Scholarly article on topic 'Simulation of dust grain charging under tokamak plasma conditions'

Simulation of dust grain charging under tokamak plasma conditions Academic research paper on "Materials engineering"

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Abstract of research paper on Materials engineering, author of scientific article — Zhuang Liu, Dezhen Wang, Gennady Miloshevsky

Abstract Dust grains in fusion devices may be radioactive, contain toxic substances, and may penetrate into the core plasma resulting in the termination of plasma discharges. Therefore, it is important to study the charging mechanisms of dust grains under tokamak's plasma conditions. In this paper, the charging processes of carbon dust grains in fusion plasmas are investigated using the developed dust simulation (DS) code. The Orbital Motion Limited (OML) theory, which is a common tool when solving dust-charging problems, is used to study the charging of dust grains due to the collection of plasma ions and electrons. The secondary electron emission (SEE) and thermionic electron emission (TEE) are also considered in the developed model. The surface temperature of dust grains (Td) is estimated for different plasma parameters. Floating potentials have been validated against the data available from the dust simulation code package DUSTT. It is shown that the dust grains are negatively charged for relatively low plasma temperatures below 10eV and plasma densities below 10 19 m − 3 . For higher plasma temperature and density, however, the charge on dust grains may become positive. The charging time depends not only on the grain's size, but also on the plasma temperature.

Academic research paper on topic "Simulation of dust grain charging under tokamak plasma conditions"

Nuclear Materials and Energy 00 0 (2017) 1-6

ELSEVIER

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Nuclear Materials and Energy

journal homepage: www.elsevier.com/locate/nme

Simulation of dust grain charging under tokamak plasma conditions

Zhuang Liua,b,*( Dezhen Wangb, Gennady Miloshevskya

a Center for Materials under Extreme Environment, School of Nuclear Engineering, Purdue University, West Lafayette, ¡N 47907, USA b Key Laboratory of Materials Modification by Laser, ¡on and Electron Beams (Ministry of Education), School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116024, People's Republic of China

A R T I C L E I N F 0

A B S T R A C T

Article history: Received 7 July 2016 Revised 15 November 2016 Accepted 26 November 2016 Available online xxx

Keywords: Dust charging Tokamak

Dust grains in fusion devices may be radioactive, contain toxic substances, and may penetrate into the core plasma resulting in the termination of plasma discharges. Therefore, it is important to study the charging mechanisms of dust grains under tokamak's plasma conditions. In this paper, the charging processes of carbon dust grains in fusion plasmas are investigated using the developed dust simulation (DS) code. The Orbital Motion Limited (OML) theory, which is a common tool when solving dust-charging problems, is used to study the charging of dust grains due to the collection of plasma ions and electrons. The secondary electron emission (SEE) and thermionic electron emission (TEE) are also considered in the developed model. The surface temperature of dust grains (T d ) is estimated for different plasma parameters. Floating potentials have been validated against the data available from the dust simulation code package DUSTT. It is shown that the dust grains are negatively charged for relatively low plasma temperatures below 10 eV and plasma densities below 10 1 9 m-3 . For higher plasma temperature and density, however, the charge on dust grains may become positive. The charging time depends not only on the grain's size, but also on the plasma temperature.

© 2016 Elsevier Ltd.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/40/).

1. Introduction

In fusion devices, the majority of the plasma escaped from the core and directed towards the divertor moves along the scrape-off layer, where plasma recombination occurs and most of the contamination is exhausted. Small portion of escaped hot plasma, however, fails to move along the magnetic lines and hits the chamber wall and the dome area [1,2]. The chamber wall or dome produces eroded material when being hit by the plasma. The eroded material may nucleate and form dust grains in relatively cold plasma regions [3]. Consequent flaking of the thin surface films, which are produced during the interaction between the plasma and chamber wall, as well as other plasma-wall interactions, are also sources of dust grains [4]. Some dust grains are radioactive and toxic, some contribute to the plasma contamination as a supplier of impurity, some could result in radiative losses, plasma instabilities and even the termination of plasma discharges when they penetrate into the core plasma [5,6].

The physical processes of charging of dust grain have been studied in various fields [7-11]. Tokamak operation and safety is-

* Corresponding author. E-mail address: lzplasma@mail.dlut.edu.cn (Z. Liu).

sue due to contamination by dust grains, intentionally injected (Li pellets) or produced during energetic plasma-wall interactions, is among the most critical issues to deal with in magnetic fusion devices. Since the dust grains could be charged, they can be located in various places in a tokamak. Most of the grain sizes are found on the order of 0.1 jam to 10 jam [12]. Some of the grains are spherical and some can be of irregular shape [13]. There are mainly four dust simulation code packages in magnetic fusion research, namely DUSTT [14], DTOKS [15], MIGRAINe [16] and DUSTTRACK [17] . All the four codes give a description of dust behavior and transport in fusion devices. The DS code, however, focuses on the dynamic charging processes as well as the influences of background plasma parameters and the dust grain property.

In this paper, to have direct and simple expressions for the currents, we consider spherical dust grains originating from the sputtering yields of the plasma facing material. The charging mechanisms of dust grains with different sizes in fusion plasma environments are studied. By means of our Dust Simulation (DS) code based on the experiments [12,18] of dust formation and basic plasma parameters in fusion devices, the Td evolution and charging processes of dust grains are studied. The developed code was also used to provide a theoretical basis for understanding dust contamination of the core plasma and optimization of the tokamak

http://dx.doi.org/10.1016/j.nme.2016.11.030

2352-1791/© 2016 Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Z. Liu et al. /Nuclear Materials and Energy 000 (2017) 1-6

vessel wall. Calculations with the DS code have been performed for various sizes of spherical dust grains with common edge plasma parameters. For most of the edge plasma conditions the results show that the charge on grains remains negative. Under relatively higher temperatures above 10 eV, however, the TEE becomes strong enough to turn the grain's charge to positive values.

The paper is organized as follows. In Section 2, the dust-charging model is presented. In Section 3, the charging processes of carbon and tungsten dust particles with various sizes are studied in detail and the conclusions are drawn in Section 4.

2. Physical model

21. Dust charging due to interactions of plasma electrons and ions

Dust charging is the result of dynamic processes of ions and electrons reaching, interacting, and leaving the surface of dust grains. The processes that contribute to dust grain charging are collection of electrons and ions from the plasma, secondary emission, thermionic emission, photoemission, and radioactivity [3]. Since the influences of the last two processes are relatively small and can be neglected [19], our DS code accounts for collection of charged particles from the plasma, secondary emission, and TEE. In our simulations, we assume that the dust grains are spherical and the plasma particle motion is gyro-centered. The dynamic charging process is described with the OML [20], which was originally used for calculating the charge on a probe immersed in the plasma [21]. For typical edge fusion plasma parameters, Te ranging from 1 eV to several tens eV and Ne ranging from 1018 m-3 to 1020 m-3 [19], the electron and ion collisions can be safely neglected in calculations of dust charging and heating [14]. For ions, the Larmor radius is much larger than the Debye screening length, which is also larger than the grain radius (from 1 to 10 jam), so that the OML approximation can be used [3]. For electrons, the Larmor radius is comparable with the Debye length. The effect of the magnetic field on dust grain charging will be considered in the future work. For Maxwellian distributed electrons and ions, characterized by temperature of electrons Te and ions T , , which are equal in thermal equilibrium, the OML currents for spherical grains can be written as [22]

Ii = Ioi]1 -

I e = Ioe exp

I i = Ioi exp

I e Io

/, Z]eçs \

V " Tk^TT ) ( J

V kfl Te T

(-Z] eÇ] J

V kBT] )

(1+S )

Ç T < 0 Ç T < 0 Ç T > 0 ç t > 0

where z, is the ion charge, Ioe and Ioi are the currents when the grain's surface potential <ps = 0, e is the elementary charge and kB is the Boltzmann constant. The terms Ioa with a = e , i can be written in the simplified form

I oa = 4n R Na qa

under tokamak plasma edge conditions, where Rd is the dust radius, Na is the number density, qa is the charge number, and ma is the mass of plasma species a, respectively. The dust grain surface potential çs is related to the dust charge Q as

Ç/ = Q/C

where C is the capacitance of a spherical dust grain which has the form C = 4ns0 Rd [23]. The charging process due to plasma ions

and electrons can be described as

dQ - T+1 dt = I]+/e

Substituting Eq. (3) into Eq. (1) and using Eqs. (2) and (4) we obtain

dQ „ n2/I I kTe dt = 4n Rd Ne qe S;

№ VkCTe T

§ = 4' RdN.q. )2 ( ' + iCT; )

+4nRdN,q, (2!^ (w) v' > 0 (5)

These equations describe the charge evolution due to collection of plasma electrons and ions for various plasma temperature and density conditions.

2.2. Secondary electron emission (SEE)

With sufficient energy of plasma electrons hitting the dust grains, electrons can be ejected from the dust grain. The SEE yield, the ratio of the emitted secondary electrons to the whole incoming electrons, is affected by the energy of incoming electrons E e and is described by the Sternglass formula [24]

+4n Rd N ] q ] ] 2nm

(S y "(' - S ) Ç] < 0

5sec = 2 / 722 Ee

-2(Ee;x)

\ Emax /

where 5max and Emax are material-dependent constants and 5sec is the SEE coefficient. For graphite, 5max = 1 0 and Emax = 300 eV [25]. Then we can integrate numerically 5sec over the energy distribution of electrons Ee = 1 me v2 using a Maxwellian distribution function /(Ee , Te ) to determine SEE as a function of temperature Te [26]. For positively charged grains, the SEE contribution is neglected, since it is assumed that the grain is sufficiently charged to absorb the secondary electron current. The SEE can also be obtained from Young-Dekker formula [27]. Its incorporation in the code and comparison to the Sternglass formula will be done in the future work.

2.3. Thermionic emission

2.3.1. Thermionic current

When Td is high enough, some electrons can escape resulting in thermionic emission. The thermionic current for negatively charged grains with the radius of R d can be obtained using the Richardson-Dushman equation [28]

j/h = -4n RdARTd exp (-^

(2) where AR = 1,20173 x 106 A m-2 K-2i s Richardson's constant, Wf is the material-dependent work function. For graphite, Wf = 4 , 8 eV [26]. For positively charged grains, just as SEE, this thermionic term is also neglected.

2.3.2. Surface temperature of dust grains Td

There are five main heating and cooling mechanisms contributing to the net energy on dust grains in a fusion plasma [26]. These are particle bombardment, electron emission, recombination processes, neutral particle emission, and radiative cooling. For

Z. Liu et al./Nuclear Materials and Energy 000 (2017) 1-6

negatively charged grains, the net energy can be expressed as [26]

2net = [ (2kBTi + Vd kBTe)(1 - Re) + 13 1 6e + 1i 1e - kBTd ] r

+(2kBTe - Wfe5ther - 3 1 0e5sec )re - aaTd (8)

and that for positively charged grains 3net = [ ( 2kBTi + Vd kBTe )(1 - IRe)

+ (13 16e + 11 1e - kB Td ) (1 - Rn ) r + 2kBTe re - aaT4

where RE is the backscattered fraction of incoming energy, ri and re are ion and electron flux, Vd is the normalized dust grain floating potential, Vd = (e0d )/(Kb Te ), where 0d is the dust potential, a is the emissivity, a material-dependent coefficient (a = 1 for a black body), and a is Stefan's constant. The variables 5ther and 5sec are TEE and SEE yields, respectively. RN represents the fraction of backscattered ions from dust particles. Then solving the equation

md c (Td ) ^ = 4n Rd 2 net

where md = 3n Rjj p is the mass of dust particle, p is the grain density and c(Td ) is the specific heat, we can then obtain the evolution of Td .

2.4. Numerical algorithm and DS code

In the above sections, we have derived the contribution to the current from plasma ions and electrons, SEE and TEE. Then we can write the charging equation as

= ^ = l] + Ie + 1SEE + If EE

To obtain as accurate results as possible during numerical simulation, the fourth-order Runge-Kutta method is implemented in the DS code for solving this charging equation.

3. Results and discussion

The charging processes of spherical carbon dust grains with radii from 10 nm to 10 jam are simulated for conditions when the grains are placed within fusion plasmas with Ne from 1016 m-3 up to 10 20 m-3 and Te from 1 eV up to 20 eV The results are as described below.

The contour plot of equilibrium Td of carbon grains in fusion plasma with typical edge parameters is shown in Fig. 1. Td depends on both plasma density and plasma temperature. It remains below 10 00 K at low plasma densities less than 1017 m-3 . In this regime, there is only a weak dependence of Td on the plasma temperature. Td increases slightly with the increase of plasma temperature. For higher plasma densities above 1019 m-3 , however, the Td can reach the sublimation point which is 3915 K for carbon. It can be seen in Fig. 1 that for dense plasmas with Ne > 1019 m-3 , Td is considerably higher in plasmas with higher temperature. It is obvious that in a plasma with higher temperature, more energetic plasma ions and electrons will heat the dust particle resulting in higher Td . According to Eqs. (8) and (9), the higher flux of plasma particles impinging the dust grain will contribute to higher energy and Td in all five of heating and cooling mechanisms, but radiative cooling, which is not influenced by the plasma density. Thus, high plasma densities result in higher Td .

The evolution of charge as a function of time on a dust grain with a radius of Rd = 1 jam placed in a plasma with N e from 1016 m-3 up to 1020 m-3 and Te = 1 eV, 10 eV and 20 eV is shown in Fig. 2(a)-2(f). In Fig. 2(a) and 2(b), we can see a great impact

Fig. 1. Isolines of Tj of carbon grains at thermal equilibrium within a range of plasma density and temperature.

of the change of plasma density on the charging time. At Te = 1 eV, the steady charge on the dust grain is around ~50 00 electronic charges for Ne ranging from 1016 m-3 up to 1019 m-3 , while for Ne = 1020 m-3 the dust charge is around ~3500 electronic charges. At this high plasma density, Td reaches 3041 K. At this temperature, the thermal current due to TEE from a grain becomes one of the dominant currents, unlike those at the smaller plasma densities. The charges on grains are all negative for the five plasma densities. The charging times needed to reach the steady-state are 13.1 jas, 1.31 jas, 0.13 jas, 13.10 ns, and 1.31 ns for Ne ranging from 10 16 m-3 (11) up to 1020 m-3 . They are decreased accordingly by an order of magnitude for each order of magnitude in density increase. The reason for this order of magnitude decrease is that the charging time is inversely proportional to the plasma density. The number of electronic charges as a function of time for the plasma temperature increased to 10 eV is shown in Fig. 2(c) and 2(d). Compared to Fig. 2(a), both the charging time and the steady-state electronic

charge have increased for plasma density of 1016 m-3 , 101 and 10 18 m-3 (Fig. 2(c)). The charging time is 0.49 ms, 4.91 jas and 0.49 jas, respectively. The steady-state number of electronic charges is increased to ~40,00 0 (Fig. 2(c) that is about 8 times greater than that in Fig. 2(a)). For Ne =1019 m-3 and 1020 m-3 in Fig. 2(d), the steady state charge number is lower compared to that at lower densities in Fig. 2(c). The number of charges on a grain shown in Fig. 2 is expressed in the absolute values. For the case of Ne = 1019 m-3 and 1020 m-3 (Fig. 2(d)), the steady-state charge on a grain becomes positive. The number of ionic charges reaches ~22,0 00 and ~50 00, respectively. The charging time is decreased to ~12 ns and 0.02 ns, respectively. For Te = 20 eV, the number of charges on a grain is shown in Fig. 2(e) and 2(f). Both the charging time and charge numbers in Fig. 2(e) have increased about ~50% for each density compared to Fig. 2(c). In Fig. 2(f), the steady-state charge becomes positive for Ne =1019 m-3 and 1020 m-3 and the charge number remains almost the same with the charging time decreased compared to Fig. 2(d). The potential of a dust grain can be found in Fig. 4, which clearly shows that the grain is positively charged for Ne = 1019 m-3 and 1020 m-3 .

Another factor affecting the dust charging is the size of the dust grain. The charging of dust grains with radius of 10 jam, 1 jam, 100 nm, and 10 nm was simulated in a plasma with the density of 1020 m-3 and the electron temperature of 1 eV The results are shown in Fig. 3.

It can be seen in Fig. 3 that the time needed to reach the steady-state charge on a grain is about 0.12 ns, 1.31 ns, 6.97 ns,

4 Z. Liu et al. /Nuclear Materials and Energy 000 (2017) 1-6

Fig. 2. Number of electronic charges as a function of time on carbon dust grains placed in a plasma with Te = 1, 10, 20 eV and Ne ranging from 10 16 m-3 up to 102 0 m-3 .

Fig. 3. Evolution of the number of electronic charges on carbon dust grains in a plasma with Te = 1 eV and Ne = 102 0 m-3 for R. ranging from 10-8 m up to 10-5 m.

and 12.05 ns for the grain's sizes mentioned above, respectively. The smaller grains require larger charging time, since the charging time is inversely proportional to the grain's radius. The steady-state charge numbers on a grain are around 35,0 0 0, 350 0, 350, and 35 electronic charges, respectively.

Fig. 4 shows the evolution of the potential of a carbon dust grain with 1 jam radius placed in the plasma with Te = 20 eV and densities ranging from 1016 m-3 up to 1020 m-3 . For plasma densities of 1016 m-3 , 1017 m-3 , and 1018 m-3 , the potential is negative decreasing to -2.32 relative units at 53.9 jas, 6.7 jas and 0.81 jas, respectively. For plasma densities of 1019 m-3 and 1020 m-3 , the potential becomes positive increasing to values of 0.87 and 0.27 of

Fig. 4. Evolution of surface potential on a dust grain with radius of R, = 1 jam placed in a plasma with T, = 20 eV and N, ranging from 10 m -3 up to 102 0 m -3 . The potential is shown in the units normalized to the electron plasma temperature.

relative units at 0.80 ns and 0.36 ns, respectively. These results on the steady-state floating potential were compared with those from the dust code DUSTT demonstrating a good agreement. DUSTT results show that for Ne from 1016 m-3 to 1018 m-3 the normalized potential is around -2.3. And for Ne = 1019 m-3 and 1020 m-3 , DS results are also very close to DUSTT data [19]. This can be used as validation of the DS code.

In Fig. 5, the comparison of charge evolution of 1 jam radius carbon and tungsten dust grains is presented. It can be seen for Ne = 1018 m-3 and 1019 m-3 , the charge evolution is almost the

Z. Liu et al./Nuclear Materials and Energy 000 (2017) 1-6

1E-4 1E-3 0.01 0.1 1 10 100

Fig. 5. Number of electronic charges as a function of time on Ri = 1 jjm carbon and tungsten dust grains placed in a plasma with Te = 1 eV and Ne ranging from 10 i 8 m-3 up to 10i 0 m-3 .

same for carbon and tungsten. When Ne reaches 1020 m-3 , however, the charge on tungsten dust increases faster and has a higher balance charge than that on carbon dust.

The OML theory has not taken into account the effects of magnetic fields. When a dust particle is placed in the plasma with a magnetic field, the electrons move along the magnetic field lines with the Larmor radius smaller than the dust radius. This will cause a 50% decrease in the electrons collected by the dust particles. In this case, the charge evolution is shown in Fig. 6. The balance charge will decrease by about 50%. This assumption needs to be validated against Particle-In-Cell simulations, which is already on the way using the Vsim particle simulation software.

4. Conclusion

ence between our dust code and some other existing dust transport codes is that we solve time dependent differential equations for the dust charge not imposing ambipolarity conditions.

It is found that in tokamak edge plasma environment, it takes from nanoseconds to microseconds for dust grains to become fully charged. The charging time is inversely proportional to dust grain radius and plasma density. For the most of cases, higher plasma temperature may also decrease the charging time. For low plasma temperature and density, it is observed that the grains are negatively charged. For higher plasma temperature and density, the charge on grains becomes positive due to fierce TEE. The number of charges on the dust grain can reach the order of 10 to 10,0 00. For lower densities, the charging evolution is almost the same for carbon and tungsten dust. For higher densities, however, the balance charge is higher for tungsten dust.

It is observed that the charging time for some dust sizes and plasma parameters can be smaller than 1 ns. In some dust transport simulation codes, the time step from 1 ns to 10 ns is used. Care should be taken to make sure that the time step is smaller than the charging time.

More work, however, needs to be done for the developed DS charging model. The SEE and TEE currents need to be adjusted when the dust grain becomes positively charged. In addition, the shapes of some dust grains found in fusion devices are not strictly spherical. The charging of these non-spherical dust grains should be studied. Besides, the OML theory has some limitations: (a) the effect of magnetic field is not taken into account. It's estimated that for common plasma parameters in edge tokamak environment, the balance charge is about half of what the OML theory suggests. (b) The OML theory assumes the absence of potential barriers in the vicinity of the dust grain [29]. Such barriers can appear when dust grains are positively charged. If so, the dust charge is changed affecting the incoming heat power. Also, the effect of vapor shielding is not included in this model [30]. A more accurate study of the charging process is already under way using Vsim simulation software, which is a popular particle in cell simulation tool. In future research, the results of this charging model can be used for integrated computational studies of dust grain transport and overall effect in plasma performance.

Dust grains in fusion devices may not only influence tokamak operation reliability, but also may cause safety issues. The physical mechanisms of dust grain charging under relevant tokamak plasma conditions are studied. The developed Dust Simulation code based on the OML theory is used for simulating the dust charging processes and predicting the grain temperature, charging times, number of charges on grains, and floating potentials. The main differ-

Acknowledgments

This work is supported by the U.S. Department of Energy, Office of Fusion Energy Sciences and the National Basic Research Program of China under Nos. 2013GB109001, NSFC Nos. 11275042.

Fig. 6. Charge evolution of 1 jjm radius carbon dust in a plasma with Ti = 1 eV and the effect of the magnetic field.

Ni ranging from 10 i 6 m-3 up to 10i 1 m-3 . Symbol B in legends means the cases with

Z. Liu et al. /Nuclear Materials and Energy 000 (2017) 1-6

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