Computational investigation of incident ion angles and material erosion at rough graphite and silicon carbide divertor surfaces

The local incident ion angle (LIIA) at a surface is determined by the ion angle distribution (IAD) in the sheath and the surface topography. The LIIA determines the sputtering and reflection coefficient at the surface and, hence, affects erosion, deposition, material migration, and plasma-facing component (PFC) lifetime. Tokamak PFCs have intrinsic surface roughness from their manufacturing process, which tends to be smoothed out by plasma exposure, but the roughness can increase with plasma exposure due to sputtering, deposition, slag formation and transport, arc tracks, and cracks from thermal stress. Non-uniform erosion and impurity deposition on PFC surfaces due to micro-roughness have been reported from tokamak experiments.^1–10 Such non-uniform impurity deposition and deuterium trapping were enhanced in pores, pits, cracks, and valleys on the micrometer scale.¹¹ Micro-structures have been seen not only in graphite PFCs but also on W surfaces^12,13 and on Be¹⁴ due to erosion, melting, and cracking after plasma exposures. Recently, fabricated micro-structured tungsten has also been proposed to reduce near-surface thermal stresses.¹⁵ 

The trajectory of ions traveling near the surface is controlled by electric (E) and magnetic (B) fields in the sheath via E-field acceleration, B-field gyration, and E × B drift. The ion trajectory in the sheath has been modeled for various incident B-field angles with equation-of-motion (EOM) models,^3,16 the ERO Monte Carlo code,^17–19 kinetic particle-in-cell methods,²⁰ and large gyro-orbit models.²¹ Earlier studies reported that the IAD in the sheath is very sensitive to the sheath width.^16,22 Previously, we reported experimental measurements of the polar and azimuthal deuterium (D) incident ion angles at the DIII-D divertor surface by using the DiMES probe to expose 30 × 30 × 2–4 μm deep micro-trenches to a D L-mode plasma.^22,23 The deposition patterns of C impurities on the trench floors were measured by energy-dispersive x-ray spectroscopy (EDS) and revealed D incident ion shadowing by the trench walls. The C deposition pattern revealed that the ion shadowed area was formed due to D ions incident with an azimuthal direction of φ = −40° on the surface plane and a polar angle of θ = 80° [Fig. 1(a)]. The polar incident angle of the toroidal magnetic field onto the surface was α = 88° [Fig. 1(b)] with an azimuthal angle of φ = 0°. A factor k characterizes the relation between magnetic pre-sheath (MPS) length (L_MPS) and the ion gyro radius (ρ_i),

and the value of k = 3 was determined by comparing the measured incident ion angles with incident IADs calculated from an EOM model¹⁶ for k factors in the range 0.5–4.

EDS analysis of our previous experimental data from DIII-D found that the concentrations of C or Al deposited in the main ion shadowed area in the micro-trenches were an order of magnitude higher than the concentrations outside of the shadowed area.²⁴ The measured deposition rate of C in the shadowed area was ∼10 nm/s.²⁴ A general surface roughness parameter, the root mean square roughness (R_RMS), has been typically used to characterize PFC surfaces; however, Ueda et al.² concluded that the detailed surface morphology needs to be taken into account to understand C deposition on micro-rough W surfaces. The importance of the inclination angle δ of the surface to incoming ions was emphasized by Cupak et al.²⁵ as the mean surface inclination angle distribution (SIAD) ⟨δ⟩ determines the mean local incident ion angle (LIIA) distribution ⟨θ′⟩ [Fig. 1(c)] and, hence, determines the sputtering yield directly compared to the general surface roughness or root mean square roughness R_RMS. Sputtering yields of rough surfaces have been reported from laboratory experiments that used an ion beam or plasma source.^25–27 Several theoretical studies employing computationally generated sinusoidal^19,28 and fractal^1,19,29,30 topologies have been performed for micro-rough surfaces. These theoretical studies reported that rough surfaces show sputtering yield suppression by up to ∼30% from the value for a smooth surface due to the change of the LIIA at the surface,^1,25 material trapping,¹⁹ and impurity deposition.²⁸ However, computational results employing the surface topology measured on actual tokamak PFCs and using IADs that take the sheath into account are limited. Schmid et al.³ performed ion trajectory and erosion pattern calculations using the surface topology of ASDEX-Upgrade (AUG) tiles as measured by atomic force microscopy (AFM). These calculations revealed enhanced erosion at leading surface areas, and W accumulation due to prompt redeposition concentrated in the ion shadowed areas on the micro-rough surfaces.

In this paper, we report D IADs in tokamak divertor plasma environments using a collisionless EOM model.^16,23 A justification for the collisionless assumption for our plasma configurations is available in Ref. 22. Those IADs were applied to a Monte Carlo sputtering simulation code^24,31 that calculated 2D erosion, deposition, and ion shadowing on 3D topographic data of graphite and SiC divertor surfaces that had been exposed in DIII-D and NSTX-U and measured by confocal microscopy.¹¹ The calculated erosion pattern was compared to an Al concentration map obtained by Auger electron spectroscopy (AES) on a DIII-D DiMES graphite surface. The LIIA, erosion, deposition, and ion shadowed area were parameterized for different surfaces. Finally, we provide empirical global formulas to estimate the mean LIIA distribution and ion shadowed area as a function of the mean SIAD.

The collisionless equation-of-motion (EOM) model described in Refs. 3, 16, and 22 was used to calculate D IADs for the plasma parameters in DIII-D, NSTX-U, and ITER attached divertors. The EOM model employs an analytical expression for the magnetic pre-sheath (MPS) potential profile,^16,22,32

where z is the distance from the plasma-facing surface and ϕ_w is the potential at the surface referenced to the potential at the pre-sheath boundary.³³ For α > 85° (critical angle), we assumed that the sheath potential was characterized by the MPS potential rather than the classical Debye sheath.^34,35 We employed the MPS length, L_MPS = 3 × ρ_i sin(α), that was determined by our previous work²² for a DIII-D D plasma configuration where the B-field incident angle α = 88.5° and ω = ω_ci/ω_pi = 0.04, where ω_ci is the ion cyclotron frequency and ω_pi is the plasma frequency.³⁶ We assume the E-field is effectively homogeneous at micro-rough surfaces since the scale of the surface roughness is much smaller than the mm-scale L_MPS.¹⁰ Trajectories of particles with parallel and perpendicular (to the B_t toroidal direction) velocities with a Maxwellian velocity distribution at the sheath entrance were traced by solving the differential equation of motion for ions undergoing electric and Lorentz forces. Figure 2 shows calculated IADs for D ions at an ideal, smooth, planar surface as a function of α. The parameters used in the calculations were in the range of B = 1 T (NSTX-U), 2 T (DIII-D), and 5.3 T (ITER), α = 85°–89°, n_e = 10¹⁹ m⁻³ (NSTX-U and DIII-D) and 10²⁰ m⁻³ (ITER), and T_e = 10–30 eV, and ω was derived to be in the range ω = 0.01–0.05. The calculated ion angles were not sensitive to this range of plasma parameters except for the B-field angle α. For the grazing B-field incident angle (α = 88.5°), the mean polar IAD ⟨θ⟩ = 78° and mean azimuthal IAD ⟨φ⟩ ∼−40° as verified by our previous micro-trench measurements in DIII-D experiments.^22,23 The azimuthal mean IAD [Fig. 1(b)] was much less sensitive to the B-field angle than the polar mean IAD. Most likely, this is because the azimuthal guiding center motion of an ion is governed by the E × B drift, which is unchanged for α = 85°–89°. The E × B drift is unchanged because the E-field, i.e., the sheath potential profile, is constant for the same gyro radius, which is constant for the same B-field strength [Eqs. (1) and (2)].

The Monte Carlo MPR code^24,31 was used to model the physical sputtering of surfaces due to D (Sec. V) or He (Sec. IV) ion bombardment on sculpted material surfaces. The surface material is C, SiC, or Al, corresponding to the sample substrate material C or SiC (Sec. V), or the redeposited layer of Al (Sec. IV). We input to the MPR code the experimentally measured 3D topographical data from four graphite and SiC sample surfaces that had been exposed to D ion irradiation in NSTX-U or DIII-D. In the MPR model, a flux of D ions, set up using data for the polar and azimuthal IAD profiles introduced in Sec. II A, was incident on a given 3D surface. The ions with input IADs were launched from 10 nm above the surface, so that the ion trajectory in the MPR calculational range of up to μm-scale can be assumed as a straight line as the ion gyro radius is in the sub-mm-scale. A flow chart of the computational method, combining the EOM model and MPR code, is summarized in Fig. 3 of Ref. 22. The incident ion energies, E_impact, employed for calculations were 100 eV by assuming T_e = T_i = 20 eV and E_impact = 3Z_ikT_e + 2kT_i.³³ The angular dependence of the C sputtering yield for E_impact = 100 eV of D ion projectiles was obtained from Ref. 37. The angular dependence of the SiC(110) sputtering yield for E_impact = 100 eV of D ion projectiles was calculated by SDTrimSP.^38–40 Both angular dependences are plotted in Fig. 3. Data for the angular dependence of the sputtering yield on Al for He ion projectiles of 100 eV are not available, and so the experimental data for Si with He ions³⁷ were used for Al, which has a similar mass to Si. In that ion incident energy regime, the estimation of the dominant C erosion mechanism was made previously²³ by showing that chemical erosion by D and physical sputtering by C were both calculated to be one order of magnitude lower than the physical sputtering by D. Based on this assumption for C, we also assume that SiC erosion is also dominated by physical sputtering. For sputtered particles, a cosine-like distribution was used, given by

where $θout$ is the angle between the surface normal and the direction of species emitted from the target surface. The coefficients g_1,2 and n_1,2 used in this calculation were assumed to be g₁ = 1.5, g₂ = −1.0, n₁ = 0.1, and n₂ = 0.9.²³ Those reflected or sputtered particles do not make further reflection or sputtering on the surface in this model.

For analysis in Secs. IV and V, we define the normalized sputtering yield as the ratio of the number of sputtered particles from the sculpted surface and the number of sputtered particles from a smooth surface at each computational cell irradiated by the same ion flux. The normalized deposition is defined as the number of deposited particles at each cell normalized by the average number of sputtered particles. The shadowed area is defined to be where the incident ion flux is less than 5% of the average over the entire surface area.

We use notations I, II, III, and IV to identify each sample surface and i and ii for sub-areas of surfaces I and II in this paper. Properties of those surfaces are summarized in Table I. Surfaces I and II are graphite samples from DIII-D DiMES and NSTX-U experiments, respectively. Surface I of a DIII-D DiMES probe head was exposed to a series of repeated lower single null, Ohmic low density attached He L-mode, T_e ∼15 eV plasmas for 6.2 s in total (“R-He” in Ref. 16) near the outer strike point on the DIII-D tokamak using the DiMES manipulator probe.⁴¹ The DiMES head (outside of Surface I) included areas coated with 80-nm thick Al to trace the Al migration.^16,42 Al was redeposited to the Surface I area, which was located at the B-field downstream region (∼1 cm distance from the Al coating), during the He plasma exposures. Surface II from NSTX-U was exposed to 1138 discharges (L-mode and H-mode) with plasma current above 50 kA, and the total integrated plasma duration was 845 s.¹¹ For both surfaces I and II, characteristic micro-sized structures were found on a surface with sub-micron general surface roughness. Surface I topological data were obtained by optical confocal microscopy (Olympus LEXT OLS4000) for a part of the surface on a DiMES graphite head sample, which was pitted with pores intrinsic to the graphite. These pores were 5–50 μm in size and covered ∼10% of the surface area. Scanning electron microscopy (SEM) images of the same sample can be seen in Fig. 1 of Ref. 11 and Fig. 9 of Ref. 16. The measured surface I topology includes pores that were 2–10 μm deep and covered 18% of the measured surface area, as shown in Fig. 4(a). Surface II topological data were also obtained by confocal microscopy (Leica DCM3D) for a NSTX-U graphite tile surface with micro-pores. A valley-like arc-track structure of ∼10-μm width was found for surface II, as shown in Fig. 7. The subareas i and ii, where R_RMS is sub-mm, were chosen in surfaces I and II, respectively. Topological data for surfaces III and IV were measured by using a WYKO optical interferometer on a SiC layer from DIII-D experiments.³⁹ The SiC layer of 100 ± 50 μm was deposited on a graphite DIII-D divertor tile via chemical vapor deposition (CVD). Surface III is from before exposure and surface IV is from after exposure to a wide variety of divertor plasma conditions, which are shown in Fig. 9. These surfaces were exposed to L- and H-mode discharges over ∼1.3 × 10⁴ s with a cumulative D fluence of about 3 × 10²⁵ m⁻² and heat fluxes ranging from approximately 0.1 to 5 MW m⁻² in steady-state and 5 to 50 MW m⁻² during ELMs. On surface III (before plasma exposure) in Fig. 9(a), we see prominent nodule-like structures with characteristic features 10–50 μm in size everywhere, which is characteristic of film growth via the CVD process. Periodic ripples appearing at ∼200-μm distances on the sample surface are due to the tile machining. After plasma exposure, the CVD nodules mostly disappear on surface IV, but the periodic ripples were still observed. Note that because of the limited x-y resolution (2.0 μm/pixel), the images of surfaces III and IV do not resolve x-y features smaller than a few μm. Due to resolution limitations, nm-scale surface features are not accounted for in any of the images of surfaces I-IV. The erosion rate of C at the graphite surface was measured as 3–15 nm/s (Ref. 43) in the DiMES experiments, and we measured previously the gross C deposition rate of ∼10 nm/s on a sample mounted on the DiMES head.²⁴ Hence, we might expect that nm-scale features will show nm-scale topological changes during the plasma operation ≫ 1 s. We assume those nm-features are smoothed due to erosion and deposition in the calculations presented herein. A similar smoothing effect due to erosion and deposition was calculated even for sub-μm features with 1000 s exposure.²⁸ We note that nm-features may become important for W as the formation of fuzz and bubbles with nm sizes due to He ion bombardment is well known.^44,45

Figure 4(a) shows a false-color profile of the topography of surface I of the graphite DIII-D DiMES sample. An SEM image of the sample surface is shown in Figs. 1(b) and 3(a) in Ref. 11. Ex-situ scanning Auger microscopy (SAM) analysis found deposition of Al impurities concentrated in those pore structures [Fig. 3(b) of Ref. 11]. We modeled the sputtering of the redeposited Al with the MPR code by bombarding the measured surface with 100-eV He ions using IADs calculated by the EOM model from the “R-He” discharge parameters.¹⁶ The mean polar and azimuthal angles calculated by the EOM model were ⟨θ⟩ = 78° and ⟨φ⟩ = −40°, respectively. The calculation showed the shadowed area consisted of ∼30% of the entire surface area, and ∼50% of the shadowed area is concentrated in the pores. This can be seen visually in the calculated erosion distribution shown in Fig. 4(b). The Al distribution measured by SAM [reported in Fig. 4(a) of Ref. 11] was re-plotted in Fig. 5(a) for comparison with the MPR sputtering calculation shown in Fig. 5(b). The calculated shadowed regions in Fig. 5(b) are consistent with the regions where increased concentrations of Al were found in Fig. 5(a). Comparison of the line profiles shown in Figs. 6(a) and 6(b) confirms that Al intensities concentrate in the He-ion shadowed area, in which no erosion by He bombardment was predicted by the MPR code. Figure 5(c) shows the normalized MPR deposition profile of secondary Al particles sputtered by He bombardment. The profile in Fig. 5(c) indicates that sputtered Al can be deposited in the He-ion shadowed floors and upstream side surface of pores. This agreement validates the calculated IADs and is consistent with our previous IAD verification using micro-trench samples.^22,23

We have computationally simulated D ion irradiation results using the topological data for surfaces I [DIII-D DiMES graphite, Fig. 4(a)], II (NSTX-U graphite, Fig. 7), III [DIII-D SiC pre-exposure, Fig. 9(a)], and IV [DIII-D SiC post-exposure, Fig. 9(b)] using the MPR code. The goal of this computational study was to examine the effects of surface topology on the erosion and impurity redeposition under D irradiations assuming L- or H-mode plasma parameters (Sec. II) and to compare these results to that from an ideal smooth surface. Calculations were done at eight different surface orientations obtained by rotating the surface (or B-field direction) azimuthally, sampled every 45°, because the micro-valley (arc track) structure on surface II and the ripple structure on surfaces III and IV are anisotropic. Those surfaces are characterized by different morphologies, e.g., the micropores on surface I from the DIII-D DiMES graphite sample, a micro-valley on surface II from the NSTX-U graphite tile, and the micronodules and/or periodic ripples for surfaces III and IV from the DIII-D SiC tile. D ions at 100 eV following the IADs calculated by the EOM model in Sec. II were used to irradiate those C and SiC surfaces using the MPR code, employing the angular dependence of the sputtering yield for 100-eV D ion bombardment of C for surfaces I and II, and SiC(110) for surfaces III and IV (Fig. 3). We assumed that the amount of deposition of secondary particles ejected by sputtering is unity. This assumption does not significantly affect the effective erosion yield since the total number of deposited secondary particles after sputtering is less than the sputtering yield of particles by an order of magnitude as reported in Sec. V B.

The relation between erosion and orientation of surface features was explored, and the results are shown in Fig. 8. The erosion patterns calculated by the MPR code for surface II (NSTX-U graphite, Fig. 7) at two hypothetical orientations of the azimuthal B-field direction (Δϕ = 90°) at α = 88° (typical B-field angle in tokamaks) are shown in Figs. 8(a) and 8(b). One can clearly see the shadowed area in the micro-valley region on upstream surfaces in Fig. 8(a), where the azimuthal direction I_D of D ions goes orthogonally toward the micro-valley. Here, I_D is taken to be the mean azimuthal IAD ⟨ϕ⟩ [Fig. 2(b)]. This result is consistent with the boron concentration on upstream surfaces measured by SAM.¹¹ On the other hand, the plasma-wetting (D ion irradiation) of the micro-valley floor is demonstrated in Fig. 8(b), where the D ion direction I_D is relatively parallel to the micro-valley. Figure 8(c) illustrates the deposition pattern of sputtered C particles on the surface for the same surface orientation of Fig. 8(a). Although the total number of deposited C particles was ∼5% of the sputtered particles, many of the deposited C particles were found close to the original sputtering locations due to sputtered particles impinging on the neighboring sub-micrometer-rough structures. Carbon deposition is apparent in the D ion shadowed area of the micro-valleys, consistent with previously measured boron concentrations on upstream surfaces of the micro-valleys.¹¹ 

We applied a similar analysis to SiC surfaces III and IV from DIII-D experiments (Sec. III). Erosion patterns were calculated for two different orientations of surface III [DIII-D SiC tile, Fig. 9(a)] at α = 88° and are shown in Figs. 10(a) and 10(b). Erosion is largest on the nodular-like micro structures regardless of orientations. A shadowed area is seen in the valley regions in Fig. 10(a). The effect of a 45° change in the azimuthal D ion incident angle is illustrated in Fig. 10(b). This caused a change in the erosion pattern, but the total shadowed area is similar. Figure 10(c) shows the distribution of deposition of C particles sputtered from the surface. The calculation revealed that 5%–10% of sputtered C particles are incident on the neighboring nodular-like structures. Deposition of carbon is also seen in the D ion shadowed area in the valley regions. The calculated erosion pattern of surface IV [DIII-D SiC tile, Fig. 8(b)] at α = 88° is shown in Fig. 10(d). Although the nodular structures disappeared due to plasma exposures on surface IV, the larger erosion on hill regions is seen. The shadowed area is still observed in the valley regions as that seen for surface III. We noticed that micro-pore structures made by the plasma exposures [seen in the left-hand side of Fig. 9(b)] are also a dominant source of the shadowed area, as was the case for micro-pores in surface I (DiMES graphite).

Figures 11(a)–11(c) show the overall effective sputtering yield, defined by

where n_phys is the total number of sputtered particles, n_redep is the total number of redeposited particles after ejection by sputtering, and n_irrad is the total number of the bombarding D ions in the area of interest. Sputtering from other incoming impurities such as C ions or neutrals was not included in this calculation in order to focus on the gross erosion due to D sputtering. For all surfaces, Y_eff was averaged over eight different surface orientations because the sample surfaces were anisotropic. The effective sputtering yields for all surfaces were 50%–100% for α = 85°–88° and ∼200% at α = 89° of an ideally smooth surface. Such erosion suppression seen for α = 85°–88° is consistent with the trends reported from many previous computational and experimental studies on rough surfaces. Higher (5%–10%) values of Y_eff are seen for both subareas i (DiMES graphite) and ii (NSTX-U graphite) than for both surfaces I and II, respectively, in Figs. 10(a) and 10(b). Calculations revealed that only 5%–10% of the sputtered C particles subsequently collide with the surface, i.e., n_redep/n_phys < 0.1, for any surface data. Hence, we conclude that trapping of sputtered species in micro- or submicro-pit structures is not the main reason for the suppression of Y_eff on the rough surfaces we investigated. The higher sputtering yield than the smooth surface at α = 89° is discussed later.

Distributions of the polar local incident ion angle (LIIA) θ′, i.e., the fraction of ions per degree at a given polar LIIA, for α = 88° are shown for surfaces I–IV in Figs. 12(a) and 12(b). The ion distributions shift toward shallower angles from the original mean polar IAD ⟨θ⟩ = 77° [Fig. 1(a)]. A broadening of the distribution is also observed with a full-width-at-half-maximum of FWHM = 15°–40° compared to a FWHM = 10° for the original polar IAD [Fig. 2(b) of Ref. 22]. We can also see a broader polar ion angle distribution on surfaces I and II than on subareas i and ii due to rougher surface topology of surfaces I and II. Figures 13(a) and 13(b) show the mean LIIA distribution ⟨θ′⟩ calculated for surfaces I–IV along with ⟨θ′⟩ at an ideally smooth surface [identical to the mean polar IADs ⟨θ⟩ in Fig. 2(a)]. The calculations revealed that surface roughness causes an increase in the polar ion incidence angle, i.e., ⟨θ′⟩ is always steeper than ⟨θ⟩. Surface I shows a steeper ⟨θ′⟩ than surface II [Figs. 13(a) and 13(b)], while surface I shows a lower value of Y_eff than surface II [Figs. 11(a) and 11(b)]. Also, a steeper ⟨θ′⟩ is seen for both surfaces I and II than for both subareas i and ii, respectively [Figs. 13(a) and 13(b)], while surfaces I and II show lower values of Y_eff than that for subsurfaces i and ii, respectively [Figs. 11(a) and 11(b)]. Hence, ⟨θ′⟩ seems to have a correlation with Y_eff. We found R_RMS is a poor predictor of the effective sputtering yield, as a similar Y_eff ratio of rough and smooth surfaces was found for surface I and for surfaces III and IV (DIII-D SiC), even though R_RMS of surface I (R_RMS = 2 μm) was quite different from that of the other surfaces III (R_RMS = 7 μm) and IV (R_RMS = 8 μm) (Table I).

The angular shifts from ⟨θ⟩ to ⟨θ′⟩ can be understood intuitively by considering the mean surface inclination angle distribution (SIAD) ⟨δ⟩, which is defined by the angle between the local surface normal direction and z-direction of our global coordinate system [Fig. 1(c)] that is perpendicular to the surface plane. The value of ⟨δ⟩ for each surface studied is summarized in Table I. Although the simple formula ⟨θ′⟩ = ⟨θ⟩ - ⟨δ⟩ seems applicable when ⟨θ⟩ is for grazing incidence at α = 89°, this is not applicable as ⟨θ⟩ becomes steeper. The B-field angle α = 85° leads to a mean impact angle ⟨θ′⟩ = 70° (Fig. 13) at which the physical sputtering yield for 100 eV D ions on C and SiC surfaces is maximized (Fig. 3). That makes the physical sputtering yield Y_smooth at α = 85° the highest among other α values for the smooth surface in Fig. 11. Figure 3 also shows that the physical sputtering yield drastically decays when θ > 70°, which makes Y_smooth decrease as α (i.e., ⟨θ′⟩) becomes shallower (Fig. 13). On the other hand, ⟨θ′⟩ is less than or equal to 75° for all surfaces investigated for α = 85–89° due to the surface roughness. Hence, rough surfaces never achieved the value seen for the smooth surface at α = 85° and never showed as low a value for Y_eff as Y_smooth at α = 89°. Y_eff of rough surfaces in Figs. 11(a)–11(c) are re-plotted over the sputtering yields in Fig. 3 at ⟨θ′⟩. Overall, values of Y_eff show good agreement empirically with the sputtering yield values that have been scaled by a factor of 75% (Fig. 3). Hence, ⟨θ′⟩ seems to be a good parameter that can be used to determine the effective sputtering yield from the sputtering yield data scaled by 0.75×. This means that we can simply estimate Y_eff, if we know ⟨θ′⟩, without computational modeling on complicated surfaces, for C and SiC surfaces.

Figures 14(a)–14(c) show the fraction of the nominal sample area that is shadowed from D ions. This is an important parameter for impurity deposition behavior because either ions or neutrals of impurity particles can be deposited on the surface and trapped in those ion shadowed areas.^3,11,23,24 Shadowed area fractions ranging from 20% to 30% were found for surfaces I, II, III, and IV at α = 88° (a typical B-field angle in tokamaks). As a general trend, the shadowed area becomes smaller as α (or ⟨θ⟩) becomes steeper, which is intuitively understandable. The shadowed area in subareas i and ii was much smaller than the shadowed area calculated for the whole surface of surfaces I and II. This indicates that the micro-pore or valley structures strongly contribute to the shadowed area, as it was visually demonstrated in the erosion patterns in Figs. 4(b), 8(a), and 10(d). 80% of the pore areas on surface I are shadowed, and 30% of the arc track valleys on surface II are shadowed for α = 88°. The micro-pores are still the significant source of the shadowed area (50% of the pore areas) even at α = 85°. We emphasize that the difference in the R_RMS values for surfaces I (1.9 μm) and IV (8.0 μm) cannot account for the shadowed area of surface I [Fig. 14(a)] being similar or larger than surface IV [Fig. 14(c)], as higher roughness is intuitively considered to make more shadowed area.

The above calculations have revealed that the R_RMS value is not a good predictor of the effective sputtering yield. Figures 15(a) and 15(b) show the mean LIIA distribution ⟨θ′⟩ and shadowed area fractions for a B-field angle α = 85°–89° replotted as a function of the mean SIAD ⟨δ⟩. The SIAD is independent from R_RMS but accounts for the topography of the surface. Both dimensionless parameters, the mean LIIA distribution ⟨θ′⟩ [Fig. 15(a)] and shadowed area fraction [Fig. 15(b)], show a reasonable correlation with the mean SIAD ⟨δ⟩. We believe that ⟨δ⟩ is a better parameter to characterize surface topography than the surface roughness R_RMS as Cupak et al. emphasized.²⁵ We also highlight the importance of having the correct ⟨θ⟩ value, which is sometimes assumed as α, for tokamak divertor surfaces in regard to ⟨θ′⟩ determination. A linear fit of the mean LIIA distribution and shadowed area fraction as a function of the mean SIAD ⟨δ⟩ was performed for each B-field angle α, and the results are plotted in Fig. 15. The fitting coefficients, a and b, for the linear formula

where y is the mean LIIA distribution ⟨θ′⟩ or shadowed area fraction, are summarized in Table II. Both ⟨θ′⟩ and the shadowed area fraction are dimensionless and only affected by the IADs and surface topology, and not by the materials or incident ion energy. Hence, those empirical formulas are applicable for different PFC materials under the plasma parameters giving the IADs in Sec. II.

We found that the effective sputtering yield is mainly affected by the local incident ion angle (LIIA) θ′, which shifts from the incident ion angle distribution (IAD) θ in the sheath due to the surface topology. Both dimensionless parameters, the mean LIIA distribution ⟨θ′⟩, and shadowed area fraction show a reasonable correlation with the mean inclination angle distribution (SIAD) ⟨δ⟩, but is not correlated with the root mean square surface roughness R_RMS as both dimensionless parameters are affected by the surface topology rather than the scale. We have provided empirical formulas to estimate the mean LIIA distribution ⟨θ′⟩ and shadowed area fraction from the mean SIAD ⟨δ⟩ for the B-field incident angle α = 85°–89°. Those empirical formulas apply to all plasma-facing component materials under the plasma parameters used to calculate the incident IADs. Using the mean LIIA distribution ⟨θ′⟩, the sputtering yield for a rough surface is approximated by a scaled value (0.75×) from the angular dependence data of the sputtering yield for the ideal smooth surface as demonstrated for carbon and SiC surfaces. Using the shadowed area fraction, a better estimation of material deposition is possible using information on the impurity flux.

The authors thank Ane Lasa and Jonathan Coburn for their advice on the MPR code. The authors thank Christopher P. Chrobak, Jerome Guterl, and Igor Bykov for their advice on IAD calculations and measurements. The authors thank Eduardo Marin for providing the SiC topological data. The authors also thank Arturo Domingues and DeeDee Ortiz for their support of the SULI program. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences, using the DIII-D National Fusion Facility, a DOE Office of Science user facility, under Award Nos. DE-AC02-09CH11466 and DE-FC02–04ER54698. This work was also supported by the DOE SULI program, U.S. DOE Contract No. DE-AC02-09CH11466. Z.L. acknowledges support by the Environmental Internship Program in Princeton's High Meadows Environmental Institute.

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

The authors have no conflicts to disclose.

Shota Abe: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Project administration (lead); Software (lead); Supervision (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). Charles Hubert Skinner: Funding acquisition (lead); Project administration (lead); Supervision (lead); Writing – review & editing (equal). Andrew Hua Liu: Data curation (equal); Investigation (equal); Software (lead). Jhovanna Garcia: Data curation (equal); Investigation (supporting); Software (supporting). Zihan Lin: Data curation (supporting); Formal analysis (supporting); Software (supporting). Stefan Bringuier: Data curation (equal); Resources (equal); Writing – review & editing (supporting). Tyler Abrams: Data curation (equal); Formal analysis (supporting); Funding acquisition (lead); Project administration (equal); Resources (equal); Supervision (lead); Writing – review & editing (supporting). Bruce E. Koel: Conceptualization (equal); Funding acquisition (lead); Project administration (equal); Resources (equal); Supervision (lead); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.