Hydrogen induced vacancy formation in tungsten

S.C. Middleburgh, R.E. Voskoboynikov, M.C. Guenette, D.P. Riley

1 Institute of Materials Engineering, Australian Nuclear Science and Technology Organisation,

Lucas Heights, New South Wales, Australia

Abstract

Atomic scale modelling methods have been used to study the change in vacancy population

when H is introduced into the W bulk matrix. Schottky defects are predicted to dominate in

pure W, and the vacancy concentration is expected to be very small. A mechanism whereby

H solutes facilitate vacancy formation has been outlined and a single H interstitial is predicted

to reduce the Schottky formation energy from 2.96 eV to 1.62 eV. Clustering of H interstitials

in W facilitates vacancy formation even further: the vacancy formation energy is predicted to

be 0.74 eV when two bound H interstitials are considered. H has also been shown to affect

the vacancy binding characteristics in W, changing the behaviour from a repelling interaction

in pure W, retarding the formation of small vacancy clusters, to an attractive interaction

when vacancy-H clusters are considered. The changes in defect behaviour predicted, will have

observable implications to operational properties and the application of W in fusion reactor

components.

1 Introduction

The use of high purity tungsten for plasma facing components in thermonuclear fusion devices,

such as first-wall and divertor armour, is well established [1, 2]. Future designs of fusion reactors

rely on tungsten’s unique physical properties of an extreme melting temperature (3410 K) and low

1

atomic sputtering yield to retain integrity in the fusion environment, whilst high values for thermal

conductivity and heat capacity maintain thermal transfer efficiencies.

Although these properties are under continuing development via improved alloy chemistries

[3], coordinated efforts of several major programs [1, 2] have successfully advanced the processing

and fabrication of tungsten components via alternative means, aimed at reducing the impact of

embrittlement and thermal shock constraints of pure tungsten [4]. Foremost amongst these ad-

vances, is the deliberate manipulation of material microstructures, facilitating the use of original

manufacturing techniques at a fraction of the cost and complexity of traditional high temperature,

refractory processing. Secondary effects of microstructural controls, in particular the elongated

grains resulting from hot rolling, have produced tungsten materials with sufficient ductility to al-

low for macroscopic deformation processes, e.g. deep drawing. This microstructural approach has

seen immediate application in the successful manufacture of thimbles as used in multi-component

divertor armour [1, 5] , but remains restricted by lower grain size limits.

A potential solution to grain size refinement is alloy development. Alloying is somewhat prob-

lematic due to the potential for increased transmutation effects [3]. Considering the H-effect on

the mechanical properties of tungsten within fusion plasma in a wider cross-disciplinary context,

it can be applied for facilitated fabrication of structural elements via the use of hydrogen during

processing similar to [6].

To date, the development of tungsten alloys for use in fusion devices has considered primary

damage formation [7]-[9], recovery [10]-[19], swelling [20]-[25], D-T induced transmutations [26]-[33],

radiation embrittlement [10], [34]-[38] and the radiation degradation of dispersions strengthened

tungsten [36], [39]-[44]. Excluding radiation damage effects, tungsten interactions with α-particles

and hydrogen isotopes have primarily focused on blistering [45]-[55], fuzz formation [56, 57] and

deuterium/tritium retention [48]-[51], [58]-[73]. Considering this extensive research effort, material

degradation on a macroscopic scale remains unexpectedly high in certain operating conditions [74]

that include the high temperature, high pressure saturation of tungsten with hydrogen, resulting in

an observed contraction of the unit cell lattice parameters [75]-[102]. This mechanism, was ascribed

to the formation of vacancy-H clusters. The increase in vacancy concentration in the presence of

2

hydrogen far exceeds the expected concentration at thermodynamic equilibrium in the pure metal.

The research that has been undertaken suggested that terminating surfaces act as a source of

Schottky vacancies, while fusion specific investigations identified that Frenkel defects dominate after

recovery from neutron induced cascades. Furthermore, additional studies outlined that residual

interstitial atoms and their clusters form highly mobile dislocation loops that migrate readily away

from damage regions [103, 104], while residual vacancies may trap hydrogen and form vacancy-H

complexes. Related simulation via 1st principles confirmed the potential for hydrogen retention at

tungsten vacancies [105, 106], suggesting a potential activation mechanism for fusion components

via retention of tritium.

There is some experimental evidence to suggest that H isotope induced vacancy formation can

occur. Thermal desorption spectroscopy results have shown 38 eV deuterium ions can be retained

with a trapping energy of 1.45 eV [48]. This energy is higher than that of previously observed low

temperature traps caused by intrinsic defects, and has been designated an ion induced trap [62].

Positron annihilation spectroscopy results have demonstrated that recrystallised tungsten exposed

to 38 eV deuterium ions leads to an increased vacancy concentration, despite 38 eV being well below

the energy required for deuterium to displace tungsten atoms [50, 107].

While vacancy concentration remain consistent under equilibrium conditions, potential mass

de-trapping of hydrogen from vacancies may result from highly non-equilibrium thermal transients.

Such transient thermal excitations may occur during fusion operations, resulting in the formation

of a super-saturation of vacancies within tungsten. These vacancies devoid of hydrogen may cluster,

forming voids. A drastic reduction of the melting point, as observed in the Fe-H system [78], is also

possible.

In this work we consider an alternative mechanism by which hydrogen affects the vacancy form-

ation energy and hence concentration of vacancies in tungsten. Using Density Functional Theory

(DFT) we investigated the energetics of point defect generation in the presence of excess of hydro-

gen relevant to conditions prevalent in fusion applications and tungsten component fabrication. In

Section 3.1 we explored the vacancy concentration in pure tungsten. We then explore the role of

hydrogen in lowering the vacancy formation energy in tungsten. Additional details are provided in

3

Sections 3.2 and 3.3, during which we have considered the role of clustering on vacancy behaviour

in tungsten under higher internal H content.

2 Approach and methodology

DFT calculations using the Vienna Ab-initio Simulation Package [79, 80] were carried out employ-

ing the supplied plane augmented wave (PAW) pseudopotentials with the Generalised Gradient

Approximation exchange correlation as described by Perdew, Burke and Ernzerhof (GGA-PBE)

supplied with the code [81]. A 4×4×4 super-cell of BCC-W was used that contained 128 atomic

sites. A 3×3×3 k-point grid was used with a 500 eV cut-off energy that provided an accuracy over

10−2 eV per unit cell.

The calculated defect energies are used in combination with each-other to evaluate the effect of

H on the formation of vacancies in W (via Schottky and Frenkel processes). They were calculated

using geometrically optimised structures with no symmetry constraints and under zero pressure.

The vacancy formation energies were evaluated by taking the energy difference between a perfect

super-cell and a super-cell harbouring a defect, considering the atoms added or removed from an

arbitrary, non-interacting position from outside of the system (N.B. this is not a defect formation

energy). The calculations employed a standard approach used previously in a number of studies

[82, 83].

The approach chosen was compared to two others: a method that fixed the super-cell containing

a defect to experimental lattice parameters and another method that fixed the lattice parameters

to that of perfect W that DFT predicts under zero pressure. Three alternate potential forms and

three alternate super-cell sizes were also tested. The comparison of the methods is given in the

Appendix. This testing highlighted the variation in results expected with each method and justified

methods used throughout this study.

4

3 Atomic scale modelling results and discussion

The defect energies are reported in Table 1. The most stable H defect clusters were identified by

populating the ( 12 , 12 ,0) sites as carried out in previous work [105].

Table 1: The calculated energy of point defects and vacancy-H clusters in tungsten.

Defect Type Description Kroger-Vink [108] Energy (eV)

W Vacancy - VW 11.23

W Interstitial - Wi 3.21

Divacancy 1st nearest neighbour {2VW}1nn 22.71

2nd nearest neighbour {2VW}2nn 22.09

3rd nearest neighbour {2VW}3nn 22.83

4th nearest neighbour {2VW}3nn 22.75

5th nearest neighbour {2VW}3nn 22.85

H interstitial ( 12, 12,0) site Hi -0.86

Bound H interstitial pair 1st nearest neighbour {2Hi}1nn -2.52

2nd nearest neighbour {2Hi}2nn -1.74

3rd nearest neighbour {2Hi}3nn -2.08

4th nearest neighbour {2Hi}4nn -1.82/-1.31

5th nearest neighbour {2Hi}5nn -1.73

6th nearest neighbour {2Hi}6nn -1.76

Bound H interstitial triplet - {3Hi} -2.18

H on W vacancy - HW 9.04

2H on W vacancy - (2H)W 6.50

3H on W vacancy - (3H)W 4.11

4H on W vacancy - (4H)W 1.82

5H on W vacancy - (5H)W -0.11

6H on W vacancy - (6H)W -1.98

H on 2W vacancies lowest energy arrangement HW 20.42

2H on 2W vacancies lowest energy arrangement (2H)W 17.67

3H on 2W vacancies lowest energy arrangement (3H)W 15.01

4H on 2W vacancies lowest energy arrangement (4H)W 12.44

3.1 Vacancy formation in ideal BCC tungsten

The defect energies reported in Table 1 used together with the energy of a unit of W solid (-8.27 eV1)

can be employed to investigate the equilibrium defect behaviour in the system. The Schottky defect

formation energy in the absence of H can simply be calculated by considering the following equation:

WW → VW + W(s) (1)

1This energy was calculated by dividing the total binding energy of the perfect W super-cell and dividing by thenumber of atoms (i.e. 128 in our case).

5

where WW is a regular W atom on its lattice site, VW is a vacant W site and W(s) is a unit

of W solid [108]. Vacancy formation via reaction (1) is calculated to proceed with Ef=2.96 eV

within a 4×4×4 super-cell. This compares to 3.16 eV in a 3×3×3 super-cell showing a system size

dependency (see Appendix for further details). The change in unit cell volume (δV) observed due

to one vacancy (0.78 % vacancy concentration) compared to the perfect W unit cell volume was

-0.117 A3, a contraction.

Using the laws of mass action [109] the concentration of vacancies in pure W is 1.09×10−8 (at

2000 K, close to 23 Tm [110]). When considering the potential for formation of vacancy clusters it

is essential to consider the binding of two vacancies. This was achieved by evaluating vacancies in

the first nearest neighbour to the fifth nearest neighbour positions. Following the same approach,

the energy of a divacancy is calculated and the corresponding equilibrium concentration varies from

2.83×10−16 (Ef = 6.17 eV) in a 1st nearest neighbour position to 1.26×10−16 (Ef = 6.31 eV) in a

5th nearest neighbour position.

The binding energies were calculated using the following reaction:

2VW → {2VW}x .n.n. (2)

The binding energies for 1st to 5th nearest neighbour vacancies were calculated to be 0.24 eV,

0.63 eV, 0.36 eV, 0.28 eV and 0.38 eV, respectively. The low concentration and positive binding

energy infer that existence of divacancies in tungsten can be neglected.

By comparison, the Frenkel defect formation energy for W was calculated to be 14.44 eV, con-

siderably higher than the vacancy formation energy via Schottky mechanism and hence Frenkel pair

defects will be at negligible concentration under thermodynamic equilibrium conditions. This con-

firms that the dominant intrinsic defect in pure W is the Schottky defect, as is normal for elemental

metallic systems [111].

Given the intrinsic defect behaviour, one expects W to perform as a typical refractory metal

with high ductile-to-brittle transition temperature. We now suggest a mechanism involving H that

affects the vacancy population, deviating the material properties from the ideal values.

6

3.2 The effect of hydrogen on vacancy formation in tungsten

H solutes, in the absence of W vacancies, will take up an interstitial position. Solution of H gas

into W by the following reaction was predicted to proceed with an energy of 2.53 eV (i.e. hydrogen

solution is not favourable under thermodynamic equilibrium conditions):

1

2H2 → Hi (δV = 0.113 A3 at 0.78 % concentration) (3)

Therefore, H can only be introduced into the system by non-equilibrium methods, for example,

an ion implantation mechanism [112]. Within a fusion device, such phenomena occur by plasma-

material interactions accompanied by formation of Frenkel defects. However, further vacancies can

be induced in the W system if one considers the role of H interstitials:

WW + Hi → HW + W(s) (δV = −0.015 A3 at 0.78 % concentration) (4)

This vacancy formation was predicted to proceed with an energy of 1.62 eV which is significantly

lower than the 2.96 eV as previously calculated for single vacancy formation in the absence of H.

As a result, the equilibrium concentration of vacancies in W following the introduction of H is

expected to be larger. The estimated decreased Schottky formation energy will result in a more

mobile W lattice, increasing diffusion, potentially lowering the melting temperature, increasing

crystal plasticity, and reducing thermal conductivity. Desorption of H from the W lattice will occur

with temperature and will be considered elsewhere.

The total strain on the lattice is also observed to be far lower when a vacancy traps H. We have

already reported that vacancies have a large negative defect volume (-0.117 A3), while isolated H

interstitial defects have a large positive defect volume (0.113 A3). When combined the vacancy-H

cluster defect volume is -0.015 A3, far smaller indicating that the crystal lattice is less frustrated

and the defects are more stable in a bound state.

The ease of vacancy formation may be increased even further if defect clustering occurs. These

potential effects are dependent on the stability of the bound defects. The binding energy between

two H interstitials as a function of separation is illustrated in Figure 1. A negative binding energy

for nearest neighbour H-H interstitials is evident and becomes significant at a separation distance

less than 3 A. Clustering is therefore considered to be thermodynamically favourable.

7

1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 4 . 5 5 . 0- 1 . 0 0

- 0 . 7 5

- 0 . 5 0

- 0 . 2 5

0 . 0 0

0 . 2 5

0 . 5 0

Bindin

g Ene

rgy (e

V)

H y d r o g e n - H y d r o g e n I n t e r s t i t i a l D i s t a n c e ( A n g s t r o m )

Figure 1: Binding energy of two hydrogen interstitial defects in tungsten as a function of theirseparation (the fourth nearest neighbour is considered from two symmetrically distinct sites at thesame distance).

8

In considering the formation of multi-H clusters in W, the energy for a vacancy to be created

in the presence of a double H interstitial can be calculated by:

WW + {2Hi} → {2H}W + W(s) (δV = 0.002 A3 at 0.78 % concentration) (5)

This reaction is predicted to proceed with an energy of 0.74 eV, which is significant in that it is

lower than both the normal Schottky formation energy in W (2.96 eV) and still lower than the

vacancy formation energy in the presence of a single H interstitial (1.62 eV). Figure 2 illustrates the

effect of the lower vacancy formation energy, plotting the equilibrium concentration of vacancies

predicted by the laws of mass action [109]. Figure 2 was compiled assuming excess of H solutes

within the W lattice and that the material had come to complete equilibrium with its environment.

The change in volume associated with the defect compared to the perfect W lattice is minimised

further indicating even less frustration of the lattice.

0 2 5 0 5 0 0 7 5 0 1 0 0 0 1 2 5 0 1 5 0 0 1 7 5 0 2 0 0 01 E - 1 51 E - 1 41 E - 1 31 E - 1 21 E - 1 11 E - 1 01 E - 91 E - 81 E - 71 E - 61 E - 51 E - 41 E - 30 . 0 10 . 1

1

Vaca

ncy C

once

ntrati

on

T e m p e r a t u r e ( K )

V a c a n c y c o n c e n t r a t i o n w i t h { 2 H } W d e f e c t s V a c a n c y c o n c e n t r a t i o n w i t h H W d e f e c t s V a c a n c y c o n c e n t r a t i o n i n i d e a l t u n g s t e n

Figure 2: Predicted equilibrium concentration of vacancies in tungsten metal with and withoutthe presence of hydrogen solutes in the lattice. Two different types of hydrogen containing defectare considered: (1) single hydrogen defects associated with each vacancy (red line) and (2) twohydrogen defects considered with each vacancy (black dash-doted line). Thermal desorption ofhydrogen from the tungsten lattice is not taken into account.

9

Figure 3 summarises the effect of H interstitials on the formation of W vacancies and provides

the stable geometries that form as a result of vacancy formation. Notice the relaxation of the H

species towards the vacant W site upon creation.

Figure 3: The formation of a vacancy via a Schottky mechanism is shown in pure tungsten (i),in tungsten in the presence of a single H (ii) and in tungsten in the presence of a two bound Hinterstitial defects (iii). Hydrogen species are shown as white spheres, the vacancy as a cube andtungsten species as dark blue spheres.

For completeness, vacancy formation via the interaction of three H interstitial defects may be

expected to proceed if the H interstitials preferentially bind by the following interaction:

Hi + {2Hi} → {3Hi} (6)

However, consideration of this reaction predicts it to proceed with a positive energy of 1.20 eV and

hence, is unlikely. It is therefore predicted that clustering of multiple H interstitials, {xHi}, where

x>3, is highly improbable.

10

Intrinsic vacancies within the W matrix or those formed by irradiation, such as plasma or neutron

interaction, can be considered as potential trap sites. These vacancies attract H interstitials via the

following reaction:

VW + Hi → HW (7)

This reaction proceeds with an energy of -1.33 eV and allows for preferential accommodation of a

H into a pre-existing vacancy. Further H interstitial defects will be attracted to this vacancy, as

previously predicted [105] via the following reactions:

HW + Hi → (2H)W + (−1.69 eV ), (8)

(2H)W + Hi → (3H)W + (−1.53 eV ), (9)

(3H)W + Hi → (4H)W + (−1.44 eV ), (10)

(4H)W + Hi → (5H)W + (−1.07 eV ), (11)

(5H)W + Hi → (6H)W + (−1.01 eV ), (12)

i.e. H solutes have a high driving force to trap around vacant W sites. Conversely, thermal

excitations will result in the opposite reactions causing the dissociation of H from the vacancy.

Although the Frenkel defect formation mechanism is far less favourable in W compared with

Schottky defects, they are still of importance when the W crystal is damaged by irradiation. The

H will have a small effect on the Frenkel formation energy, lowering it to 13.34 eV from an original

value of 14.44 eV. Although this is not a large energy difference, the additional damage imposed

within the structure may be of significance when considering the total life-time structural integrity

of W components.

3.3 Vacancy clustering due to hydrogen

As previously shown in Section 3.1 the concentration of divacancies is negligible. This would ensure

that single vacancies form a uniform distribution throughout the W crystal. In order to investigate

11

the role of H on the potential clustering of vacancies, calculations were performed by introducing 1–

4 H atoms into a divacancy (populating nearest neighbour interstitial sites). The following general

reaction was used to calculate the binding energy:

{x H}W + {y H}W → {{x H}W : {y H}W} (13)

These calculations show the binding energies between two W vacancies become negative only when

associative number of H atoms >2 (as shown in Figure 4). Note then that a single associative H

does not induce binding of vacancies.

1 2 3 4- 0 . 6- 0 . 5- 0 . 4- 0 . 3- 0 . 2- 0 . 10 . 00 . 10 . 2

Bindin

g Ene

rgy (e

V)

N u m b e r o f H y d r o g e n A t o m s

Figure 4: Binding energy of two tungsten vacancies to a nearest neighbour position with increasingnumbers of hydrogen associated with the cluster.

The lowest energy cluster arrangement that was identified when four hydrogen atoms are asso-

ciated with two vacancies is illustrated in Figure 5. Two hydrogen atoms coordinate in a nearest

and second nearest neighbour position to each vacancy forming a highly symmetric defect cluster.

12

Figure 5: The lowest energy arrangement of four hydrogen atoms (white spheres) associated withtwo tungsten vacancies (cubes) in body centred cubic tungsten.

4 Summary

Atomic scale computer simulations have predicted a decrease in the W vacancy formation energy

in the presence of H from an original 2.96 eV in the perfect material to 1.62 eV. The most apparent

effect of lowering the vacancy formation energy is an increase in vacancy population.

Further consideration of multiple vacancy-H interactions was shown to conclude that two H

interstitial defects will attract and lower the vacancy formation energy further from 1.62 eV to

0.74 eV.

Findings of this work suggest that H not only promotes vacancy formation in W but once formed

the vacancy will also accommodate further H clustering. De-trapping of H from the vacancy to an

interstitial position has been predicted to be energetically unfavourable (>1 eV).

Of importance, this work has shown that there is potential for vacancies in W to bind to one-

another in the presence of H. This is not the case for vacancies in W in the absence of H. The

coalescence of vacancies to larger vacancy clusters may be the driving force for larger void/bubble

formation which could in turn lead to a significant micro-structural evolution, not expected in pure

W. Macroscopically these effects may manifest as gross variance in physical properties.

13

Thermally induced dissociation of vacancy-H clusters will result in a finite lifetime for any

resultant vacancy population that exceeds the equilibrium concentration (i.e. a rise in vacancy

formation energy will result in fewer vacancies in the bulk).

Appendix

We tested the effect of system size, method and pseudopotential on the Schottky vacancy formation

energy, Eqn. 1, the results of which are reported in Table 2. Three different super-cell sizes were

considered: 2×2×2, 3×3×3 and 4×4×4 containing 16, 54 and 128 lattice sites, respectively. Four

different pseudopotentials were considered (all supplied with the VASP package [79, 80]): two that

use the GGA-PBE exchange correlation with 6 and 12 electrons treated as valance, and two that

use the LDA exchange correlation with 6 and 12 electrons treated as valance. Three methods were

used to calculate the energies for the perfect and defective super-cells: the first being calculations

carried out at zero pressure allowing both the internal coordinates of the atoms and the volume

of both perfect and defective super-cells to vary; the second being set have the lattice parameter

fixed to an experimental value (3.16475 A [113]) and the third having the lattice parameter of the

defective cell fixed to the values calculated of the perfect super-cell at zero pressure.

Table 2: Schottky vacancy formation energies (eV) calculated using a range of methods (allowinglattice parameter ‘a’ to relax at zero pressure or fixing it to either the experimental value or perfectcrystal DFT value), pseodopotentials (two different valance electron values for both GGA-PBE andLDA exchange correlations) and system sizes (2×2×2, 3×3×3 and 4×4×4).

super-cell Method GGA-PBE GGA-PBE LDA LDA

(12 valance electons) (6 valance electons) (12 valance electons) (6 valance electons)

2×2×2 Fully Relaxed 3.40 3.53 3.50 3.62

Experimental ‘a’ 3.37 3.62 3.92 4.17

DFT ‘a’ 3.53 3.66 3.71 3.87

3×3×3 Fully Relaxed 3.15 3.34 3.19 3.24

Experimental ‘a’ 3.12 3.33 3.65 3.80

DFT ‘a’ 3.13 3.36 3.46 3.50

4×4×4 Fully Relaxed 2.95 3.32 3.13 3.18

Experimental ‘a’ 3.07 3.26 3.61 3.76

DFT ‘a’ 2.74 3.26 3.58 3.62

14

It is clear that all variables have a distinct effect on the Schottky vacancy formation energy.

Increasing the number of electrons that are considered to be bonding lowers the vacancy formation

energy in using all super-cell sizes and exchange correlation ‘flavours’. We have chosen to use the

greater number of valance electrons as possible for our defect energy calculations as this will allow

a more realistic approach to metallic bonding to be reproduced by the calculations.

Table 2 reports a trend of decreasing Schottky vacancy formation energy with increase in system

size. As the larger cell lowers the extent of defect-defect interactions due to the periodic boundary

conditions used in this super-cell approach, the defect calculations were carried out in the 4×4×4

super-cell.

The GGA-PBE calculations are consistently lower than the LDA calculations - this is to be

expected as the LDA is known to over-estimate the bonding in a system (and as such overestimate

the Schottky defect formation energy) whereas GGA type exchange correlations are known to

slightly under-estimate bonding [114].

The three methods used are an interesting commentary on the ‘philosophy’ of defect energy

calculations - discussed in detail in previous articles [115, 116]. We have chosen to use the zero

pressure calculations in this work to fully understand the role of extrinsic defects (here H) in W

without imposing excess external pressures that may influence the enthalpies of formation that

we intend to calculate. External pressures will necessarily be imposed on the calculations when

fixing the lattice parameter in any way. Although all methods have their drawbacks (ours being

the influence of defect concentration of cell parameter - which can also be a useful measure - see

REF) the method presented will provide useful values to guide experimental observations.

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