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Influence of the magnetised sheath on the redeposition location of sputtered tungsten and its effect on the net erosion
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Nuclear Materials and Energy 0 0 0 (2017) 1–6
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Influence of the magnetised sheath on the redeposition location of sputtered tungsten and its effect on the net erosion N. Mellet a,∗, J.P. Gunn b, B. Pégourié b, A. Hakola c, M. Airila c, Y. Marandet a, P. Roubin a a
Aix-Marseille Université, CNRS, PIIM, UMR 7345, F-13397 Marseille, France CEA, IRFM, F-13108 Saint-Paul-lez-Durance, France c VTT Technical Research Centre of Finland Ltd., Espoo, Finland b
a r t i c l e
i n f o
Article history: Received 15 July 2016 Revised 16 December 2016 Accepted 9 March 2017 Available online xxx Keywords: Redeposition Magnetised sheath Net erosion Impurity migration Tungsten
a b s t r a c t The effect of the magnetised sheath on tungsten migration and its consequences on net erosion of tungsten plasma facing components are investigated. This study points out a strong effect of the E × B drift on the redeposition location as well as a rapid redeposition of the ejected particles that are ionised close to the surface. A plasma parameter study shows that the migration direction is changed at ≈ 10 eV from opposite to E × B drift at low electron temperature to the same direction as E × B at high electron temperature. Different models have been considered in the case of plasma parameter gradients that are also at the source of a component of the electric field parallel to the surface. New features are observed like an enhanced net erosion peak at the strike point if only the component of the electric field towards the surface is taken into account. When the component parallel to the surface that drives the particles away from or towards the surface is considered, a very different net erosion picture is obtained: a net erosion peak exists between two net deposition regions. This points out the importance of incorporating this effect in simulations. © 2017 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/)
1. Introduction The erosion of Plasma Facing Components (PFCs) is a subject of importance: it determines their lifetime and the generation of impurities that have potential negative effects on the plasma operation. The net erosion of PFCs, which is observable after plasma exposure, is determined by the balance between gross erosion and redeposition, and depends strongly on the distance travelled by redeposited particles. The migration of impurities can be investigated locally for example with ERO [1], REDEP/WBC [2] or, on a larger scale, with WALLDYN [3]. Here we focus on the region close to the surface that corresponds to the magnetised sheath and where a strong electric field exists. This electric field has been shown to increase strongly the number of sputtered particles that return to the wall [4,5]. A strong effect on the redeposition location is thus also expected, especially in tungsten (W) divertors where the atoms are ionised close to the surface. The subject has been investigated experimentally in ASDEX-Upgrade [6] and DIII-D [7] by the use of special sample markers inserted as a PFC. In the first
∗
Corresponding author. E-mail address: [email protected] (N. Mellet).
experiment [6], measurements by Rutherford Backscattering Spectroscopy (RBS) have shown the presence of a net erosion peak close to the strike point surrounded by two net redeposition regions. In the second experiment [7], very small markers have been studied, where the erosion and deposition components can be separated and for which modelling using ERO for different magnetised sheath electric field models has been carried out [8]. This work concentrates on investigating the basic mechanisms due to the magnetised sheath and to the resulting changes of the redeposition location. The latter is composed of two characteristic lengths: the Debye length λD that is linked to the Debye sheath that exists even without magnetic field and a length of the same order of magnitude as the main ion Larmor radius rL that is linked to the magnetic pre-sheath that is due to the grazing incidence of the magnetic field. At low electron density (when rL λD ), the magnetised sheath width scales with the Debye length, while at high electron density (when rL λD ), the magnetised sheath width scales according to the main ion Larmor radius. This effect is accounted in the present work through the use of a Particle-In-Cell (PIC) code [9] that determines the sheath electric field in a selfconsistent manner and makes no assumption on the sheath width. The use of such a method in comparison to analytic formulas has
http://dx.doi.org/10.1016/j.nme.2017.03.009 2352-1791/© 2017 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Please cite this article as: N. Mellet et al., Influence of the magnetised sheath on the redeposition location of sputtered tungsten and its effect on the net erosion, Nuclear Materials and Energy (2017), http://dx.doi.org/10.1016/j.nme.2017.03.009
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Fig. 1. Schematic view of the geometry and the simulations steps: 1) Sheath electric field computation with D+ and electrons. 2) Impact angle of Ne4+ . 3) Redeposition simulation of W ejected from the wall.
been shown to be of importance in [10]. Once it is determined, the sheath electric potential and electron density profile can be used in both 1D and 2D situations, which permits to study different model grades for the particle local migration. The paper is organized as follows. In Section 2 the different models used in this paper are presented. In Section 3, the results are displayed. We first focus on the simple case where quantities only vary in the direction perpendicular to the surface before considering the complete case with plasma parameter gradients. Finally conclusions are drawn in Section 4. 2. Models Two models are presented in this paper. The first one considers that plasma parameters only vary in the direction perpendicular to the surface (1D model). The second one incorporates effects due to the variation of parameters in the direction perpendicular to the magnetic field and parallel to the surface (2D model). In all cases a flat surface is considered. 2.1. 1D Model The simulations are performed in three steps as described in [10] and schematically shown in Fig. 1 with the corresponding geometry. The first step is to determine the sheath potential as well as the temperature and electron density profiles with a PIC code [9] that uses Poisson’s equation and the equation of motion of the electrons and the main ion species, which we will consider to be D+ . The injection of particles is made at the magnetised sheath entrance with a Maxwellian distribution in the direction perpendicular to the magnetic field and with the parallel distribution calculated with a Vlasov code that solves the model described in Chung and Hutchinson [11]. The sheath potential depends on three parameters: the ion to electron temperature ratio τ , the angle of the magnetic field with respect to the surface α B
and the ratio between the main ion Larmor radius at the sound speed for Ti = 0 and the Debye length at the sheath entrance, √ ζ = rL /λD ∝ ne /B, which scales as the square root of the electron density. We focus here on the case τ =1 (Ti = Te ), αB = 3◦ and B = 4.2 T so that a direct relation between ζ and ne exists. Additionally the sheath potential drop is prescribed as φ0 = 2.84Te (eV) [12], the electric potential scaling linearly with the electron temperature Te . The electric sheath potential, temperature and density profiles are thus only calculated for a set of values of ζ , (ζ = (0.1, 0.2, 0.3, 0.5, 1, 2, 3, 5, 7, 10, 15, 20, 25, 30, 40, 50, 100, 150)). The second step is the injection of Ne4+ at the magnetised sheath entrance, which will be the impinging impurity considered in this work. This choice is motivated by the fact that neon can be used as an inert seeding gas to reduce the heat flux on the PFCs. In present machines, lighter impurities like N are used leading to a smaller erosion than in the present case. The injection is made with the same procedure as in Mellet et al. [10] using the technique described in M. Kocˇ an and Gunn [13] for the parallel velocity distribution. Additional we considered TNe = TD . The test particle simulations in the electric field provided by the PIC code give the impact angle and energy of Ne4+ on the surface, which is used to determine the gross erosion. The concentration of Ne4+ (1%), as well as that of the ionised W, is considered to be sufficiently small not to affect the sheath electric potential making the test particle approach valid. The third step is the injection of W atoms from the wall with a cosine angular and a Thompson energy distribution that includes a square root cut-off determined by parameters linked to the impinging particle (Ne4+ ). W atoms may be ionised by electrons according to the ionisation rate provided in ADAS [14] and affected by the electric field. The possibility of multiple ionisation is taken into account up to the charge state 8+. The particles are then followed so that the distance they travelled can be determined at the moment they hit the surface. An important distinction between two kinds of trajectories is made [5]: prompt and long-range redeposition. In the first case the particles are redeposited during their first Larmor gyration. In the second case the particles undergo several Larmor gyrations and can travel a longer way even if they stay close or in the sheath. 2.2. 2D Extension The model presented in the previous subsection can be extended by considering the variation of the plasma parameters also in the x-direction (see Fig. 1). Consider a case where the electron temperature is linearly decrasing from the strike line along the xdirection (see Fig. 5(a)), which leads to a variation of the sheath potential in the x-direction and thus to an additional component of the electric field Ex that is displayed in Fig. 2(c). The first observation is that Ez is much larger close to the surface than Ex . Ex extends however further in the sheath than Ez , their amplitude
Fig. 2. a) Sheath electric potential calculated from the high Te case (Fig. 5), b) z-component and c) x-component of the electric field.
Please cite this article as: N. Mellet et al., Influence of the magnetised sheath on the redeposition location of sputtered tungsten and its effect on the net erosion, Nuclear Materials and Energy (2017), http://dx.doi.org/10.1016/j.nme.2017.03.009
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Fig. 3. Redeposition location distribution of W ejected at (0,0) without (left) and with (right) sheath electric field. Top: view from above. Bottom: lateral distributions.
being comparable from ∼ 1 mm above the surface. The electron density is sufficiently largeso that rL λD making the sheath width proportional to rL ∝ Ti = Te . This dependence is visible in Fig. 2(b). Concerning Ex , it is oriented opposite to the x axis for x < 0 and in the direction of the x axis for x > 0. The simulation region is a box of infinite dimension in the x and y direction (s(x, z ) = s(−0.04, z ) for x < −0.04 and s(x, z ) = s(0.04, z ) for x > 0.04 with s corresponding to any background quantity) and twice as the largest sheath width in the z direction. Particles are ejected along the x-axis where the number of particles is proportional to the corresponding gross erosion rate along x. 3. Results 3.1. 1D sheath Simulations of the redeposition location of tungsten ejected at a given point are performed first in the frame of the 1D sheath model (i.e. where the quantities vary only in the z-direction). Altogether 4 × 105 particles have been ejected at the (0,0) point in the magnetised sheath potential and electron density calculated for Te = 30 eV and ne = 1.87 × 1019 m−3 (ζ = 20) given at the sheath entrance. The magnetic field is oriented towards negative y and with an angle with respect to the surface of αB = 3◦ . Fig. 3 displays the comparison between the redeposition location obtained with and without the magnetised sheath so that its effect can be observed. In the case without sheath, flat density and temperature profiles have been considered. In both cases, the prompt and longrange contributions can be easily distinguished. When no sheath electric field is considered, prompt redeposition is centered at a distance from the ejection point of the same order as W+ Larmor radius and corresponds to the expected scenario of the ion being
3
redeposited during the first Larmor gyration. The long-range deposition exhibits no lateral displacement, the ionised particles following the magnetic field before being redeposited. When the sheath is added, the picture is changed. The prompt redeposition distribution is this time located around the ejection point. Additionally it represents now 90% of the total redeposition (50% without sheath). This shows the strong effect of the electric field that tends to bring back the W ions towards the surface, even at a distance from the surface larger than the Larmor radius. This can be understood by the fact the sheath potential is ∼ 87 eV in the present case, which is very large compared to the average ejection energy (∼ 15 eV, of the order of W surface binding energy Us = 8.68 eV). Concerning the long-range contribution, the redeposition location distribution is shifted of 8.3 mm in the direction of the E × B drift, pointing out its influence. A ne and Te (given at the sheath entrance) parameter study is performed next in order to see the influence of electron density and temperature on the redeposition location. Except for the scanned quantity, the same parameters as those used previously are employed. Fig. 4 shows the average lateral displacement for all the redeposited tungsten and for the two contributions taken separately as a function of ne and Te . The prompt redeposited ratio (# of prompt redeposited W ions/# of redeposited W ions) is also plotted. Finally the average z coordinate and electric field at the first ionisation for prompt redeposited W are displayed. This has not been done for long-range redeposition as it directly depends on the size of the simulation box. When ne is increased, the average lateral migration is generally reduced for both contributions (Fig. 4(a)). In the case of prompt redeposition, this can be explained as follows. Once ionised the particle would have a Larmor gyration if no electric field was present. The stronger the electric field the faster the particle is brought back to the surface and the less it can travel in the x direction. The electric field when the particle is first ionised increases with ne (Fig. 4(g)) partially because the ionisation gets closer to the surface (Fig. 4(e)) and explains why the migration is reduced. The bump in Fig. 4(e)) is due to the modification of the electron density profile with ne . Concerning long-range deposition, the mechanism is more complex and related to the sheath electric field profile dependence on ne . The prompt redeposition ratio strongly increases with electron density (Fig. 4(c)) resulting in a zero total average lateral displacement at large ne . When Te is increased, the average prompt redeposition distribution is shifted towards the ejection location (Fig. 4(b)). This can be attributed to the larger electric field that accelerates the ejected W strongly towards the surface as φ0 = 2.84Te is increased (Fig. 4(h)). It is also due to the average W Larmor radius that is only slightly increased with the Thompson distribution even if the impact energy of Ne4+ is proportional to Te (Fig. 4(f)) . About the long-range redeposition, the E × B drift increases as Te and the average redeposition location is the most strongly shifted at the largest electron temperature. The effect on the prompt redeposition fraction (Fig. 4(d)) is moderate making however the total average displacement pass from positive to negative x. Finally, in Fig. 4(eand f), it can be seen that the ionisation of prompt redeposited W can happen farther than their Larmor radius showing that the sheath can strongly modify their trajectory. 3.2. Effect of gradients on net erosion Four different models are compared. The first two models are based on the 1D treatment of the sheath region. The simplest is to consider a non-redeposition fraction fnon−redep = # of escaping W/# of ejected W. The local net erosion is then calculated as follows:
vnet = fnon−redep vgross
(1)
Please cite this article as: N. Mellet et al., Influence of the magnetised sheath on the redeposition location of sputtered tungsten and its effect on the net erosion, Nuclear Materials and Energy (2017), http://dx.doi.org/10.1016/j.nme.2017.03.009
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Fig. 4. Parameter study of the redeposition average location (a,b). The prompt redeposition ratio is plotted in (c,d). The first ionisation average distance with respect to the surface in (e,f) and the average electric field at ionisation in (g,h) are shown only for prompt redeposited W. ne is varied at Te = 30 eV (a,c,e,g). Te is varied at ne = 1.87 × 1019 m−3 (b,d,f,h).
The disadvantage of the method is that it does not incorporate W migration. It can thus be used as a reference to quantify the effect of the redeposition location. Note also that no net deposition is possible. The second model is an extension of the 1D model whose results are displayed in Section 3.1. The individual lateral displacement distributions are interpolated to fit the parameters at each location and summed to obtain a net erosion profile. The shortcoming is that the particles that are ejected at a given point will feel the electric field and the density calculated at this point even if they travel laterally where different plasma parameters exist. Besides, Ex is not retained. The last two models are based on the 2D treatment of the sheath region. Ejected particles will thus feel the full 2D variation of the electron density and the electric field. The difference between them is that one considers only Ez while the most sophisticated considers also Ex . For the 2D based models, 4 × 106 ejected W atoms have been used. In order to investigate the effect of temperature with the same particle flux on the surface, two cases have been defined (Fig. 5). They roughly represent the profiles around the divertor strike point but with a linear decay of the parameters (electron temperature and density at the sheath entrance) in order to keep the situation as simple as possible. The corresponding gross erosion obtained with the energy and angle determined by the test particle simulations of Ne4+ and the physical sputtering yields of [15] are displayed in Fig. 5(c). Both curves are surprisingly close to each other. The differences observed in net erosion between both cases will thus be attributed almost exclusively to the redeposition mechanisms. The results of all simulations are displayed in Fig. 6(a) for the high Te case and in Fig.e 6(b) for the low Te case. A general observation is that net erosion is larger at high Te . Even if the ionisation cross section and the sheath width are larger at twice larger Te , the dominant effect is the reduction of the mean free path before ionisation in a plasma that is 4 times denser. If we consider the nonredeposition fraction model, we see without any surprise that it is symmetric as the parameters are also symmetric. The extended 1D model breaks this symmetry and several new features can be observed. The first is a strong reduction of net erosion for x < 0. If we refer to Fig. 4(b), we see that the average displacement at Te > 10 eV is towards negative values of x. This means that the particles redeposited at a given location of x < 0 originate from right where the gross erosion is larger. The fact that more particles are redeposited than in the case they would have been ejected at the same location explains the reduction of net erosion. The opposite effect is seen for x > 0. The last feature of the extended 1D model is the presence of a net erosion peak at x = 0 larger than predicted
Fig. 5. Two cases with the same particle flux on the surface: a) electron temperature, b) electron density at the sheath entrance and c) gross erosion rate.
Please cite this article as: N. Mellet et al., Influence of the magnetised sheath on the redeposition location of sputtered tungsten and its effect on the net erosion, Nuclear Materials and Energy (2017), http://dx.doi.org/10.1016/j.nme.2017.03.009
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Fig. 6. Comparison of the net erosion rate between 4 different models for the high (a) and low (b) Te cases: based on the non-redeposition fraction (red, dashed), 1D model (black, full), 2D model without Ex (blue, dotted), full 2D model with Ex (green, dashed-dotted). c) Redeposition rate for the two contributions in the high Te case with the full 2D model. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
by the non-redeposition fraction model. The explanation is similar to that given for x > 0: the redeposited particles come from a place where the gross erosion is smaller. The effect is however enhanced as the migration of prompt-redeposited W from x < 0 adds itself to the migration of long-range redeposited W from x > 0, while at x = 0 the different directions of those two contributions are counterbalancing each other. Finally note that the extended 1D model is very fast to run as it interpolates already computed cases. On the other hand, it requires a very fine resolution to be sufficiently precise and thus a large database of cases is required. When the full 2D electric field and electron density are considered without Ex , the picture is not very different than with the extended 1D model. In particular the net erosion peak is still present. Net erosion is however smaller in the x < 0 region. This difference can be explained by separating the ejected W atoms, which are oriented towards lower values and higher values of x. In the first case, they will be ionised further from the surface as the electron density is decreased and lead to an increased migration towards lower x values. In the second case, they will be ionised closer to the surface as the electron density is increased and lead to a decreased migration towards higher x values. This explains why net erosion is diminished in this region. The opposite phenomenon applies in the x > 0 region but is more moderate. The final case is the full 2D electric field model. The net erosion pattern is here totally modified. For x > 0, net erosion is strongly reduced. The Ex field induces a E × B drift that is oriented towards the surface in this region thus increasing redeposition. For x < 0, the opposite effect is observed: the E × B drift is oriented towards the plasma. While this considerably reduces redeposition close to the strike point leading to an enhanced net erosion peak, W ions still manage to return to the surface at lower x. In fact Ex × B diminishes the velocity towards the surface that is due to the inclination of the magnetic field. To have an idea of this effect, the average displacement in the direction along the magnetic field direction is y = 51 cm with Ex while only y = 3.7 cm without Ex . For that reason, the profiles shown in Fig.e 6(ab) consider the redeposited particles until y = 1 m. The fact that W ions are redeposited at x < −0.02 m is due to the Ez × B drift, which is oriented towards lower x. A plot of the redeposition location for both contributions (prompt and long-range) is shown in Fig. 6(c). While prompt redeposition is only little affected by Ex , a strong effect is observed for long-range redeposition that is responsible for the observed modification of the net erosion pattern. Long-range W ions travel indeed in a region where Ex is comparable in magnitude to Ez and are thus prone to the effects described above. Finally the
Table 1 Average net erosion rate (nm/s) on the interval −0.04 m < x < 0.04 m for the 4 models and both cases. Case
f non−redep
1D model
2D w/o Ex
2D model
high Te low Te
0.182 0.101
0.181 0.100
0.168 0.092
0.177 0.078
average net erosion rate for the all models are provided in Table 1. No significant difference exists between them. Except from the two 1D models, for which about the same value is obtained, no clear conclusions can be deduced. For example, while the Ex field decreases the average net erosion at low Te , it reduces it at high Te . 4. Conclusion The effect of the magnetised sheath on the local migration of sputtered tungsten has been investigated. The normal component of the electric field has been shown to bring the promptly redeposited ions close to the ejection point. The long-range contribution is shifted in the direction of the E × B drift. A parameter study shows that the displacement is strongly reduced with increasing electron density as the prompt redeposition contribution becomes dominant. By increasing the electron temperature, the redeposition average is moved from the direction opposite to the E × B drift to the direction of the E × B drift. In the case of electron density and temperature gradients, four models have been studied. Generally speaking net erosion obtained with the high temperature case is larger than with the low temperature case even if the gross erosion is similar. It has however to be put in the perspective that at lower temperature ionisation by electrons drops and larger net erosion could be expected. A collisional model of the sheath would however be required for consistent simulations at larger densities. The comparison of the different models with a model that incorporates only a simple redeposition fraction and does not take the impurity migration into account is an evidence of this effect on net erosion. A second element that affects the net erosion pattern is the presence of Ex that has a strong effect on particles as it tends to bring them towards the wall or towards the plasma depending on its orientation. The presence of two net deposition regions and an enhanced net erosion peak at the left of the strike point are the main features. The effect of Ex is mainly seen on the long-range contribution, for which W ions travel in regions where the magnitude of Ex is comparable to that of Ez . Finally the full
Please cite this article as: N. Mellet et al., Influence of the magnetised sheath on the redeposition location of sputtered tungsten and its effect on the net erosion, Nuclear Materials and Energy (2017), http://dx.doi.org/10.1016/j.nme.2017.03.009
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2D model presented here has been compared to experimental data [6] and was able to reproduce the net erosion peak surrounded by the two net redeposition regions [16]. References [1] [2] [3] [4]
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Please cite this article as: N. Mellet et al., Influence of the magnetised sheath on the redeposition location of sputtered tungsten and its effect on the net erosion, Nuclear Materials and Energy (2017), http://dx.doi.org/10.1016/j.nme.2017.03.009
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Nuclear Materials and Energy 0 0 0 (2017) 1–6
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Influence of the magnetised sheath on the redeposition location of sputtered tungsten and its effect on the net erosion N. Mellet a,∗, J.P. Gunn b, B. Pégourié b, A. Hakola c, M. Airila c, Y. Marandet a, P. Roubin a a
Aix-Marseille Université, CNRS, PIIM, UMR 7345, F-13397 Marseille, France CEA, IRFM, F-13108 Saint-Paul-lez-Durance, France c VTT Technical Research Centre of Finland Ltd., Espoo, Finland b
a r t i c l e
i n f o
Article history: Received 15 July 2016 Revised 16 December 2016 Accepted 9 March 2017 Available online xxx Keywords: Redeposition Magnetised sheath Net erosion Impurity migration Tungsten
a b s t r a c t The effect of the magnetised sheath on tungsten migration and its consequences on net erosion of tungsten plasma facing components are investigated. This study points out a strong effect of the E × B drift on the redeposition location as well as a rapid redeposition of the ejected particles that are ionised close to the surface. A plasma parameter study shows that the migration direction is changed at ≈ 10 eV from opposite to E × B drift at low electron temperature to the same direction as E × B at high electron temperature. Different models have been considered in the case of plasma parameter gradients that are also at the source of a component of the electric field parallel to the surface. New features are observed like an enhanced net erosion peak at the strike point if only the component of the electric field towards the surface is taken into account. When the component parallel to the surface that drives the particles away from or towards the surface is considered, a very different net erosion picture is obtained: a net erosion peak exists between two net deposition regions. This points out the importance of incorporating this effect in simulations. © 2017 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license. (http://creativecommons.org/licenses/by-nc-nd/4.0/)
1. Introduction The erosion of Plasma Facing Components (PFCs) is a subject of importance: it determines their lifetime and the generation of impurities that have potential negative effects on the plasma operation. The net erosion of PFCs, which is observable after plasma exposure, is determined by the balance between gross erosion and redeposition, and depends strongly on the distance travelled by redeposited particles. The migration of impurities can be investigated locally for example with ERO [1], REDEP/WBC [2] or, on a larger scale, with WALLDYN [3]. Here we focus on the region close to the surface that corresponds to the magnetised sheath and where a strong electric field exists. This electric field has been shown to increase strongly the number of sputtered particles that return to the wall [4,5]. A strong effect on the redeposition location is thus also expected, especially in tungsten (W) divertors where the atoms are ionised close to the surface. The subject has been investigated experimentally in ASDEX-Upgrade [6] and DIII-D [7] by the use of special sample markers inserted as a PFC. In the first
∗
Corresponding author. E-mail address: [email protected] (N. Mellet).
experiment [6], measurements by Rutherford Backscattering Spectroscopy (RBS) have shown the presence of a net erosion peak close to the strike point surrounded by two net redeposition regions. In the second experiment [7], very small markers have been studied, where the erosion and deposition components can be separated and for which modelling using ERO for different magnetised sheath electric field models has been carried out [8]. This work concentrates on investigating the basic mechanisms due to the magnetised sheath and to the resulting changes of the redeposition location. The latter is composed of two characteristic lengths: the Debye length λD that is linked to the Debye sheath that exists even without magnetic field and a length of the same order of magnitude as the main ion Larmor radius rL that is linked to the magnetic pre-sheath that is due to the grazing incidence of the magnetic field. At low electron density (when rL λD ), the magnetised sheath width scales with the Debye length, while at high electron density (when rL λD ), the magnetised sheath width scales according to the main ion Larmor radius. This effect is accounted in the present work through the use of a Particle-In-Cell (PIC) code [9] that determines the sheath electric field in a selfconsistent manner and makes no assumption on the sheath width. The use of such a method in comparison to analytic formulas has
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Fig. 1. Schematic view of the geometry and the simulations steps: 1) Sheath electric field computation with D+ and electrons. 2) Impact angle of Ne4+ . 3) Redeposition simulation of W ejected from the wall.
been shown to be of importance in [10]. Once it is determined, the sheath electric potential and electron density profile can be used in both 1D and 2D situations, which permits to study different model grades for the particle local migration. The paper is organized as follows. In Section 2 the different models used in this paper are presented. In Section 3, the results are displayed. We first focus on the simple case where quantities only vary in the direction perpendicular to the surface before considering the complete case with plasma parameter gradients. Finally conclusions are drawn in Section 4. 2. Models Two models are presented in this paper. The first one considers that plasma parameters only vary in the direction perpendicular to the surface (1D model). The second one incorporates effects due to the variation of parameters in the direction perpendicular to the magnetic field and parallel to the surface (2D model). In all cases a flat surface is considered. 2.1. 1D Model The simulations are performed in three steps as described in [10] and schematically shown in Fig. 1 with the corresponding geometry. The first step is to determine the sheath potential as well as the temperature and electron density profiles with a PIC code [9] that uses Poisson’s equation and the equation of motion of the electrons and the main ion species, which we will consider to be D+ . The injection of particles is made at the magnetised sheath entrance with a Maxwellian distribution in the direction perpendicular to the magnetic field and with the parallel distribution calculated with a Vlasov code that solves the model described in Chung and Hutchinson [11]. The sheath potential depends on three parameters: the ion to electron temperature ratio τ , the angle of the magnetic field with respect to the surface α B
and the ratio between the main ion Larmor radius at the sound speed for Ti = 0 and the Debye length at the sheath entrance, √ ζ = rL /λD ∝ ne /B, which scales as the square root of the electron density. We focus here on the case τ =1 (Ti = Te ), αB = 3◦ and B = 4.2 T so that a direct relation between ζ and ne exists. Additionally the sheath potential drop is prescribed as φ0 = 2.84Te (eV) [12], the electric potential scaling linearly with the electron temperature Te . The electric sheath potential, temperature and density profiles are thus only calculated for a set of values of ζ , (ζ = (0.1, 0.2, 0.3, 0.5, 1, 2, 3, 5, 7, 10, 15, 20, 25, 30, 40, 50, 100, 150)). The second step is the injection of Ne4+ at the magnetised sheath entrance, which will be the impinging impurity considered in this work. This choice is motivated by the fact that neon can be used as an inert seeding gas to reduce the heat flux on the PFCs. In present machines, lighter impurities like N are used leading to a smaller erosion than in the present case. The injection is made with the same procedure as in Mellet et al. [10] using the technique described in M. Kocˇ an and Gunn [13] for the parallel velocity distribution. Additional we considered TNe = TD . The test particle simulations in the electric field provided by the PIC code give the impact angle and energy of Ne4+ on the surface, which is used to determine the gross erosion. The concentration of Ne4+ (1%), as well as that of the ionised W, is considered to be sufficiently small not to affect the sheath electric potential making the test particle approach valid. The third step is the injection of W atoms from the wall with a cosine angular and a Thompson energy distribution that includes a square root cut-off determined by parameters linked to the impinging particle (Ne4+ ). W atoms may be ionised by electrons according to the ionisation rate provided in ADAS [14] and affected by the electric field. The possibility of multiple ionisation is taken into account up to the charge state 8+. The particles are then followed so that the distance they travelled can be determined at the moment they hit the surface. An important distinction between two kinds of trajectories is made [5]: prompt and long-range redeposition. In the first case the particles are redeposited during their first Larmor gyration. In the second case the particles undergo several Larmor gyrations and can travel a longer way even if they stay close or in the sheath. 2.2. 2D Extension The model presented in the previous subsection can be extended by considering the variation of the plasma parameters also in the x-direction (see Fig. 1). Consider a case where the electron temperature is linearly decrasing from the strike line along the xdirection (see Fig. 5(a)), which leads to a variation of the sheath potential in the x-direction and thus to an additional component of the electric field Ex that is displayed in Fig. 2(c). The first observation is that Ez is much larger close to the surface than Ex . Ex extends however further in the sheath than Ez , their amplitude
Fig. 2. a) Sheath electric potential calculated from the high Te case (Fig. 5), b) z-component and c) x-component of the electric field.
Please cite this article as: N. Mellet et al., Influence of the magnetised sheath on the redeposition location of sputtered tungsten and its effect on the net erosion, Nuclear Materials and Energy (2017), http://dx.doi.org/10.1016/j.nme.2017.03.009
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Fig. 3. Redeposition location distribution of W ejected at (0,0) without (left) and with (right) sheath electric field. Top: view from above. Bottom: lateral distributions.
being comparable from ∼ 1 mm above the surface. The electron density is sufficiently largeso that rL λD making the sheath width proportional to rL ∝ Ti = Te . This dependence is visible in Fig. 2(b). Concerning Ex , it is oriented opposite to the x axis for x < 0 and in the direction of the x axis for x > 0. The simulation region is a box of infinite dimension in the x and y direction (s(x, z ) = s(−0.04, z ) for x < −0.04 and s(x, z ) = s(0.04, z ) for x > 0.04 with s corresponding to any background quantity) and twice as the largest sheath width in the z direction. Particles are ejected along the x-axis where the number of particles is proportional to the corresponding gross erosion rate along x. 3. Results 3.1. 1D sheath Simulations of the redeposition location of tungsten ejected at a given point are performed first in the frame of the 1D sheath model (i.e. where the quantities vary only in the z-direction). Altogether 4 × 105 particles have been ejected at the (0,0) point in the magnetised sheath potential and electron density calculated for Te = 30 eV and ne = 1.87 × 1019 m−3 (ζ = 20) given at the sheath entrance. The magnetic field is oriented towards negative y and with an angle with respect to the surface of αB = 3◦ . Fig. 3 displays the comparison between the redeposition location obtained with and without the magnetised sheath so that its effect can be observed. In the case without sheath, flat density and temperature profiles have been considered. In both cases, the prompt and longrange contributions can be easily distinguished. When no sheath electric field is considered, prompt redeposition is centered at a distance from the ejection point of the same order as W+ Larmor radius and corresponds to the expected scenario of the ion being
3
redeposited during the first Larmor gyration. The long-range deposition exhibits no lateral displacement, the ionised particles following the magnetic field before being redeposited. When the sheath is added, the picture is changed. The prompt redeposition distribution is this time located around the ejection point. Additionally it represents now 90% of the total redeposition (50% without sheath). This shows the strong effect of the electric field that tends to bring back the W ions towards the surface, even at a distance from the surface larger than the Larmor radius. This can be understood by the fact the sheath potential is ∼ 87 eV in the present case, which is very large compared to the average ejection energy (∼ 15 eV, of the order of W surface binding energy Us = 8.68 eV). Concerning the long-range contribution, the redeposition location distribution is shifted of 8.3 mm in the direction of the E × B drift, pointing out its influence. A ne and Te (given at the sheath entrance) parameter study is performed next in order to see the influence of electron density and temperature on the redeposition location. Except for the scanned quantity, the same parameters as those used previously are employed. Fig. 4 shows the average lateral displacement for all the redeposited tungsten and for the two contributions taken separately as a function of ne and Te . The prompt redeposited ratio (# of prompt redeposited W ions/# of redeposited W ions) is also plotted. Finally the average z coordinate and electric field at the first ionisation for prompt redeposited W are displayed. This has not been done for long-range redeposition as it directly depends on the size of the simulation box. When ne is increased, the average lateral migration is generally reduced for both contributions (Fig. 4(a)). In the case of prompt redeposition, this can be explained as follows. Once ionised the particle would have a Larmor gyration if no electric field was present. The stronger the electric field the faster the particle is brought back to the surface and the less it can travel in the x direction. The electric field when the particle is first ionised increases with ne (Fig. 4(g)) partially because the ionisation gets closer to the surface (Fig. 4(e)) and explains why the migration is reduced. The bump in Fig. 4(e)) is due to the modification of the electron density profile with ne . Concerning long-range deposition, the mechanism is more complex and related to the sheath electric field profile dependence on ne . The prompt redeposition ratio strongly increases with electron density (Fig. 4(c)) resulting in a zero total average lateral displacement at large ne . When Te is increased, the average prompt redeposition distribution is shifted towards the ejection location (Fig. 4(b)). This can be attributed to the larger electric field that accelerates the ejected W strongly towards the surface as φ0 = 2.84Te is increased (Fig. 4(h)). It is also due to the average W Larmor radius that is only slightly increased with the Thompson distribution even if the impact energy of Ne4+ is proportional to Te (Fig. 4(f)) . About the long-range redeposition, the E × B drift increases as Te and the average redeposition location is the most strongly shifted at the largest electron temperature. The effect on the prompt redeposition fraction (Fig. 4(d)) is moderate making however the total average displacement pass from positive to negative x. Finally, in Fig. 4(eand f), it can be seen that the ionisation of prompt redeposited W can happen farther than their Larmor radius showing that the sheath can strongly modify their trajectory. 3.2. Effect of gradients on net erosion Four different models are compared. The first two models are based on the 1D treatment of the sheath region. The simplest is to consider a non-redeposition fraction fnon−redep = # of escaping W/# of ejected W. The local net erosion is then calculated as follows:
vnet = fnon−redep vgross
(1)
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Fig. 4. Parameter study of the redeposition average location (a,b). The prompt redeposition ratio is plotted in (c,d). The first ionisation average distance with respect to the surface in (e,f) and the average electric field at ionisation in (g,h) are shown only for prompt redeposited W. ne is varied at Te = 30 eV (a,c,e,g). Te is varied at ne = 1.87 × 1019 m−3 (b,d,f,h).
The disadvantage of the method is that it does not incorporate W migration. It can thus be used as a reference to quantify the effect of the redeposition location. Note also that no net deposition is possible. The second model is an extension of the 1D model whose results are displayed in Section 3.1. The individual lateral displacement distributions are interpolated to fit the parameters at each location and summed to obtain a net erosion profile. The shortcoming is that the particles that are ejected at a given point will feel the electric field and the density calculated at this point even if they travel laterally where different plasma parameters exist. Besides, Ex is not retained. The last two models are based on the 2D treatment of the sheath region. Ejected particles will thus feel the full 2D variation of the electron density and the electric field. The difference between them is that one considers only Ez while the most sophisticated considers also Ex . For the 2D based models, 4 × 106 ejected W atoms have been used. In order to investigate the effect of temperature with the same particle flux on the surface, two cases have been defined (Fig. 5). They roughly represent the profiles around the divertor strike point but with a linear decay of the parameters (electron temperature and density at the sheath entrance) in order to keep the situation as simple as possible. The corresponding gross erosion obtained with the energy and angle determined by the test particle simulations of Ne4+ and the physical sputtering yields of [15] are displayed in Fig. 5(c). Both curves are surprisingly close to each other. The differences observed in net erosion between both cases will thus be attributed almost exclusively to the redeposition mechanisms. The results of all simulations are displayed in Fig. 6(a) for the high Te case and in Fig.e 6(b) for the low Te case. A general observation is that net erosion is larger at high Te . Even if the ionisation cross section and the sheath width are larger at twice larger Te , the dominant effect is the reduction of the mean free path before ionisation in a plasma that is 4 times denser. If we consider the nonredeposition fraction model, we see without any surprise that it is symmetric as the parameters are also symmetric. The extended 1D model breaks this symmetry and several new features can be observed. The first is a strong reduction of net erosion for x < 0. If we refer to Fig. 4(b), we see that the average displacement at Te > 10 eV is towards negative values of x. This means that the particles redeposited at a given location of x < 0 originate from right where the gross erosion is larger. The fact that more particles are redeposited than in the case they would have been ejected at the same location explains the reduction of net erosion. The opposite effect is seen for x > 0. The last feature of the extended 1D model is the presence of a net erosion peak at x = 0 larger than predicted
Fig. 5. Two cases with the same particle flux on the surface: a) electron temperature, b) electron density at the sheath entrance and c) gross erosion rate.
Please cite this article as: N. Mellet et al., Influence of the magnetised sheath on the redeposition location of sputtered tungsten and its effect on the net erosion, Nuclear Materials and Energy (2017), http://dx.doi.org/10.1016/j.nme.2017.03.009
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Fig. 6. Comparison of the net erosion rate between 4 different models for the high (a) and low (b) Te cases: based on the non-redeposition fraction (red, dashed), 1D model (black, full), 2D model without Ex (blue, dotted), full 2D model with Ex (green, dashed-dotted). c) Redeposition rate for the two contributions in the high Te case with the full 2D model. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
by the non-redeposition fraction model. The explanation is similar to that given for x > 0: the redeposited particles come from a place where the gross erosion is smaller. The effect is however enhanced as the migration of prompt-redeposited W from x < 0 adds itself to the migration of long-range redeposited W from x > 0, while at x = 0 the different directions of those two contributions are counterbalancing each other. Finally note that the extended 1D model is very fast to run as it interpolates already computed cases. On the other hand, it requires a very fine resolution to be sufficiently precise and thus a large database of cases is required. When the full 2D electric field and electron density are considered without Ex , the picture is not very different than with the extended 1D model. In particular the net erosion peak is still present. Net erosion is however smaller in the x < 0 region. This difference can be explained by separating the ejected W atoms, which are oriented towards lower values and higher values of x. In the first case, they will be ionised further from the surface as the electron density is decreased and lead to an increased migration towards lower x values. In the second case, they will be ionised closer to the surface as the electron density is increased and lead to a decreased migration towards higher x values. This explains why net erosion is diminished in this region. The opposite phenomenon applies in the x > 0 region but is more moderate. The final case is the full 2D electric field model. The net erosion pattern is here totally modified. For x > 0, net erosion is strongly reduced. The Ex field induces a E × B drift that is oriented towards the surface in this region thus increasing redeposition. For x < 0, the opposite effect is observed: the E × B drift is oriented towards the plasma. While this considerably reduces redeposition close to the strike point leading to an enhanced net erosion peak, W ions still manage to return to the surface at lower x. In fact Ex × B diminishes the velocity towards the surface that is due to the inclination of the magnetic field. To have an idea of this effect, the average displacement in the direction along the magnetic field direction is y = 51 cm with Ex while only y = 3.7 cm without Ex . For that reason, the profiles shown in Fig.e 6(ab) consider the redeposited particles until y = 1 m. The fact that W ions are redeposited at x < −0.02 m is due to the Ez × B drift, which is oriented towards lower x. A plot of the redeposition location for both contributions (prompt and long-range) is shown in Fig. 6(c). While prompt redeposition is only little affected by Ex , a strong effect is observed for long-range redeposition that is responsible for the observed modification of the net erosion pattern. Long-range W ions travel indeed in a region where Ex is comparable in magnitude to Ez and are thus prone to the effects described above. Finally the
Table 1 Average net erosion rate (nm/s) on the interval −0.04 m < x < 0.04 m for the 4 models and both cases. Case
f non−redep
1D model
2D w/o Ex
2D model
high Te low Te
0.182 0.101
0.181 0.100
0.168 0.092
0.177 0.078
average net erosion rate for the all models are provided in Table 1. No significant difference exists between them. Except from the two 1D models, for which about the same value is obtained, no clear conclusions can be deduced. For example, while the Ex field decreases the average net erosion at low Te , it reduces it at high Te . 4. Conclusion The effect of the magnetised sheath on the local migration of sputtered tungsten has been investigated. The normal component of the electric field has been shown to bring the promptly redeposited ions close to the ejection point. The long-range contribution is shifted in the direction of the E × B drift. A parameter study shows that the displacement is strongly reduced with increasing electron density as the prompt redeposition contribution becomes dominant. By increasing the electron temperature, the redeposition average is moved from the direction opposite to the E × B drift to the direction of the E × B drift. In the case of electron density and temperature gradients, four models have been studied. Generally speaking net erosion obtained with the high temperature case is larger than with the low temperature case even if the gross erosion is similar. It has however to be put in the perspective that at lower temperature ionisation by electrons drops and larger net erosion could be expected. A collisional model of the sheath would however be required for consistent simulations at larger densities. The comparison of the different models with a model that incorporates only a simple redeposition fraction and does not take the impurity migration into account is an evidence of this effect on net erosion. A second element that affects the net erosion pattern is the presence of Ex that has a strong effect on particles as it tends to bring them towards the wall or towards the plasma depending on its orientation. The presence of two net deposition regions and an enhanced net erosion peak at the left of the strike point are the main features. The effect of Ex is mainly seen on the long-range contribution, for which W ions travel in regions where the magnitude of Ex is comparable to that of Ez . Finally the full
Please cite this article as: N. Mellet et al., Influence of the magnetised sheath on the redeposition location of sputtered tungsten and its effect on the net erosion, Nuclear Materials and Energy (2017), http://dx.doi.org/10.1016/j.nme.2017.03.009
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2D model presented here has been compared to experimental data [6] and was able to reproduce the net erosion peak surrounded by the two net redeposition regions [16]. References [1] [2] [3] [4]
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Please cite this article as: N. Mellet et al., Influence of the magnetised sheath on the redeposition location of sputtered tungsten and its effect on the net erosion, Nuclear Materials and Energy (2017), http://dx.doi.org/10.1016/j.nme.2017.03.009