocity with electron temperature T_e,j and electron mass m_e, u_ce,j is cutoff velocity with and this cutoff velocity yields the electron normalization coefficient . The subscript w denotes the sheath-wall boundary located at z = w. Based on Eq. (3), the profile of the electron density in the sheath and electron flux to the wall can be written respectively as 4 5 On the assumption that the secondary emission at the material surface has a negligible speed, the energy balance and flux conservation equations in the sheath can be written as 6 7 Here, Γ_es = Γ_es,1 + Γ_es,2 is the flux of the secondary emission electron including Γ_es,1 caused by background electrons and Γ_es,2 caused by the energetic electrons. We assume that SEE due to ion impact is negligible because it is not important unless ion impact energies are greater than 1Kev while the electron induced SEE is significant even at rather modest energies greater than 30 eV.^[26] With SEE coefficient δ_e,j, Γ_es,j can be expressed as 8 Here, E_p is the kinetic energy of the incident electron. We use the Young–Dekker formula^[27] in which x_m = [1 − (1/k)][exp (x_m − 1)], δ_m is the maximum of δ, and E_m is the characteristic energy corresponding to δ_m. For the carbon, δ_m = 0.708 and E_m = 350 eV.^[28]

Then, a profile of secondary electron density is 9 The sheath system of equations is closed by the Poisson equation 10 where, ε₀ is the permittivity of the free space. The Poisson’s equation boundary conditions are given as follows: at the wall the floating potential φ_w is determined from the balance between electron flux and ion flux, i.e., 11 at the sheath edgez = 0, the electric potential is taken to be zero, where the electric field is assumed to be very small, i.e., 12 and the quasi-neutrallity condition is satisfied 13 For the multi-ion-component plasma collisionless sheath, it is suggested from both analytical and numerical results that each ion species should satisfy its own Bohm criterion at the sheath edge.^[13,29,30] When there are more than the population of negatively charged particles in the plasma sheath, by following Riemann,^[31] the Bohm criterion can be written as 14 where T_scr is the electron screening temperature, defined as T_scr = en_e(φ)/dn_e/dφ|_{φ = 0} in which n_e (φ) can be obtained from the electron density distributions Eqs. (4) and (8) including thermal background electrons, energetic electrons and secondary emission electrons. In the following discussion, we only consider v_i0 = C_s, i.e., Bohm criterion in a marginal form.

Next, we normalize Eqs. (1), (2), (4), (9), and (10) with the following set of dimensionless variables 15 where is the electron Debye length. The parameter is the velocity on a normalized reference scale. At the sheath edge, according to Eqs. (5) and (9), the density ratio of secondary emission electron to thermal electron is and the density ratio of ions to thermal electron satisfy α_T = [(1 + α_e) + α_e,s]/(1 + α_i) under the quasi-neutrality condition [Eq. (13)]. The φ_w will be obtained under the current balance condition [Eq. (11)]. By using the above normalization, the following set of coupled equations can be obtained: 16 17 18 19 20 To solve Eqs. (16)–(20), the fourth-order Runge–Kutta method is used under the appropriate boundary conditions at the plasma-sheath edge. Starting from ξ = 0, we proceed in space until the sheath potential ϕ = ϕ_w. In this process, the normalized sheath length thickness D is determined and the distance from the wall to any point in the sheath x = D − ξ can be obtained. In all the results reported in this paper, C is considered as the wall material and deuterium is considered to be the background gas by adopting n_e,1 (0) = 10¹⁷ m⁻³ and T_e,1 = 10 eV, which are in the range of parameters of the EAST tokamak near the low hybrid wave antenna. The energetic ion component is considered to be the deuterium ion due to NBI or the hydrogen ion due to ion cyclotron resonance heating (ICRH). The deuterium ion is used as default energetic ion and T_e,1 = T_i,1 unless otherwise stated.

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 α_i is greater than .^[13] Under the same parameters, due to the smaller mass and the resulting larger energetic ion flux, the potential drop across the sheath in the case of the hydrogen ion as an energetic ion component is smaller than in the case of the deuterium ion as the energetic ion component. When the energetic electron component is present in the sheath as shown in Fig. 1(a), the sheath potential can show a very large drop even with a small fraction.^[8–11] Once the sheath potential presents a sharp transition, the energetic electron begins to dominate the sheath, which indicates that the ion energy flux toward the wall becomes very large and then plasma-wall interaction may become serious. According to the current balance condition, the sheath potential varies as the temperature of background ion is changed. This is the case appearing in the fusion plasma, in which the background ion temperature may be higher than the background electron temperature during the NBI or ICRH, while it may be lower than the background electron temperature during the LHCD. Figure 1(b) shows the dependence of the sheath potential on the temperature of background ions and the concentration of energetic electrons.

Fig. 1.

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	Fig. 1. (color online) Normalized sheath potentials, respectively, as a function of (a) energetic electron concentration and energy and (b) energetic electron concentration and temperature of background ions with βe,2 = 50.

With the given values of α_i, α_e, β_i,j, β_e,j, and β_T, 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. (16)–(20). Figures 2 and 3 show the profiles of the sheath potential, background electron and ion density, and energetic particle density in the sheath. By comparing the effects of the hydrogen ion and deuterium ion serving as the energetic ion components on the sheath, we can find in Fig. 2 that the profiles of the sheath potential, and thermal background ion and electron densities almost have no difference among each other, except energetic ion density. Figure 3 shows that the sheath structure depends strongly on energetic electrons. Even a small fraction of energetic electrons can lead to a large difference in profile between the sheath potential and the plasma parameters.

Fig. 2.

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	Fig. 2. Profiles of the normalized sheath potentials, background electron and ion densities, and energetic particle densities in the sheaths as a function of distance from the plasma sheath to the wall.

Fig. 3.

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	Fig. 3. Profiles of the normalized sheath potentials, background electron and ion densities, and energetic particle densities in the sheaths as a function of distance from the plasma sheath to the wall for βe,2 = 50.

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. 4. For the high temperature of background ion, sheath potential is smaller while the plasma parameters are larger when the energetic ion or electron component is present.

Fig. 4.

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	Fig. 4. (color online) Profiles of the normalized sheath potentials, background electron and ion densities, and energetic particle densities in the sheaths, with (a) energetic ion component and (b) energetic electron component, as a function of distance from the plasma sheath to the wall.

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 21 where R_d is the dust particle radius, ϕ_d is the dust surface potential, and λ_screen is the screening length near the dust grain. Here, by following the method in Ref. [14], λ_screen = λ_lin is used when R_d > λ_lin, where . While R_d < λ_lin, λ_screen = λ_De is assumed. The dust surface potential is determined by a zero net current onto the dust particle, i.e., 22 where the dust charging currents of the thermal plasma, energetic particle and emission electron are calculated in the framework of the orbital motion limited (OML) theory. We assume that the dust temperature is low and thermal radiation causes the dust particles to sufficiently cool so that the thermionic electron emission and mass sublimation/evaporation may not be taken into account.

In Eq. (22), the ion collection currents are^[14] and 23 where u_j = v_i,j/v_thi and , .

The thermal and energetic electron collection currents are 24 The secondary emission current due to electron impact is given by^[14] 25 with and T_es being the temperature of the secondary emitted electron. Here, T_es = 1 eV is used in the calculation.

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 R_d << λ_De is satisfied.^[17,19] This implies that the dust particle radius R_d should be less than Debye length λ_De; 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 R_d/λ_De = 5. With the above sheath model parameters, we have λ_De ≈ 7.0 × 10⁻⁵ m. On the other hand, as mentioned in Section 1, the study in this paper is motivated due to the fact that the micron-sized charged grains can be utilized as a probe to study the sheath characteristics experimentally. In the present work, the maximum value of R_d is examined to be smaller than 20 μm, i.e., R_d < 20 μm and then R_d/λ_De < 0.3. Therefore, our default parameters satisfy the condition R_d/λ_De < 5.

Making use of Eqs. (16)–(20), we can obtain numerically the local electric field, the ion density and velocity, and the electron density in the sheath, and then can calculate the dust surface potential in the sheath by using Eq. (22). Figures 5(a) and 5(b) show the spatial profiles of the normalized dust surface potential in the sheath for different energetic ion concentrations and energies, respectively. The dust charging caused by the energetic ion is higher (but not much higher) than by no energetic ion in the sheath because energetic ions do not play an important part in the charging process of dust in the range of our calculation parameters. With the increase of energetic ion concentration or energy, the dust surface potential becomes higher due to the larger ion flux following into the dust. For the same calculation parameters as shown in Fig. 5(c), the dust charges are higher in the case of a hydrogen ion as the energetic ion component than in the case of a deuterium ion as the energetic ion component. As seen from Fig. 2, either the hydrogen ion or deuterium ion is the energetic ion component, the profiles of the sheath potential and plasma variables are almost the same in the sheath, except that hydrogen energetic ion density is larger. Therefore, in the case of hydrogen ion as the energetic ion component, the larger energetic ion flux causes the higher dust surface potential. When the ratio of the thermal ion temperature to the electron temperature is changed, it is shown in Fig. 5(d) that the dust charge increases monotonically with increasing the temperature of the thermal background ion because ion flux increases.

Fig. 5.

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	Fig. 5. (color online) Spatial profiles of the normalized dust surface potentials in the sheaths with (a) different energetic ion concentrations, (b) different energetic ion energies, (c) hydrogen ion and deuterium ion as energetic ion components, and (d) different ratios of the thermal ion temperature and electron temperature, respectively.

According to Fig. 3, the profiles of the sheath potential and plasma variables in the sheath depend strongly on the energetic electron, which indicate that the dust surface potential is a function of energetic electron. When an energetic electron component is present, we compare the spatial profiles of the normalized dust surface potential in the sheaths. As seen in Fig. 6, we can see a large negative dust surface potential near the wall in each of the cases where the energetic electrons have a large concentration or energy. This is due to the fact that at higher energetic electron concentration or energy, the sheath is dominated by the energetic electron, and then there are almost no thermal electrons near the wall. As a result, the zero net current flowing onto the dust surface requires the small energetic electron current. From Fig. 6, it is indicated that we can discuss the sheath properties once obtaining the information about the dust charging since the energetic electron with a small concentration or energy can change significantly the dust surface potential in the charging process of dust.

Fig. 6.

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	Fig. 6. (color online) Spatial profiles of the normalized dust surface potentials in the sheaths with (a) different energetic electron energies, (b) different energetic electron concentrations, and (c) different ratios of thermal ion temperature and electron temperature, respectively.

The effect of the background ion temperature on normalized dust surface potential is also shown in Fig. 6(c). We can see that the dust surface potential decreases with the increase of thermal ion temperature. The higher the thermal ion temperature, the lower the sheath potential will be since the thermal electron flux and energetic electron flux become larger. As a result, the repulsion of the less negatively charged wall allows the collection of less positive ions closer to the wall.

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: 26 The electric force in the sheath is 27 The gravitational force can be expressed as 28 where ρ_d and g are the mass density of dust particle and acceleration of gravity, respectively. For the carbon dust particle ρ_d = 2.25 × 10³ kg⋅m ⁻³.

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 F_id,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 F_id,orb). For a negatively charged grain, the drag collection is given by^[14] 29 and for a positively charged grain^[14] 30 For the orbital part of the ion drag force,^[14] 31 where is the Chandrasekhar function, and is the Coulomb logarithm, from which the scattering Λ is calculated, for a negatively charged grain with the impact parameter is the screening length, and For a positively charge grain, the Coulomb logarithm is given by Before discussing the levitating of dust particle inside the sheath, we consider where it happens that electric force balances ion drag force and gravity. Figure 7(a) shows the profiles of electric force, ion drag force and gravity on a dust particle with R_d = 1 μm. We can see that the electric force and ion drag force are the dominant force in the sheath. In a sheath of fusion plasma, the gravity is negligible unless the radius of dust grain R_d > 100 μm.^[14] The profile of total force which is the sum of electric force, ion drag force and gravity is also shown in Fig. 7(a). It is found that there are two balancing points corresponding to F_total = 0. Whether the balancing point for the dust is stable is determined by the trapping potential energy. The dust particle trapping potential energy relative to the sheath edge with the dust particle trapping is where z = D − x. From Fig. 7(b), we can see that the positions of F_total = 0 are the peak and valley of the dust particle trapping potential energy and the stable position is only the valley of the dust particle trapping potential. Therefore, based on the forces balance, we can obtain the levitating position of the dust particle inside the sheath by analyzing the trapping potential energy.

Fig. 7.

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	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 8 shows the plots of levitation positions of the dust particles versus radius when the energetic ion component is present. With the increase of energetic ion concentration or energy, the radius of each of the dust particles levitating in the sheath decreases, and its levitating position is much closer to the wall. This is due to the fact that the effect of energetic ion on the dust charging causes the dust surface potential to become less negative and the electric force on dust is always negative in the whole sheath. Moreover, F_e ∝ R_d and .

Fig. 8.

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	Fig. 8. (color online) Plots of levitation position versus radius of the carbon dust particle for (a) three different values of the energetic ion concentration, and (b) three different values of the energetic ion energy, respectively.

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. 9 it shows that the levitating position of the dust is much closer to the wall and the bigger dust particle can be trapped by the sheath in the case of deuterium ion as energetic ion. For the sheath of deuterium energetic ions, the dust particle feels the larger electric force since it has more negative dust surface potential (seen in Fig. 5(c)). Meanwhile the ion drag force is also larger due to the deuterium ion mass being larger than the hydrogen ion mass. However, with increasing the radius of the dust particle, the ion drag force ( ) increases faster than the electric force (F_e ∝ R_d). As a result, with increasing the radius of dust, the difference between levitating positions of dust in two cases decreases and the bigger dust particle can be trapped by the sheath in the case of a deuterium ion as the energetic ion than in the case of a hydrogen ion as an energetic ion.

Fig. 9.

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	Fig. 9. (color online) Plots of the levitation position versus radius of carbon dust particle inside the sheath for the carbon wall in two cases of hydrogen ion and deuterium ion as energetic ions with αi = 0.05 and βi,2 = 50.

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. 10 for the different concentrations and energies of the energetic electrons. We can see that the levitating position of dust particle depends strongly on the energetic electron concentration or energy. Once the energetic electron dominates the sheath, the drop of sheath potential becomes very large and the dust surface potential near the wall region becomes very negative as shown in Figs. 6(a) and 6(b). Therefore, the deep well of the trapping potential energy is formed due to the large electric field force. As a result, the large radius dust particle can levitate inside the sheath and variation of its levitating position is small for R_d < 20 μm. When the energetic electron concentration or energy increases, the levitating position is away from the wall due to the variation of electric force.

Fig. 10.

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	Fig. 10. (color online) Plots of levitation position versus balancing dust radius in the sheaths with (a) different energetic electron concentration and (b) different energetic electron energy, respectively.

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. 11(a). For the same size dust particles, with the increase of the thermal ion temperature, their levitating position moves toward the wall, and the radius of the dust particle levitating inside the sheath decreases. The reason is that the less negative dust charges cause the less negative electric force acting on the dust while the ion drag force increases with the increase of background thermal ion temperature. Figure 11(b) shows the plots of levitating position against dust radius in the sheath for energetic electron with different ratios of the thermal ion temperature and electron temperature. We can see that with the increase of thermal ion temperature, the levitating position moves toward the wall for the same size dust particles, and the larger size dust particle cannot stay in levitation equilibrium for the case of higher thermal ion temperature. The reason is that the ion drag force increases with the increase of background thermal ion temperature, while the electric force acting on the dust is less negative due to the smaller electric field as indicated in Fig. 4, although the dust surface potential is less negative as shown in Fig. 6(c).

Fig. 11.

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	Fig. 11. (color online) Plots of levitation position versus balancing dust radius in the sheath for three different ratios of the thermal ion temperature to electron temperature.