Hydrogen in Tungsten: Absorption, Diffusion, Vacancy Trapping, and Decohesion

Hydrogen in tungsten: Absorption, diffusion, vacancy trapping, and decohesion

Donald F. Johnson Department of Chemistry, Princeton University, Princeton, New Jersey 08544 Emily A. Cartera) Department of Mechanical and Aerospace Engineering, and Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544

(Received 14 August 2009; accepted 3 November 2009)

Understanding the interaction between atomic hydrogen and solid tungsten is important for the development of fusion reactors in which proposed tungsten walls would be bombarded with high energy particles including hydrogen isotopes. Here, we report results from periodic density-functional theory calculations for three crucial aspects of this interaction: surface-to-subsurface diffusion of H into W, trapping of H at vacancies, and H-enhanced decohesion, with a view to assess the likely extent of hydrogen isotope incorporation into tungsten reactor walls. We find energy barriers of (at least) 2.08 eV and 1.77 eV for H uptake (inward diffusion) into W(001) and W(110) surfaces, respectively, along with very small barriers for the reverse process (outward diffusion). Although H dissolution in defect-free bulk W is predicted to be endothermic, vacancies in bulk W are predicted to exothermically trap multiple H atoms. Furthermore, adsorbed hydrogen is predicted to greatly stabilize W surfaces such that decohesion (fracture) may result from high local H concentrations.

I. INTRODUCTION thin (1–3 mm) coating on graphite or carbon fiber com- posite tiles, but future PFCs might be composed simply of One of the most crucial aspects to the development of 2,5 a viable fusion power reactor is the choice of materials bulk W. Tungsten retains strength at high temperatures to act as plasma-facing components (PFCs). A large and can sufficiently conduct heat away from the surface, although underlying Cu-based components will likely act amount of current research is aimed at characterizing 2 how various materials can withstand the high-energy as future heat sinks. Ingress, transport, and retention of particle flux imposed on the first walls. For example, radioactive tritium in PFMs is of great concern and the amount of hydrogen isotopes permanently retained in the divertor plate (in divertor-type reactors) may be 6 subjected to particle fluxes up to 1024 m2s1 and tungsten is low compared to carbon-based materials. heat fluxes up to 10 MWm2,1–3 while fluxes at other This is one of the main justifications for using W as a plasma-facing walls can be orders of magnitude smaller. PFM. Another issue is that bombardment by high energy High energy particles can sputter surface atoms or pene- particles causes sputtering of surface atoms and displace- trate micrometers deep, leading to defect formation (e.g., ments of bulk atoms (vacancy/interstitial formation). The sputtering rate of W is low, and W atoms promptly re- vacancies or interstitials) and transmutations (due to 4 neutron bombardment). These processes will slowly deposit when sputtered, resulting in decreased overall erosion. Several groups have investigated blistering on weaken and corrode the plasma-facing material (PFM). 7–9 In addition, PFCs must retain structural integrity at high W surfaces caused by D, but the mechanisms leading temperatures (up to 900 C). While both low-Z (e.g., to blister formation are still debated. Cracking, blistering, C, Be) and high-Z (e.g., W) materials have been consid- and mechanical properties will be affected by hydrogen ered for use in PFCs, tungsten has emerged as one of the concentration and mobility. In light of these concerns, it is most promising materials for use in PFCs.2,4,5 necessary to understand the thermodynamics and kinetics Tungsten has several properties that make it well suited of hydrogen absorption into tungsten surfaces and the for use as a PFM. Current technology employs W as a subsequent behavior of hydrogen in bulk tungsten, including diffusion and trapping. a)Address all correspondence to this author. Tungsten surfaces have been well studied experimental- e-mail: [email protected] ly, since clean, low-Miller-index surfaces are easily ob- DOI: 10.1557/JMR.2010.0036 tainable. Special attention has been paid to the W(001)

pffiffiffi pffiffiffi surface,10–15 which exhibits a ( 2 2)R45 reconstruc- interatomic potentials.37 Recently, DFT-GGA calculations tion (see Fig. 1) below room temperature. At higher were used to obtain potential energy surfaces for H2 on temperatures, a bulk-terminated c(1 1) structure is W(001) and W(110), to predict dissociative sticking prob- observed,10 although ion-scattering experiments have abilities.39 Molecular dynamics simulations predicted a suggested a disordered surface.16 On the reconstructed sticking probability <0.6 for low-energy (3–300 meV) hy- surface, the top layer of atoms displace 0.27 A˚ in the drogen atoms sticking to W(001).37 On W(110), surface [110] direction to form zigzag chains, which is thought diffusion of H/D via tunneling plays an important role to be caused by a strong-coupling charge density wave17 below 130 K.42,43 that stabilizes the nearly half-filled d-bands. Density- Mass spectroscopy measurements detecting hydrogen functional theory (DFT) employing the local-density degassing from bulk tungsten by Frauenfelder44 found approximation (LDA)18 and tight-binding theory19 cal- that H sits at tetrahedral interstitial sites, characterized culations have confirmed this reconstruction. X-ray dif- by an endothermic dissolution energy (1.04 eV) and a fraction experiments20 and DFT calculations21 have also relatively high activation energy (0.39 eV) for H diffu- investigated interlayer relaxation of W(110). Additional- sion [e.g., in another body-centered cubic (bcc) metal, ly, tight-binding theory with a modified fourth moment Fe, the H diffusion activation energy is 0.1 eV45]. approximation was used to calculate relaxation and re- Ultrasoft pseudopotential (USPP)-DFT-GGA calcula- construction on seven different W surfaces.19 tions recently confirmed the tetrahedral site preference46 Numerous experiments have studied the adsorption of hy- and recent PAW-DFT-GGA calculations47 found a low drogen on tungsten surfaces: W(001),22–28 W(110),21,26–30 energy barrier for diffusion between these sites using the W(111),26,28,31 and W(211).32–35 Additionally, theoretical drag method (n.b., this is an unreliable algorithm for investigations of hydrogen adsorption have focused on the finding transition states48). Ion-channeling experi- W(001)36–39 and W(110)38–41 surfaces. Field-emission- ments49 found that D also sits at tetrahedral sites in W. fluctuation measurements29,30 and projector augmented Recent experiments have focused on H, D, and T disso- wave (PAW)-DFT-GGA (generalized gradient approxima- lution and long term retention in tungsten, especially tion) calculations40 found that diffusion of H on the W(110) trapping behavior.50–55 Initial experiments using a In-111 surface occurs between threefold sites with transition states probe estimated that W vacancies can trap one or two (TSs) at bridge sites. However, H prefers lower coordina- Hatoms.55 Recent PAW-DFT-GGA predictions suggest tion bridge sites on the W(001) surface, as observed with vacancies can trap perhaps up to 10 H atoms simulta- low-energy electron diffraction (LEED)22 and confirmed neously, or up to eight atoms sequentially, with binding with molecular dynamics simulations employing reactive energies ranging from 0.4 to 1.2 eV,56 while self-trapping was ruled out.47 Thermal desorption spectroscopy (TDS) experiments have been used to derive trapping energies of around 1.4 eV for H in bulk tungsten,53 while thermal desorption of D in W found three traps with energies of 1.03 eV, 1.34 eV, and 2.1 eV.52 The two lower values were attributed to trapping of the first (1.34) and second (1.03) D atoms at vacancies and the higher value was assumed to be due to trapping near voids. In the current work, we investigate via Kohn-Sham DFT calculations the thermodynamics and kinetics of hydrogen absorption into tungsten and its subsequent bulk interactions. Specifically, we calculate surface adsorption and subsequent subsurface absorption of hydrogen on W(110) and W(001), since these are low-energy, low-index surfaces likely to be present as a wall surface and have been studied experimentally. We identify surface-to-subsurface diffusion pathways on these surfaces and evaluate associated energy barriers and rate FIG. 1. Reconstructed W(001) surface with hydrogen (small spheres) constants. We analyze bulk diffusion, H-V binding, in short-bridge (SB) and long-bridge (LB) adsorption sites. Surface and energy barriers into and out of vacancy trapping layer tungsten atoms (large spheres) shift in the ½110 direction to sites, as well as overall diffusivities. We also estimate produce zigzag chains (dotted line) in the [110] direction. Second free energies and equilibrium constants for H isotope and third layer atoms are dark gray. Schematics of surface diffusion pathways are shown for SB-SB, SB-LB, and LB-LB. H is predicted to absorption into defect-free and defective bulk W. Lastly, diffuse along zigzag chains and to hop occasionally to adjacent chains we analyze hydrogen adsorption at different cover- via LB intermediates; hopping from LB to LB sites is unlikely. ages to predict (via a Born-Haber cycle57) the ease of

316 J. Mater. Res., Vol. 25, No. 2, Feb 2010 D.F. Johnson et al.: Hydrogen in tungsten: Absorption, diffusion, vacancy trapping, and decohesion hydrogen-enhanced decohesion at (110) and (001) planes The second layer atoms also shift 0.05 A˚ ,or20% of in bulk W. We provide calculation details in Sec. II and the displacement of top layer atoms, which is also con- then present and discuss our results in Secs. III and IV. sistent with previous work.18 Experiments at high tem- Finally, we summarize our work in Sec. V. peratures (>400 K) observe no surface reconstruction, so we also simulate this high-temperature phase. On both the reconstructed and unreconstructed surfaces, we II. CALCULATION DETAILS placed H in high and low symmetry sites to determine The Vienna ab-initio simulation package (VASP)58,59 adsorption site preferences. was used to perform spin-polarized DFT calculations. We calculate hydrogen adsorption energies, Eads, (and 60,61 We used the all-electron PAW method within the absorption energies, Eabs, and dissolution energies, Es) frozen core approximation, and treated electron ex- according to: change and correlation with the GGA.62 Spin-polarized ðÞ¼ ðÞðÞ ðÞ; ð Þ calculations were performed due to the presence of an DEads H EWnH EWn ½ EH2 1 unpaired electron on H. Standard PAW ion-electron where W H refers to the cell with an adsorbed potentials were used in which one and six electrons are n (absorbed) H atom, and W is the energy of a W surface treated self-consistently for H and W, respectively. n periodic cell. However, when calculating dissolution The kinetic energy cutoff for the planewave basis set energies, E ,W refers to a bulk periodic cell containing was set to 500 eV; increasing the cutoff resulted in s n n W atoms (n = 16, 54, or 128) and W H contains an variations in the total energy of <1 meV/atom. We used n interstitial H atom. Total energies for D (or T) are iden- k-meshes of 8 8 8, 6 6 6, and 4 4 4 for tical to H, for example, DE (H) = DE (D). However, calculations of bulk 16-atom (2a 2a 2a ), 54-atom ads ads 0 0 0 frequencies (and consequently zero-point energy correc- (3a 3a 3a ), and 128-atom (4a 4a 4a ) 0 0 0 0 0 0 tions and entropy factors) are affected by the mass dif- cells, respectively, where a is the lattice constant. 6 0 ference, as explained below. 6 1 and 8 6 1 k-meshes were used for the W(001) With adsorption energies and clean W surface ener- and W(110) surface calculations, respectively. As a re- gies, we can estimate the ideal fracture energy as a sult, the k-point spacing for the surface cells was <0.028 function of H coverage via a Born-Haber cycle. Follow- (2p/A˚ ), consistent with the bulk calculations. These ing the scheme of Jiang and Carter,57 we calculate the k-meshes converged the total energy to <2 meV/atom. coverage-dependent surface energy (half the ideal frac- The Methfessel-Paxton (MP) smearing method63 was ture energy), g(Y ), according to: used to integrate the Brillouin zone and account for H partial occupancies of the metal near the Fermi level gðYHÞ¼EsðHÞYH=A þ gð0Þ with a smearing width of 0.1 eV, which keeps total þ E ðY ÞY =A ; ð2Þ energy errors below 1 meV. The MP method also allows ads H H accurate forces to be calculated for metals. Structures where YH is the fractional coverage of H, A is the area were considered relaxed when all forces on the ions of the surface periodic cell, g(0) is the surface energy of ˚ were less than 0.025 eV/A. Finally, the energy of H2 the clean surface, and Eads(YH) is the adsorption energy (H) was obtained by placing the molecule (atom) in a per H atom at coverage YH. Here, Es(H) is the dissolu- cubic box with sides of 12 A˚ . tion energy per hydrogen atom. The ground state of tungsten is nonmagnetic with We define the hydrogen-vacancy (H-V) binding ener- a bcc structure. The converged numerical parameters gy, DEb(V+H), as the energy difference between an H discussed above applied within PAW-DFT-GGA pro- atom initially dissolved at a bulk interstitial site and an ˚ duce a lattice constant a0 = 3.171 A and a bulk modulus H atom then dissolved near a vacancy. The H-V binding B = 316 GPa. These are in good agreement with previous energy, DEb(VHn+H), for sequential additional hydro- ˚ 64 DFT-LDA calculations (a0 =3.163A ) and experiments gen atoms is then calculated as: ˚ 65 (a0 =3.16A, B = 323.2 GPa ). Surface slab models consisted of seven layers, with the top four layers allowed DEbðÞ¼VHn þ H EðW53VHnþ1ÞþEðW54Þ to relax and the bottom three layers held fixed to the bulk EWðÞ53VHn EðW54HÞ ; ð3Þ structure to mimic a semi-infinite crystal. At least 10 A˚ of vacuum is added to separate slabs from periodic images. where E(W53VHn) refers to the total energy of n As mentioned above, the W(001) surface undergoes a H atoms surrounding a vacancy in bulk W, E(W ) refers 54 reconstruction in which atoms shift slightly in the ½110 to the total energy of bulk W, and E(W54H) is the total direction to form zigzag chains (Fig. 1). This is the low- energy of an H atom residing at tetrahedral interstitial temperature phase. Our calculations predict displace- site of bulk W. In each case, the energy of bulk W refers ments of 0.27 A˚ with an energy gain of 0.14 J/m2, to a 54-atom periodic cell of W, whereas the cases with which agrees well with previous DFT calculations.18,39 vacancies are calculated with 53 W atoms in the periodic

J. Mater. Res., Vol. 25, No. 2, Feb 2010 317 D.F. Johnson et al.: Hydrogen in tungsten: Absorption, diffusion, vacancy trapping, and decohesion cell, with the volume fixed at the equilibrium volume of perfect bulk W. The binding energy for the first trapped H, DEb(V+H) is given simply by Eq. (3) with n set to zero. To compare to later experiments, we also consider the trapping energy at a vacancy as the sum of the activation energy for bulk diffusion, Ea, and the vacancy binding energy, DEb, as explained below. The climbing-image nudged elastic band (CI-NEB) method was used to determine minimum energy pathways (MEPs) for diffusion of H into surfaces and through bulk W.48 Atomic positions are linearly interpolated to set up initial guess images (interpolated structures) along the pathway that are connected by artificial springs to construct a “band.” The band is relaxed according to the CI-NEB algorithm to obtain the MEP and the TS. The energy difference between the minimum FIG. 3. Energy profile for hydrogen (small sphere) diffusion into W(001) from the long-bridge surface site via Path II (see text). The and the TS is Ea. As a post-relaxation analysis, the movement of the H atom (neglecting W atom displace- structure of the surface at critical points is displayed in the insets. ments) is used to define the reaction progress, as plotted in Figs. 2–7. The absolute displacement of the H atom between neighboring images is summed to give the cu- mulative distance of the nth image. Vibrational frequencies are calculated at critical points to confirm minima and characterize the nature of each TS; more than one imaginary frequency indicates a higher order saddle point. Finite displacements of 0.015 A˚ are made on the H atom to construct a numerical second derivative (Hessian) matrix from finite differences of the forces induced by these displacements. Tungsten atoms are kept fixed under the assumption that lattice vibrations of the heavy metal atoms are sufficiently decoupled from the vibrations of the lighter H atom. Therefore, upon diagonalization of the Hessian we obtain three real frequencies at minima and two real frequencies at TSs. FIG. 4. Energy profile for hydrogen (small sphere) diffusion into bulk-terminated W(001), Path III (see text). The structure of the surface at several positions along the path is displayed in the insets.

FIG. 2. Energy profile for hydrogen (small sphere) diffusion into FIG. 5. Energy profile for hydrogen (small sphere) diffusion into W(001) from the short-bridge surface site via Path I (see text). The W(110) from the threefold surface site. The structure of the surface structure of the surface at critical points is displayed in the insets. at critical points is displayed in the insets.

318 J. Mater. Res., Vol. 25, No. 2, Feb 2010 D.F. Johnson et al.: Hydrogen in tungsten: Absorption, diffusion, vacancy trapping, and decohesion

Q 3N initial n ð = Þ khTST ¼ Q i i e Ea kBT ; ð4Þ 3N1 saddle i ni where ninitial and nsaddle correspond to frequencies of the H atom at the initial minimum and saddle point, respectively. As mentioned above, only H is displaced in the frequency calculations, so N = 1 in Eq. (4), giving rise to three frequencies at the minimum and two frequencies at the saddle point. ZPE-corrected Ea are used and entropic contributions are neglected since they are small, as explained below. The pre-exponential frequency factor is correspondingly adjusted for D and T isotopes, as described above. Bulk diffusivity is calculated from the empirical expression for the Arrhenius diffusion constant: FIG. 6. Energy profile for diffusion of H (small sphere) through bulk W (gray) where the diffusion distance is the absolute displacement of = n = D ¼ D e Ea kBT ¼ a2n e Ea kBT ; ð5Þ H. Only the six nearest atoms are displayed. At the transition state H 0 6 0 is coplanar with the three nearby W atoms. Inclusion of ZPE-corrections lowers the barrier to Ea =0.38eV. where n is the number of distinct escape pathways from a given minimum, a is the jump length, and n0 is the pre- exponential factor in Eq. (4). Thermodynamic equilibrium constants, K, are calcu- ðÞ = lated from K ¼ e DG kBT where the Gibbs free energy is DG = DH – TDS. We approximate DH = DU since our calculations are at constant volume. We approximate the entropy changes as entirely vibrational entropy, Svib, which is expressed as68:

X3N hni=kBT Svib ¼ kB ln 1 e i = 1 þ hni e hni kBT 1 ; ð6Þ kBT

where ni are the frequencies. Since the H concentration is small, configurational entropy changes are small and FIG. 7. Energy profile for hydrogen (small sphere) diffusion from a can be safely neglected. Here, the initial state is nearby tetrahedral site to an octahedral site at a vacancy in bulk H adsorbed on either surface and the final state is either tungsten. Only atoms nearest the defect are shown. H dissolved in bulk or H dissolved at a vacancy.

III. HYDROGEN ON TUNGSTEN SURFACES Zero-point energy (ZPE) corrections were calculated at A. Adsorption of hydrogen on W(001) and W(110) critical points from the vibrational frequencies: EZPE = ½ Shnj,wherenj are the real frequencies. For example, On the W(001) surface, we simulate adsorption of the vibrational frequency of the H2 molecule was cal- H on sites on the reconstructed surface (low-temperature culated to be 4325 cm1, which is close to the ex- phase) and the bulk-terminated, or “unreconstructed” perimental value 4301 cm 1,66 and corresponds to surface (high-temperature phase). Surface reconstruction EZPE(H) = 134 meV. Frequencies for D and T are derived creates short-bridge (SB) sites and long-bridge (LB) by multiplying by thep squareffiffiffiffiffi root ofp theffiffiffiffi ratio of sites with W-W distances of 2.82 A˚ and 3.57 A˚ , respec- 1= masses. Therefore, nD ¼ p½ffiffiffiffiffinH and nT ¼ 3 nH,which tively. H is predicted to preferentially adsorb on the presultsffiffiffiffiffi in: EZPEðDÞ¼ ½ EZPEðHÞ and EZPEðTÞ¼ SB site on reconstructed W(001) (Fig. 1) with Eads = 1= 3 EZPEðHÞ. 0.92 eV/atom (0.88 eV/atom with ZPE corrections) Rate constants (k) for diffusion (bulk and surface-to- [Eq. (1)]. The SB site contracts further when H is subsurface) are calculated within the harmonic transition adsorbed, decreasing to a W-W distance of 2.64 A˚ . state theory (hTST) according to67 The LB site was also found to be a local minimum, with

J. Mater. Res., Vol. 25, No. 2, Feb 2010 319 D.F. Johnson et al.: Hydrogen in tungsten: Absorption, diffusion, vacancy trapping, and decohesion

Eads = 0.49 eV/atom (0.46 eV/atom w/ZPE) and a with the second desorption peak. The peak correspond- contracted W-W distance of 3.21 A˚ . ing to the smaller activation energy is likely due to Adsorption on the bridge site of the bulk-terminated desorption at higher coverages (again, lateral repul- surface was found to be more exothermic, Eads = sions between adsorbed H atoms will lower the de- 1.20 eV/atom (1.16 eV/atom w/ZPE), but the sorption activation energy). adsorbed H atom induces some surface reconstruction which could be responsible for the large energy gain. B. Surface diffusion of hydrogen The W-W distance on the clean, unreconstructed surface Surface diffusion of H on W(110) has been well stud- is 3.171 A˚ , but the W-W distance contracts to 2.61 A˚ ied theoretically40 and experimentally30; we therefore across the bridge site when H adsorbs, i.e., when the W focus our attention on the W(001) reconstructed surface. atoms relax toward the adsorbed H. Diffusion on the unreconstructed surface is difficult to Previous molecular dynamics (MD) simulations using model since the structure is metastable in our calcula- a reactive interatomic potential predicted weaker bind- tions, which are performed at 0 K. We identify three ing: E = 0.38 eV/atom37 while DFT-GGA calcula- ads possible pathways to describe hops between sites: SB- tions predicted a molecular adsorption energy of 1.9 SB, LB-LB, and SB-LB (Fig. 1). The energy barrier for eV,41 equivalent to a dissociative adsorption energy per SB to SB hopping is predicted to be 0.40 eV (0.35 eV H atom of E = 0.95 eV/atom, which agrees well ads w/ZPE). The energy barrier for an SB to LB hop is with our results. 0.60 eV (0.53 eV w/ZPE) and the reverse process, LB To compare with experimental data, we assume that to SB, has an energy barrier of 0.17 eV (0.11 eV w/ZPE) the activation energy for desorption is equal to the due to the thermodynamic stability difference of 0.43 eV adsorption energy and we consider our ZPE-corrected between LB and SB sites. An NEB calculation of LB-LB desorption energies per molecule: 1.76 eV and 2.32 eV diffusion predicted a barrier of 0.44 eV, but the saddle for the reconstructed and bulk-terminated surface, re- point is second-order, suggesting direct LB-LB hops will spectively. This assumption of equality between ad- not occur. Overall, surface diffusion in the low coverage sorption and desorption energies should be valid here, regime will consist of SB-SB hopping and occasional since H dissociation barriers on most transition metal 2 SB-LB hops. As a result, H is expected to diffuse along surfaces are small or zero.69 TDS experiments for high zigzag chains of surface W atoms in [110] directions coverages of hydrogen on W(001) observed a desorp- (Fig. 1). A higher energy, two-step process, SB-LB-SB, tion peak corresponding to 1.40 eV.26 Our values are allows an H atom to move to an adjacent zigzag chain. higher, but they reflect ¼ ML coverage where H-H repulsions are much smaller. At 1.0 ML coverage (one H per surface W), we predict weaker binding due to C. Hydrogen absorption into W(001) surfaces increased H-H repulsions: Eads = 0.68 eV/atom Three possible pathways were considered for surface- (0.63 eV/atom w/ZPE) and Eads = 0.80 eV/atom to-subsurface diffusion of H into W(001). Paths I and II (0.75 eV/atom w/ZPE) for reconstructed and bulk- involve diffusion into the reconstructed surface from the terminated W(001), respectively. The resulting molecu- SB and the LB surface sites (Figs. 2 and 3, respectively). lar desorption energy for the reconstructed surface Path III considers diffusion into the bulk-terminated (1.26 eV) agrees fairly well with the high coverage data, surface from a bridge site (Fig. 4). Stable subsurface 1.40 eV. However, since TDS experiments obtain these minima were identified and MEPs were calculated to desorption peaks at 800 K, one would expect to see determine inward and outward energy barriers. In all the high-temperature phase, that is, the bulk-terminated three pathways, diffusion is simulated for H penetrating surface. The predicted molecular desorption energy past the first and second layers. (1.50 eV) for the bulk-terminated surface is in slightly Stable subsurface sites for H were found to be tetrahe- better agreement with the experimental data (1.40 eV). dral interstitials in all cases. H atoms placed in subsur- The most favorable adsorption site on the W(110) face sites between the surface layer and second layer of surface is predicted to be the threefold hollow site reconstructed W(001) surface were unstable and rose to with Eads = 0.75 eV/atom (0.71 eV/atom w/ZPE). the surface. However, a stable subsurface site was found This is more exothermic than previous PAW-DFT-PBE in the first subsurface layer of bulk-terminated W(001) calculations which predicted an atomic adsorption en- (Fig. 4, top left inset). Energy differences (Eabs – Eads) ergy of 2.846 eV at y = 0.5 ML coverage,40 equiva- and associated barriers with ZPE-corrected values in lent to a dissociative adsorption energy of Eads = parentheses are listed in Table I. For the purposes of 0.59 eV/atom. TDS exhibited desorption peaks cor- comparison, the energy differences listed in Table I are responding to activation energies of 1.17 eV/H2 and reflective of H absorbed to the same penetration depth 26 1.42 eV/H2. Our calculated Eads per molecule with into the subsurface in all three pathways. For example, ZPE corrections is 1.42 eV, in perfect agreement the “First step” refers to diffusion from the surface to the

320 J. Mater. Res., Vol. 25, No. 2, Feb 2010 D.F. Johnson et al.: Hydrogen in tungsten: Absorption, diffusion, vacancy trapping, and decohesion

TABLE I. Adsorption energy Eads, first subsurface absorption energy Eabs(sub1), second subsurface absorption energy Eabs(sub2), endothermicities of the first step Eabs(sub1) Eads(surf) and the second step Eabs(sub2) Eabs(sub1), and associated energy barriers for surface-to-subsurface diffusion of H into W (see also Figs. 25). All values reported in eV. Values in parentheses are ZPE corrected.

First step Second step

Eabs Eabs Eabs Inward Outward Eabs (sub2) Inward Outward Surface Eads (sub1) (sub2) (sub1)Eads barrier barrier Eabs (sub1) barrier barrier W(001) Path I 0.92 1.07 0.99 1.99 2.00 0.01 0.08 0.14 0.22 (from SB) (0.88) (1.20) (1.12) (2.08) (2.02) (0) (0.09) (0.08) (0.17) W(001) Path II 0.49 0.74 0.87 1.23 1.24 0.01 0.13 0.42 0.29 (from LB) (0.46) (0.85) (1.01) (1.31) (1.29) (0) (0.15) (0.41) (0.25) W(001) Path III 1.20 0.95 0.87 2.16 2.16 0 0.08 0.10 0.17 (unreconstructed) (1.16) (1.09) (1.00) (2.25) (0.09) (0.03) (0.12) W(110) 0.75 0.93 1.11 1.69 1.74 0.06 0.18 0.26 0.08 (0.71) (1.05) (1.24) (1.76) (1.77) (0.01) (0.19) (0.20) (0.01)

subsurface site coplanar with second layer W atoms in corresponding Path I site. An overall mechanism in- all three (001) pathways. volving SB-LB equilibration (endothermicity DE = Absorption sites in-plane with the second layer (sub1 0.43 eV) followed by absorption into the subsurface in Table I) were found to have Eabs = 1.07 eV, 0.74 eV, along Path II (first step endothermicity, 1.23 eV, plus and 0.95 eV for sites along Path I, Path II, and Path III, second step barrier, 0.42 eV, see Table I) has a slightly respectively. Surface-to-subsurface diffusion along Paths smaller total barrier (2.08 eV) than subsurface absorp- I and III is energetically costly, with zero-point- tion following Path I (2.13 eV). The overall barrier is the corrected endothermicities > 2 eV. The endothermicity energy difference between the initial state and the high- along Path II is less only because the initial LB adsorp- est energy TS. In Path II, the Eabs for H in the subsurface tion site has higher energy than the other initial adsites. are both close to the bulk dissolution energy Es (0.96 eV) The absorption energies for H in the final sub-subsurface and the second step barrier (0.42 eV) is close to the bulk sites (sub2, between the second and third layer) are still diffusion barrier (see Sec. IV. A), which indicates that endothermic: Eabs = 0.99 eV (1.12 eV w/ZPE), Eabs = once H penetrates past the second layer, diffusion behav- 0.87 eV (1.01 eV w/ZPE), and Eabs = 0.87 eV (1.00 eV ior is bulk-like. w/ZPE) along Paths I, II, and III, respectively. These Rate constants for H diffusion into the subsurface values are similar to the bulk dissolution energy, Es = were calculated according to Eq. (4), where the activa- 0.96 eV (1.09 eV w/ZPE), suggesting that surface effects tion energy is the ZPE-corrected energy difference be- persist only a few layers deep. Still, the 0.1 eV differ- tween the initial state (surface adsorption) and highest ence between H absorbed under the SB site and under energy TS along the pathway. We predict k =6.72 1012 –1 12 the LB site indicates that the surface reconstruction has a exp(–2.17 eV/kBT)s , k =5.24 10 exp(–1.72 eV/kBT) –1 12 –1 fairly large effect on the sub-subsurface sites, and indeed s ,andk =7.38 10 exp(–2.28 eV/kBT)s for Paths I, the second layer W atoms shift parallel to the surface II, and III, respectively. However, if we consider Path II to plane as mentioned earlier. proceed starting from an initial SB adsorption site, then In the Path I scheme, H starts at the SB site and moves the rate constant becomes k =5.66 1012 exp(–2.14 eV/ –1 into the surface by passing between the two surface kBT)s . Equilibrium constants for H moving from a SB W atoms forming the short bridge (see Fig. 2 insets). site into the bulk are reported in Table II; we see that To accommodate the H, the two W atoms move apart H dissolution into perfect bulk W is thermodynamically slightly, from an initial separation of 2.64 A˚ to a maxi- very unfavorable. mal separation of 3.75 A˚ when H is in-plane with the surface layer atoms. By comparison, there is less dis- D. Hydrogen absorption into W(110) placement of surface W atoms in the Path II scheme Two stable interstitial sites were identified in the sub- since H is diffusing into an LB site where the W-W surface layer of W(110). The surface-to-subsurface diffu- distance is large (3.21 A˚ ) initially. sion of H into W(110) is also endothermic (Fig. 5 and Even though the SB site is the more stable adsorption Table I). From the threefold surface site, H first moves site, the subsurface site under the LB site is the more to a subsurface site halfway between the surface layer stable absorption site. The second step of the process is and second layer (Fig. 5 insets), with Eabs =0.93eV exothermic for Path I, but endothermic for Path II [1.05 eV w/ZPE; Eq. (1)]. H then moves to a nearby (Table I). Still, the second subsurface site (between layer subsurface site closer to the second layer, with Eabs = two and three) along Path II is more stable than the 1.11 eV (1.24 eV w/ZPE). At the first subsurface site,

J. Mater. Res., Vol. 25, No. 2, Feb 2010 321 D.F. Johnson et al.: Hydrogen in tungsten: Absorption, diffusion, vacancy trapping, and decohesion

TABLE II. Equilibrium constants for H moving from an initial adsorbed state to a final dissolved state (bulk or vacancy). Gibbs free energies, DG (eV), and equilibrium constants, K, are reported at 500 K, 900 K, and 1450 K.

W(001) (001) Surface ! bulk (001) Surface ! vacancy

T (K) DGK DGK 500 2.02 4.45 1021 0.693 1.04 107 900 2.01 2.16 1013 0.773 1.35 105 1450 1.96 1.55 107 0.968 4.31 104 W(110) (110) Surface ! bulk (110) Surface ! vacancy T (K) DGK DGK 500 1.85 2.09 1019 0.527 4.88 106 900 1.85 2.13 1012 0.615 1.34 104 1450 1.82 4.79 107 0.828 1.33 103

H is coordinated to two surface W and two subsurface W surement (1.04 eV).44 ZPE-corrected dissolution ener- ˚ ˚ atoms, 1.01 A below the surface layer and 1.22 A above gies for D and T are Es = 1.05 eV and Es = 1.04 eV, the second layer. At the second subsurface site, H is co- respectively. The octahedral interstitial site was deter- ordinated to one surface W and three subsurface W atoms mined to be a higher order saddle point. The data that anditis1.56A˚ belowthesurfacelayerand0.66A˚ follow for bulk diffusion, vacancy formation, and H-V above the second layer. Even at this shallow depth, Eabs is trapping were obtained using a 54-atom cell (3ao per close to Es, similar to the energetics of H in the (001) side). subsurface layers. The barrier for the first step is 1.74 eV Hydrogen diffusion through tungsten is modeled as (1.77 w/ZPE) and the second step barrier is 0.26 eV jumps between tetrahedral interstitial sites in the crystal (0.20 eV w/ZPE) (Table I). Energy barriers for further lattice. A diffusing H atom passes through an asymmet- penetration into W are expected to approach bulk diffusion ric triangle to reach a neighboring interstice and is behavior discussed below. coplanar with these three atoms at the TS (Fig. 6). We As was done for W(001), we calculate the rate predict an energy barrier for diffusion of Ea = 0.42 eV constants [Eq. (4)] and equilibrium constants for absorp- (0.38 eV w/ZPE) (Fig. 6). Recent PAW-DFT-GGA cal- tion into W(110). For the first step we predict k1 = culations predicted an energy barrier of 0.20 eV for this 12 –1 47 9.07 10 exp(–1.77 eV/kBT)s . Combining the two pathway, but we believe this result is questionable 13 steps together results in a rate constant k12 = 1.07 10 because the drag method was used, which is known to –1 48 exp(–1.96 eV/kBT)s , which suggests H uptake into the often miss the actual saddle point. Moreover, no vibra- W(110) surface will be enhanced relative to the W(001) tion frequency analysis was reported in Liu et al.,47 so surface. Here, k12 = k1k2/k–1, the product of the rate we do not know whether they found an actual transition constants of the two inward steps (k1 and k2) divided by state to which a barrier can be associated. the rate constant for outward diffusion of the first step Following Wert and Zerner’s formulation of diffusion 70 (k–1). Finally, overall equilibrium constants for H moving through a bcc lattice and the harmonic transition state 67 7 2 from the W(110) surface to bulk at several different tem- theory, we calculate D0 = 8.93 10 m /s from peratures are reported in Table II. Although again insig- Eq. (5). This gives an overall diffusivity of D = 8.93 7 2 nificant quantities of H are predicted to dissolve in 10 exp(0.38 eV/kBT)m/s, which is close to Frauen- perfect bulk W, the favorable thermodynamics of H dis- felder’s experimental value: D = 4.1 107 exp(0.391 2 71 solution near vacancies increases H absorption by many eV/kBT)m/s. This good agreement lends credence to orders of magnitude, as we shall discuss further below. our predictions for the kinetics of surface diffusion and diffusion into the subsurface discussed above. IV. HYDROGEN IN BULK TUNGSTEN Entropic barriers are usually negligible, especially when the diffusing atom has little effect on the metal A. Bulk diffusion lattice. Using Eq. (6) to calculate the entropy barrier þ= Dissolution energies were calculated [Eq. (1)] to be resulted in entropy contributions, eDS kB , that range be- Es = 0.95 eV, 0.96 eV, and 0.87 eV for H in a tetrahedral tween 0.93 and 1.25 for all isotopes and temperatures interstitial site in a 16-atom, 54-atom, and 128-atom considered. Since our entropy factors, which simply periodic cell, respectively. Frequency analysis con- scale the diffusion coefficients, are all 1, we neglect firmed the tetrahedral site to be a minimum, in agree- them in our diffusivity calculations. ment with experiment,44,49 and recent USPP- and Upon adjusting frequencies and ZPE corrections PAW-DFT-GGA calculations.46,47 Incorporating ZPE- to account for isotope mass, we obtain diffusivities 7 corrections results in an Es = 1.09 eV for the 54-atom for D and T, respectively, as D =6.32 10 exp(0.39 2 7 2 cell, in excellent agreement with Frauenfelder’s mea- eV/kBT)m /s and D =5.16 10 exp(0.40 eV/kBT)m /s.

322 J. Mater. Res., Vol. 25, No. 2, Feb 2010 D.F. Johnson et al.: Hydrogen in tungsten: Absorption, diffusion, vacancy trapping, and decohesion

2 TABLE III. Predicted diffusivity, D (m /s), of H, D, and T through W TABLE IV. Hydrogen-vacancy binding energies DEb [eV; Eq. (3)] calculated from Eq. (5) at relevant temperatures. for increasing numbers of hydrogen atoms. Also given are the H-H ˚ distances dH-H (A) for nearest neighbors (n) and atoms on opposite Temperature Hydrogen Deuterium Tritium sides of the vacancy (o). Up to six H atoms are predicted to exother- 500 1.38 1010 7.32 1011 5.27 1011 mically bind to a vacancy. Values in parentheses are ZPE corrected. 900 6.81 109 4.11 109 3.13 109 Cluster Process DE (VH +H) d 1450 4.33 108 2.78 108 2.17 108 b n H-H VH V + H 1.24 (1.41) N/A

VH2 VH + H 1.24 (1.40) 2.55 (o) VH3 VH2 +H 1.12 (1.14) 2.12 (n), 2.59 (o) Overall diffusivities for H, D, and T are listed in Table III VH4 VH3 +H 1.00 (1.12) 2.07 (n), 2.65 (o) for several temperatures. The diffusivity increases by VH5 VH4 +H 0.94 ( 0.91) 1.91 (n), 2.23 (n), 2.70 (o) nearly three orders of magnitude over a 1000 K tempera- VH6 VH5 +H 0.54 ( 0.79) 1.95 (n), 2.76 (o) ture range for all three isotopes. The differences in diffusivity between the three isotopes are greatest at the lowest As seen in other bcc metals, vacancies can trap multi- temperature, as expected. However, they only differ at ple H atoms.73,74 Additional H atoms placed into octahe- most by roughly a factor of two. dral sites surrounding the vacancy are predicted to bind exothermically to the vacancy (Table IV). In other B. Vacancy trapping words, it is energetically favorable for an H atom to A vacancy was created by removing one W atom from a move from a normal bulk interstitial site to a vacancy, 54-atom cell and the vacancy defect formation energy was even if there is already an H atom near the vacancy. The calculated to be 3.44 eV according to the conventional six H-V binding energies listed are all negative, or exo- expression of Gillan,72 which employs a periodic cell con- thermic, suggesting that a vacancy can trap up to at least taining the vacancy with a volume scaled by (N –1)/N, six H atoms. Recent work predicted up to 10 H atoms where N is the number of atoms. This differs only slightly could be trapped simultaneously at a single vacancy, 56 from the constant volume vacancy formation energy, ultimately forming H2. Since it is much more energeti- which is predicted to be 3.32 eV, slightly larger than cally favorable for H to sit near a vacancy rather than at recent USPP- and PAW-DFT-GGA predictions.46,47 Thus, a bulk interstitial site, H atoms can stabilize vacancies to mimic a macroscopic bulk sample with a small concen- (and vice versa), an effect also seen in other metals.75–77 tration of vacancies and hydrogen, we keep the cell lattice With just three atoms binding to a vacancy, the energy parameters fixed at perfect bulk W equilibrium values gain (3.60 eV) outweighs the energy cost to create a when adding H atoms to the vacancy defect. We therefore vacancy (3.32 eV). calculate the vacancy binding energies [Eq. (3)] using We can understand the trends in H-V binding energies constant volume energies throughout (where the lattice by considering H-H distances and subsequent repulsions parameters are those of perfect bulk W). that result. The second H placed into an octahedral When trapped near a vacancy, H is predicted to sit at site opposite the first H, has a binding energy Eb = an octahedral-like site where it is coordinated to five W 1.24 eV, equal to the first binding energy. This config- atoms, the sixth site being the vacancy (Fig. 7, right uration is 0.05 eV lower energy than two H atoms in inset). It is energetically favorable for an H atom to reside neighboring octahedral sites around the vacancy. The near a vacancy relative to a bulk tetrahedral site with a distance between octahedral sites surrounding a vacancy ˚ ˚ binding energy of Eb = 1.24 eV (1.41 eV w/ZPE). with no H atoms is 2.222 A (3.142 A) for neighboring Equilibrium constants for adsorbed H in equilibrium (opposite) sites. This suggests some H-H repulsion with vacancy-trapped H (Table II) suggest the pres- occurs when atoms are close, but there is little interac- ence of vacancies will dramatically increase overall tion when atoms sit far apart, explaining the identical H uptake. binding energy for the first and second H atoms to reach We also characterized an MEP for H hopping into and the vacancy. When three, four, five, or six atoms sur- out of a vacancy trap from a nearby interstitial site round the vacancy, H-H repulsion causes some atoms to (Fig. 7). The dissolution energy at this initial tetrahedral move out of pure octahedral positions. For example, in site (0.66 eV) is already lower than a normal bulk tetra- the stable five-atom configuration, four atoms (forming hedral site (0.96 eV). We predict that H diffuses between a square) move away from the fifth atom and toward the three W atoms (similar to bulk diffusion) to reach the empty octahedral site. The four-atom configuration is a octahedral site at the vacancy with a negligible barrier square of H atoms—distorted slightly into a flattened of 0.013 eV (0 w/ZPE) and an overall exothermicity, tetrahedron—with nearest-neighbor distances dH-H = DE = 0.94 eV (1.10 eV w/ZPE), consistent with re- 2.07 A˚ . The nearest-neighbor distance is an average cent PAW-DFT-GGA predictions.56 Therefore, if an H of 2.07 A˚ for five H atoms and decreases to 1.95 A˚ for atom is close to a vacancy, it will be readily trapped. six H atoms. The crowding of these H atoms produces

J. Mater. Res., Vol. 25, No. 2, Feb 2010 323 D.F. Johnson et al.: Hydrogen in tungsten: Absorption, diffusion, vacancy trapping, and decohesion successively larger repulsions and hence successively (0.50 eV/A˚ 2). With reconstruction, 2g is reduced to 7.75 weaker binding energies. J/m2 (0.48 eV/A˚ 2). At 0.50 ML of H, three different Previous thermal desorption studies assumed a maxi- surface arrangements were consideredp inffiffiffi thep recon-ffiffiffi mum of two H atoms (or D or T) being trapped per structed W(001) surface cell: two in a ( 2 2 2)R45 vacancy. Previous measurements found the trapping cell and one in a (22) cell. A rotation was performed to energy for H in bulk W to be 1.40 eV.53 For D, account for zigzag chains in the [110] direction (Fig. 1) two trapping energies were measured to be 1.03 eV and and k-point spacing for this cell was keptpffiffiffi belowpffiffiffi 0.028 1.34 eV.52 These values reflect an overall energy barrier (2p/A˚ ). A “ridged” arrangement in a p( 22 2)R45 to move H from a trap to the bulk, which is estimated configuration, where all SB sites on one zigzag chain from the sum of the vacancy binding energy and contain an adsorbed H while the adjacent chain contains the migration energy (activation energy for diffusion): zero adsorbed H atoms, resulted in thep highestffiffiffi p energyffiffiffi Eb + Ea. Using our calculated values (with ZPE) of (Eads = 0.73 eV/atom). A staggered p( 22 2)R45 1.41 eV and 0.38 eV for these quantities, we predict an configuration, where each zigzag chain contains parallel energy barrier of 1.79 eV associated with the release of W-H-W bridges (i.e., alternating SB sites are occupied on one H atom from a vacancy trap. A trapping energy of each zigzag chain) that are not parallel to W-H-W 1.40 eV would therefore be equivalent to a vacancy bind- bridges of adjacent chains, was more stable (Eads = ing energy of 1.02 eV, which could correspond with the 0.90 eV/atom). The lowest energy arrangement (Eads = release of the third, fourth, or fifth H atom (Table IV). 0.92 eV/atom) involves SB absorption in a c(22) Thus, the measured trapping energy may be an average configuration with all W-H-W bridges parallel on all of the three different values. Likewise, successive release chains, which maximizes the H-H distances. of a sixth and fifth D atom from a vacancy corresponds to The embrittling effects of H in W are predicted to be trapping energies of 1.10 eV and 1.32 eV, respectively, large compared to Fe and Al.57 In fact, the W(110) which agrees well with the previous measurement.52 surface is predicted to cleave exothermically for high concentrations of H. The origin of this effect is the large C. Hydrogen-enhanced decohesion Es for H in bulk W, which appears in Eq. (2). High energy H atoms incident on a W surface will penetrate We next consider the energetics involved in hydrogen- and come to rest deep in the bulk. If the local concentra- induced fracture of W between (001) or (110) planes, tion of H atoms is high enough, the system can lower its by construction of a Born-Haber cycle [see Eq. (2)]. total energy by cleaving along (110) planes, as predicted The exothermic adsorption of H atoms serves to stabi- by the negative gY/g0 in Fig. 8. This may be an explana- lize the surface and decrease the fracture energy, as tion for the surface-orientation dependence of blistering seen in Fig. 8. The ideal cleavage energy (2g)ofW 2 observed on W surfaces under exposure to deuterium along a (001) plane without reconstruction is 8.04 J/m plasma beams.7 However, while exothermic cleavage is a driving factor, it is not the only factor to consider, since there is an energy cost associated with plastic deformation that occurs with blistering. As noted previously,57 the Born-Haber cycle is a valid means to calculate ideal fracture energies as a function of local hydrogen concentration, if the dissolution energy Es does not vary with H concentration and diffusion barriers are small. The outward diffusion barriers (Table I) are small, but the diffusion barrier through bulk W is fairly large. Therefore, cleavage is only likely to occur when H atoms are concentrated locally. Due to the favorable trapping near vacancies, pockets of high H concentration are likely to be present. Thus, under high neutron fluxes that produce vacancies and high fluxes of H (or D/T) atoms, our calculations suggest vacancies can become nu- cleation sites for spontaneous crack growth and cleavage, FIG. 8. Ideal fracture energies as a function of hydrogen coverage in possibly leading to blistering. (001) and (110) planes in bulk W [Eq. (2)], as compared to fracture energies in Fe(110) and Al(111) (*Ref. 57). Values are normalized to the ideal fracture energy of the metal in the absence of hydrogen. Due V. SUMMARY AND CONCLUSIONS to the high dissolution energy, cleavage along W(110) planes is energetically favorable for high local H concentrations, that is, when The kinetics of hydrogen absorption into tungsten, enough atoms are available to form 1.0 ML on both cleaved surfaces. trapping near vacancies, and the effect hydrogen has on

324 J. Mater. Res., Vol. 25, No. 2, Feb 2010 D.F. Johnson et al.: Hydrogen in tungsten: Absorption, diffusion, vacancy trapping, and decohesion the cohesive properties of bulk tungsten have been Energy) for funding and to NAVO DRSC for computer analyzed with PAW-DFT-GGA calculations. Predicted time. Figures 1–7 were created with the help of VESTA adsorption energies are in good agreement with experi- visualization software.80 ment, as are bulk diffusion constants. We also investigated diffusion of H on the reconstructed W(001) surface, which REFERENCES is predicted to occur along zigzag chains. Absorption energies within the first three layers of W 1. R. Toschi, P. Barabaschi, D. Campbell, F. Elio, D. Maisonnier, and D. Ward: How far is a fusion power reactor from an experi- are close to the highly endothermic bulk dissolution en- mental reactor. Fusion Eng. Des. 56–57, 163 (2001). ergy. Energy barriers for surface-to-subsurface diffusion 2. H. Bolt, V. Barabash, W. Krauss, J. 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