A first-principles study of helium storage in oxides and at oxide--iron interfaces

0.46 MB·English

by  Paul Erhart

A first-principles study of helium storage in oxides and at oxide–iron interfaces
Paul Erhart1,a)
Physical and Life Science Directorate, Lawrence Livermore National Laboratory, Livermore,
California, USA and Department of Applied Physics, Chalmers University of Technology, Gothenburg,
Sweden
Density-functionaltheorycalculationsbasedonconventionalaswellashybridexchange-correlationfunctionals
have been carried out to study the properties of helium in various oxides (Al O , TiO , Y O , YAP, YAG,
2 3 2 2 3
YAM, MgO, CaO, BaO, SrO) as well as at oxide-iron interfaces. Helium interstitials in bulk oxides are
2 shown to be energetically more favorable than substitutional helium, yet helium binds to existing vacancies.
1 The solubility of He in oxides is systematically higher than in iron and scales with the free volume at the
0 interstitial site nearly independently of the chemical composition of the oxide. In most oxides He migration
2 is significantly slower and He–He binding is much weaker than in iron. To quantify the solubility of helium

n at oxide-iron interfaces two prototypical systems are considered (Fe|MgO, Fe|FeO|MgO). In both cases the
a He solubility is markedly enhanced in the interface comparedto either of the bulk phases. The results of the
J
calculations allow to construct a schematic energy landscape for He interstitials in iron. The implications

4 of these results are discussed in the context of helium sequestration in oxide dispersion strengthened steels,
1
including the effects of interfaces and lattice strain.

]
i PACS numbers: 61.72.J- 68.35.-p68.35.Dv 28.52.Fa
c
s
-
rl I. INTRODUCTION nanometer-sized oxide particles10–12 that act as obsta-
mt clesfordislocationmotionandaremetastableuptovery
Fusion environments are characterized by an abun- hightemperatures. Theoxideparticlesandevenmoreso
.
at dance of energetic helium ions (α-particles, He2+) that the oxide-matrix interfaces are expected to act as sinks
m are produced in the fusion reaction as well as via for He interstitials. The very high density of these par-
ticles should lead to a fine distribution of He bubbles
nuclear transmutation reactions in the plasma facing
-
d components.1,2 Candidate materials for structural appli- and thereby effectively limit the formation of larger su-
n cations in fusion reactors must therefore be able to tol- percritical voids that lead to mechanical failure.3 Sev-
o eral experiments provided evidence that ODS steels are
eratelargeconcentrationsofheliumwithoutdegradation
c much more swelling resistant than their non-oxide con-
of their mechanical properties under extreme conditions
[
ofupto200displacementsperatomand2000appmHe.3 taining counterparts, raising hopes that these materials

2 Mostconventionalsteelsareunabletosustainthesecon- willeventuallysatisfythedesigncriteriaforfuturefusion
v reactors.
ditionssincetheysufferfromHemediatedbubbleaswell
3
asvoidformationandgrowth(“swelling”)leadingtoem- DesigningODSsteelsthatcansustaintheextremecon-
3
4 brittlement and mechanical failure.3 ditions inside a fusion reactor requires a close collabora-
0 Using first-principles calculations it has been shown tion between experiment and modeling. As there are
. that in iron (as a model system for steels) He prefers currently no neutron sources available that can repro-
9
to occupy substitutional rather than interstitial sites.4,5 ducethe intenseneutronspectrumresultingfromfusion,
0
1 For dynamical reasons, however, during irradiation the a combination of (experimental and numerical) simula-
1 majorityof He is introduced in the formof interstitials.6 tionsmustbeemployedtopredictmaterialperformance.
: Since the effective interaction between He interstitials in In this context, numerical modeling of the time evolu-
v
the ironmatrix is attractive5 they can readilyform clus- tion of defect populations (vacancies, interstitials, inter-
i
X ters. The latter are further stabilized by the addition of stitial loops, dislocations, bubbles, voids etc.) plays a
r vacancies, which eventually leads to the formation and pivotalrole.13,14 Such models rely ona databaseofrates
a
growth of bubbles and voids. for various microscopic processes that occur in the ma-
To overcome the shortcomings of conventional steels terial, which typically comprises both experimental and
for applications in fusion environments, it has been atomic scale modeling data. Given the importance of
suggested to employ nano-structured ferritic alloys,3 in oxides particles in improving swelling resistance, the mi-
particular oxide dispersion strengthened (ODS) steels. croscopic mechanisms that govern their interaction with
These materials are obtained by mechanical alloying of He deserve particular attention, yet at present our mi-
oxide particles with steel powders (see e.g., Refs. 7– croscopic understanding of these interactions is limited.
9). They are characterized by a fine distribution of
A recent very extensive transmission electron mi-
croscopy study15 revealed that in ODS steels bubbles
and voids form preferentially in the close vicinity of ox-
ide particles,leadinginmany casesto the formationofa
a)Electronicmail: [email protected] “ring”ofbubblessurroundingaparticle. Thesamestudy
1
also demonstrated that oxide particles in ODS steels ex- Heinterstitialformationenergiesamongdifferentoxides,
hibit a broad variety of size, chemical composition, bulk in Sect. IVA their volume dependence is studied addi-
and interface structure including amorphous, partially tionallyincludingtheoxidesY Al O (YAM),Y Al O
4 2 9 3 5 12
amorphous and crystalline particles, core-shell configu- (YAG), YAlO (YAP), MgO, CaO, SrO, and BaO. Sec-
3
rations as well as chemical gradients. It was further- tion IVB concerns the migration barriers for He inter-
more shown that materials with smaller particles can stitials in oxides, which are found to be systematically
tolerate more helium providing direct evidence for the higherthaniniron. ThebindingbetweenHeinterstitials
importance of the interface area.15 In this context, an is the subject of Sect. IVC, where it is shown that most
oxide “particle” can be as small as a few ˚Angstr¨oms, oxides are less prone to He cluster formation than Fe,
correpondingtojustacoupleofatoms.10,12 Whilesizeof a behavior that results from a lower density of intersti-
these“nanofeatures”3canbebelowtheresolutionobtain- tial site in these materials. The solubility of He intersti-
able in transmission electron microscopy, they nonethe- tialsintworepresentativeoxide-ironinterfaces(Fe|MgO,
less contribute to He sequestration due to their sheer Fe|FeO|MgO) is quantified in Sect. V, where solubilities
number and very large effcetive interface area. at interfaces are found to be systematically higher than
in the bulk phases. Finally all these data are combined
Atomic scale modeling is in principle ideally suited
to sketch a typical energy landscape for He interstitial
to complement these experiments and to provide de-
migration across an oxide-iron interface and discuss the
tailed insight as well as quantitative information regard-
results in the context of He sequestration in ODS steels.
ing the behavior of helium inside and near oxide parti-
cles. The enormous chemical and structural complexity
present in ODS steels, however, renders a direct simu-
lation of these systems at present impractical. On the II. METHODOLOGY
otherhandmacroscopicmeasurementsofHepopulations
indicate that the most important parameter is particle A. Thermodynamics
sizeandthusinterfacearea.15 This suggeststhatthesol-
ubility of helium follows more general trends that are
Whereas in elemental metals the formation energy of
independentofvariationsinthelocalchemistry. Theob-
anintrinsicdefect is constant,inoxidesdefect formation
jective of the present paper is to demonstrate that the
energies depend both on the electron chemical potential
behaviorof He in oxides and in the vicinity ofoxide-iron
(also referred to as Fermi level) and the chemical envi-
interfaces can to a large extent be described by simple
ronment. Theexpressionforthedefectformationenergy
scaling relations. This is achieved by means of density-
makes these dependencies explicit,19,20
functional theory calculations using both conventional
andhybridexchange-correlation(XC)functionalsofbulk
oxides, bulk iron, and representative oxide-iron inter- ∆ED =(ED −EH)−q(EVBM+µe)−X∆niµi, (1)
faces. The thus obtained data provides not only valu- i
able insightinto the microscopicmechanisms but can be
where µ is the chemical potential of component i, E
i D
furtherutilizedtoparametrizeforexamplerateequation
is the total energy of the system containing the defect
models.14 Note that even though nanosized oxide nuclei
and E is the total energyof the idealreference system.
H
ofthetypedescribedinRefs.10and12arenotexplicitly
Thedefectformationenergyislinearlydependentonthe
studied in the presentwork,the arguments thatindicate
defect charge state q and the electron chemicalpotential
that the governing parameter is free volume and in par-
µ which is measured with respect to the valence band
e
ticularfreevolumeatoxide-ironinterfacesingeneralalso
maximum E . Here, ∆n denotes the difference in
VBM i
transpireto the caseof verysmalloxide inclusion. Com-
the number of atoms of element i between the system
paredtoearlierinvestigationsthatconsideredHedefects
with and without the defect, for example in the case of
in select oxides and carbides,16–18 the present work aims
an isolated oxygen vacancy ∆n = −1 whereas all the
O
to provide a more general perspective, including a vari-
other ∆n are zero. The chemical potentials of the con-
i
ety of oxides that representdifferent local environments,
stituents µ are conveniently expressed with respect to
i
chemistry, and covalent-vs-ionic character. the bulk chemical potentials µbulk of the respective el-
i
The paper is organized as follows. In the next section emental ground states, µ = µbulk + ∆µ . Metal and
i i i
computationalmethodsandparametersaresummarized. oxygen-rich conditions correspond to ∆µ = 0 and
metal
The results of a comprehensive study of He-related de- ∆µ = 0, respectively. The values of the chemical po-
O
fects in three prototypical oxides (Al O , TiO , Y O ) tentials can be translated to partial pressures enabling
2 3 2 2 3
aredescribedinSect.III. ItisdemonstratedthatHe oc- direct comparison with experiments.21 For simplicity, in
curs predominantly in the form of interstitials and that the presentworkHe-richconditions are assumedalways,
solubilities are much larger than in iron. Furthermore i.e. ∆µ =0eV. ForanelementalmetalEq.(1)reduces
He
it is found that formation energies of He interstitials to the usual expression
determined using conventional XC functionals are very
closetovaluesobtainedfrom(moreelaborate)hybridXC N +∆n
∆E =E − E , (2)
D D H
functional calculations. To rationalize the variation of N
2
TABLE I. Overview of computational parameters used in calculations of properties of the ideal bulk systems as well as point
defects. Migration barrier calculations forY O and therocksalt structuredoxideswere carried usingtheparameters given in
2 3
brackets.
Al O TiO Y O YAP YAG YAM (Mg,Ca,Ba,Sr)O
2 3 2 2 3
Idealcell calculations
Numberof atoms 10 6 40 20 80 60 2
k-point sampling Γ4×4×4 Γ6×6×6 Γ2×2×2 Γ4×4×4 Γ2×2×2 Γ2×2×2 Γ8×8×8
Defect calculations
Numberof atoms 270 216 320a 160 160 60 216b
Typeof supercell rhombohedral tetragonal body-centered cubic orthorhombic simple cubic monoclinic simple cubic
Supercell size 3×3×3 3×3×4 2×2×2 2×2×2 1×1×1 1×1×1 3×3×3
k-point sampling Γ1×1×1 Γ1×1×1 Γ1×1×1 Γ1×1×1 Γ1×1×1 Γ1×1×1 Γ1×1×1
XCfunctional PBE, HSE06 LDA,HSE06 PBE PBE PBE PBE PBE
a Migration barriers calculated using 80-atom cells and a Γ2×2×2 k-point sampling.
b Migration barriers calculated using 64-atom cells and a Γ4×4×4 k-point sampling.
where N denotes the number of atoms in the ideal cell. theconstraintsofsuchascenario,itisnonethelessuseful
Intermsoftheirformationenergies,thebindingenergy toconsiderformationenergiesasthey willdetermine the
of two defects A and B is given by driving forces, which determine the long time evolution
of the system.
∆E (AB)=∆E (AB)−∆E (A)−∆E (B). (3)
b f f f
Following this convention negative binding energies cor- B. Computational details
respond to exothermic defect reactions and imply an at-
tractive interaction between A and B. Note that the
Calculations were carried out within density-
binding energy can change as a function of the chemical
functional theory using the projector augmented
potentialbutisindependentofthechemicalenvironment
wave formalism23,24 as implemented in the Vienna
since the ∆n terms that appear in Eq. (1) cancel each
i ab-initio simulation package.25–28 Ti-3p, Y-4s, Y-4p,
other in Eq. (3).
Ca-3p, Sr-4s, Sr-4p, Ba-5s as well as Ba-5p states were
Assumingindependentdefects(lowdensitylimit)22the
treated as part of the valence. The plane-wave energy
equilibriumconcentrationofadefectisrelatedtoits free
cutoffwassetto 500eVforallcalculations. Torepresent
energy of formation ∆G according to
f exchange and correlation effects, we employed the local
density approximation (LDA), the generalized gradient
c =c exp[−∆G /k T], (4)
eq 0 f B
approximation as parametrized by Perdew, Burke and
Ernzerhof (PBE)29 as well as range-separated hybrid
where the pre-factor c denotes the density of potential
0
functionals30 obtained by mixing conventional XC
defect sites per unit volume. For extrinsic defects the
functionals (LDA or PBE) with 25% exact exchange
equilibrium concentration corresponds to the solubility
of the foreign element in the matrix. The free energy of at short-range with a screening parameter of 0.2˚A−1.
defect formation ∆G can be decomposed into the for- These functionals are refered to as HSE06 and
f LDA
mation energy ∆E as well as the vibrational T∆S HSE06, respectively.
f f,vib
and electronic T∆S formation entropies. (Note that For LDA and PBE the electronic contributions to the
f,el
Eq.(4) already incorporatesthe configurationalentropy, dielectric constants reportedin Table III were computed
seeRef.22). Formaterialswithabandgap(E ≫k T) within the linear response approach taking into account
G B
the electronic contribution to the formation entropy is local field effects. For the hybrid functionals the elec-
virtually zero and even for metals this term is usually tronic contribution was calculated from matrix elements
small compared to the other contributions. The vibra- of the dipole operator in the velocity gauge and the lo-
tional formation entropy can become important at el- cal field effect correction from either LDA or PBE was
evated temperatures. The present work is, however, added. The ionic contribution was computed for LDA
mainly concernedwith comparingdefects with very sim- and PBE using linear response theory.
ilar characteristics and therefore relative changes of the Details regarding the defect calculations in oxides,
formation entropy between different defects can be ex- specificallyBrillouinzone sampling,supercellcellshapes
pected to be small compared to the formation energy and sizes, are summarized in Table I. For pure iron the
term. In the following, therefore only formation energies PBE functional was used as well as 4×4×4 supercells
are considered and ∆G ≈∆E . containing 128 atoms and a 3×3×3 Monkhorst-Pack
f f
It should be stressed that formation energies are equi- gridforsampling the Brillouinzone. For chargeddefects
librium quantities while a material under intense irradi- the monopole-monopole correction according to Makov
ation is obviously a non-equilibrium system. Yet within and Payne31 was applied using the calculated static di-
3
electric constants given in Table III. Migration barri- dence ofthe Al vacancysafefor aconstantupwardshift.
ers were obtained via the climbing image-nudged elas- In contrast, while substitutional He on an oxygen site
tic band method32,33 using three intermediate images to exhibits the same charge states as the oxygen vacancy,
represent the transition path. The convergence of the the 2+/0equilibrium transitionlevel for He is strongly
O
calculations with respect to Brillouin zone sampling and shifted with respect to V .
O
supercell size was carefully tested, based on which the This behavior can be rationalized by considering the
error in the He interstitial formation energies due to the relaxation patterns of different defect charge states.
computational parameters is estimated to be less than Since He has a closed shell the formation energy differ-
0.1eV. ence between a vacancy and the corresponding substi-
Interfacesweremodeledusingslabgeometriesemploy- tutional defect results predominantly from strain. For
ingasimilarapproachasinRef.34. FortheidealFe|MgO the aluminum vacancyallchargestatesexhibit the same
interfacethesupercellcontainedsixFe andsixMgOlay- type of outward relaxation pattern and the formation
ers equivalent to 36 atoms. The Fe|FeO|MgO interface energy difference between V and He is only weakly
Al Al
model was composed of four Fe, two FeO, and six MgO affected by the charge state. In the case of the oxygen
layers equivalent to 40 atoms. Both systems were fully vacancy, however, the first neighbor shell relaxes inward
relaxedincludingcellshapeandvolumeuntilforceswere for the neutral and outward for the positive (2+) charge
below 20meV/˚A and the components of the stress ten- state. This behavior is typical for “deep” oxygen vacan-
sor less than 1kbar. For calculations involving He in- ciesandalsoobservedinotheroxidesincludinge.g.,ZnO
terstitials the supercell size was doubled parallel to the and In O (see Ref. 35).36 As a result the strain energy
2 3
interface leading to supercells with 144 and 160 atoms, contributiontothe formationenergiesofHe is strongly
O
respectively. Helium interstitial positions were system- charge state dependent, which explains the strong shift
atically sampled across the interface as well as the two of the equilibrium transition levels from V to He that
O O
“bulk” parts corresponding to 18 distinct configurations is observable in Fig. 1.
for each interface model. Each defect configuration was The absolute values of vacancy and substitutional He
relaxedatfixedcellshapeandvolumeuntilthemaximum formation energies depend on the treatment of XC. The
forcefellbelow30meV/˚A.InthesecalculationstheBril- HSE06 hybrid functional, which provides a much im-
louin zone was sampled using a 10×10×1 Monkhorst- provedvalueforthebandgapcomparedtothePBEfunc-
Pack grid for the ideal cells and a 5×5×1 grid for the tional (compare Table III), yields larger formation ener-
defect cells. gies approximately in accord with the increase in band
Results for bulk oxides considered in this study and gap. Yet PBE and HSE06 yield qualitatively the same
an extensive comparison of different XC functionals is picture with the equilibrium transition levels of V /He
O O
provided in the appendix. and V /He tracking the conduction and valence band
Al Al
edges, respectively.
Figures1(b,c)illustratenotonlythedependenceofde-
III. HELIUM AND INTRINSIC DEFECTS fectformationenergiesontheelectronchemicalpotential
but also on the chemical environment (compare Eq. (1)
and discussion thereafter). Moving from Al-rich to O-
Helium can be incorporated either substitutionally or
rich conditions, which can be achieved by regulating the
as an interstitial defect. In the next three sections, the
oxygen partial pressure, shifts the balance between He
thermodynamics of these two forms of He in three dif- O
(∆n = −1) and He (∆n = −1) but leaves the for-
ferentprototypicaloxides(Al O ,TiO ,Y O )arecom- O Al Al
2 3 2 2 3
mation energy of He unchanged (∆n = ∆n = 0).
pared. It is shown that interstitial He is the most stable i Al O
Figure 1 shows that the formation energies of both va-
formundermostconditionsandisalsothemostrelevant
canciesandsubstitutionalHecanbecomenegativeunder
form with regard to sequestration in ODS steels.
certainconditions. As negativeformationenergiesimply
the material being unstable with respect to defect for-
mation, the Fermi level is constrained to the range in
A. Alumina
which all formationenergies of intrinsic defects are posi-
tive. Consideringthethermodynamicallyallowedregions
Figure 1 shows the formation energies of interstitial
inFig.1,itcanbeconcludedthatformostconditionsHe
andsubstitutionalHeaswellasseveralintrinsicpointde-
interstitialsarethermodynamicallythe preferredformof
fects in alumina. According to the calculations He inter-
He in alumina. Yet He interstitials do bind to existing
stitialspreferentiallyoccupypositionsthatareequivalent
vacancies, since the reaction
to2bWyckoffsitesoftheidealstructure. Theinterstitial
formationenergyis independentofthe electronchemical He +V →He
i X X
potential,whichisexpectedbasedontheclosed-shellna-
tureofHe,andaffectedonlyslightlybythechoiceofXC is exothermic andthe binding energyE (He ) [compare
b X
functional (PBE: 2.15eV, HSE06: 2.23eV). Eq. (3)] is negative as shown in Fig. 1(d,e). The figure
WhenHe is substitutedfor Althe formationenergyof also demonstrates that the impact of the XC functional
theresultingdefectcloselyfollowstheFermileveldepen- on the binding energies is small.
4
16 2
14 (a) Al−rich, PBE (b) Al−rich, HSE06 VHOeO (c) O−rich, HSE06 1 (d) HeO
ergy (eV) 1102 E−G3PBE HVHAeelAil gy (eV) − 10 PBE
Formation en 468 +2 EGHSE06 Binding ener − 20 (e) HeAl HSE06
0
−1
2
Hei + VX fi HeX
0 −2
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8
Electron chemical potential (eV) Electron chemical potential (eV) Electron chemical potential (eV) Electron chemical potential (eV)
FIG. 1. (a-c) Formation energies for point defects in alumina (a) calculated using the PBE XC functional under Al-rich
conditions, and calculated using the HSE06 hybrid functional under (b) Al-rich and (c) O-rich conditions, respectively. The
line slopes correspond to the defect charge states according to Eq. (1). (d-e) Binding energies of He interstitials to vacancies
for (d) oxygen and (e) aluminum.
20 10
(a) LDA, Ti−rich (b) HSE06, Ti−rich (c) HSE06, O−rich (a) Y−rich (b) O−rich 0.3 (c) HeO
HeY1
Formation energy (eV) 11 505 EGLDA EGHSE06 HVHVHOTeeeiOTii Formation energy (eV) 2468 EGPBE VHVHVHHOYYeeee12YYiO12 Binding energy (eV) −00..03 HeY2
0 0 −0.6
0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
Electron chemical potential (eV) Electron chemical potential (eV)
FIG.2. Formation energies for point defectsin titania calcu- FIG. 3. Formation energies for point defects in yttria calcu-
lated using (a) the PBE XC functional under Al-rich condi- lated using the PBE XC functional under (a) Y-rich and (b)
tionsaswellastheHSE06hybridfunctionalunder(b)Ti-rich O-rich conditions. (c) Binding energies of He interstitials to
and (c) O-rich conditions. vacancies.
B. Titania
(LDA: 1.55eV, HSE06: 1.98eV). The same applies to
binding energies. As in the case of alumina He inter-
The formation energies of several forms of He in tita-
stitials are thermodynamically the most stable form of
niaaswellasrelatedintrinsicdefectsareshowninFig.2.
He.
ThelowestinterstitialformationenergyisobtainedifHe
occupies positions equivalent to the 4c Wyckoff site of
the ideal structure. Similar to the case of alumina sub-
stitution on the cation sublattice leads to a defect with
characteristicsthatareverysimilartothecationvacancy.
Unlikealuminathisresemblanceisalsoobservedforoxy-
C. Yttria
gen vacancy and He . Once again this behavior can be
O
relatedto the relaxationpatterns ofthe oxygenvacancy.
In contrast to alumina, the oxygen vacancy in titania is ThedefectformationenergiesforyttriashowninFig.3
“shallow”,relaxationoccurs inwardfor all chargestates, confirm the trends that were already observed for alu-
and the strain energy associated with He insertion at mina and titania. The oxygen vacancy in yttria resem-
the vacantoxygensite is virtually independent of charge blesitscounterpartinaluminainsofarasitalsoexhibits
state. The binding energy between a He interstitial and the characteristics of a deep defect. Helium interstitials
a vacancy is −1.0eV (LDA)/−1.1eV (HSE06) for HeTi occupypositionsthatareequivalentto16cWyckoffsites
and −1.4eV (LDA)/−1.6eV (HSE06) for HeO. in the perfect lattice, which correspondto the structural
The He interstitial formation energy is only weakly vacanciesofthebixbyitestructure(seeSect.3),andthey
affected by the treatment of exchange and correlation bind to vacancies.
5
5
eV) Fe Al2O3 TABLE II. Migration barriers in eV for He interstitials in
y ( TiO2 several oxides. Note that jump directions are approximate.
erg 4 Y2O3 ∆Vrel: change in Voronoi volume of He site between initial
en YAP state and saddle point normalized by volume of initial state;
on 3 YAG ∆rrel: changeinHe–nearestneighbordistancebetweeninitial
mati Y(MAgM,Ca,Sr,Ba)O stateandsaddlepointnormalizedbyinitialneighbordistance.
al for 2 Material Direction Barrier (eV) ∆Vrel (%) ∆rrel (%)
nterstiti 1 Al2O3 01011¯1 32..8166 −−129..52 −−1116..29
e i
H 0 TiO2 001 0.11 3.6 2.5
6 8 10 12 14 16 18 20 110 1.22 −9.7 −8.5
Voronoi volume of He interstitial (Å3)
Y O ¯1¯11 0.70 1.4 −15.0
2 3
111 0.29 4.0 −9.4
FIG. 4. Formation energies of He interstitials in various ox- 0¯21 2.73 0.0 −23.2
ides as a function of the free volume at the interstitial site.
Filled symbols indicate values at the respective equilibrium YAP 1¯20 1.22 24.6 −5.6
volumina. 101 0.98 3.6 −3.4
MgO 100 0.76 16.1 −10.2
CaO 100 0.84 17.2 −9.6
IV. HE INTERSTITIALS IN BULK OXIDES SrO 100 0.78 17.4 −9.0
BaO 100 0.61 17.6 −8.6
Inthepreviousthreesectionsitwasdemonstratedthat
within the thermodynamically allowed range of Fermi
compression expansion
levels He interstitials prevail. The actual He intersti-
V) 4 4
tialformationenergiesfordifferentoxides,however,vary e
o0v.7e1reaVw,iadlelvraalnugees(fArolm2Oc3o:n2v.e1n5teioVn,aTlXiOC2:fu1n.5ct5ioeVna,lYs)2.OT3o: nergy ( 3 123
e
resolve this variation He interstitial formation energies on 0
wdiesrceuscsoemdpaubtoevdeaass waeflulnacsttiohne yotftrvioulmumaelufmorintuhme ooxxiiddeess migrati 2 −10 D 0Vre l 1(0%) 20
aonfdthreocakpspaeltndstirxu.cStuinrecde othxeidXesCdefuscnrcitbieodnainl wSeacstss.h4owanndin5 erstitial 1 ATYil2O2OO233 Fe: 60 meV
Smeactti.oInIIentoerhgaievse,caamlcuinlaotrioinnflsuweenrceecoanrrHieedionutetrustsiitniaglcfoonr-- He int 0 Y(MAgP,Ca,Sr,Ba)O
−30 −20 −10 0 10
ventionalXC functionals that arecomputationally much
Relative He−nearest neighbor distance at saddle point (%)
more efficient than their hybrid relatives.
FIG.5. MigrationbarriersforHeinterstitialsinseveraloxides
as a function of the relative change in He–nearest neighbor
A. Scaling relation for formation energies separation. The inset shows the same data as a function of
therelativechangeintheVoronoivolumeoftheHesite. The
light gray stripe is intended as a guide for theeye.
AsshowninFig.4the formationenergiesarefoundto
scale remarkably well with the free volume at the inter-
stitial site, where the latter is measured by the Voronoi
ples chemistry from geometry and provides the basis for
volume of the He site in the relaxed configuration.37 It
a simplified treatment of interfaces in Sect. V.
turns out that other measures such as the distance from
theHesitetothenearestneighboratom(whichisequiv-
alent to constructing the smallest sphere around the He
B. Migration barriers
interstitial) do not yield such favorable scaling relations.
Figure 4 also demonstrates that the formation energies
of He interstitials in iron do not fall in the range of the To compare the mobility of He interstitials in oxides
oxides and for the same free volumes are systematically with iron their migration barriers in several oxides were
higher. determined. For each oxide we included paths that in-
The scaling relation in Fig. 4 is almostindependent of cludeduptothreeneighborshellsonthelatticeofWyck-
the chemistry of the oxide involved and provides ample off sites that correspond to the most stable He position.
evidencethatthemajorsourceofvariationinthe forma- The results are compiled in Table II.
tionenergiesofHeinterstitialsinoxidesisthevolumetric To establish a scaling relation for migration barriers
compression of the He atom. It thus effectively decou- similar to the one for formation energies, we considered
6
of He interstitial sites in a given host and thus the av-
(a) (b) TiO2 erage distance between nearestneighbor He interstitials.
0.0
SinceallHe interstitialsimposearelativelyshort-ranged
YO MgO
eV) 2 3 BaO compressive strain on the surrounding lattice, one can
y ( −0.5 expect two He interstitials to interactweakly repulsively
energ AFel2O3 atatnmceosdetwraoteHseepianrtaetrisotnitsi.alIsnccaonnterffaesctt,ivaetlyverayttsrhacotrtedaicsh-
g TiO2
din −1.0 Y2O3 othersincethetotallatticestrainisconfinedtoasmaller
n
Bi MgO volume and thus the total strain energy is less for a He
CaO He2 pair than for two separate interstitials.
−1.5 SrO He3
As shown in Fig. 6(b) the binding energies for small
BaO He4
1 2 3 4 0 2 4 6 8 10×10−2 He clusters do indeed exhibit a pronounced dependence
Number of He atoms Interstitial density (sites/Å3) on the density of interstitial sites with larger densities
beingassociatedwithstrongerbinding.38 Foriron,which
FIG. 6. Binding energy of He interstitial clusters in several exhibits astrongtendency to formHe clustersbutis not
oxidesaswellasiron(FedatafromRef.5)asafunctionof(a) included in Fig. 6(b), the interstitial site density is 0.53
the number of He atoms in the clusters and (b) the density per˚A3 andthusmuchhigherthaninanyoftheoxides.39
of He interstitial sites in thehost material. ThereasoningaboveandthedatainFig.6suggestthat
ingeneralone canexpectHe clustering to be strongerin
oxides with a higher He interstitial site density and that
the change in free volume (again measured using the
most oxides are less prone to He cluster formation than
Voronoi construction) from the initial He position V
ini iron.
to the saddle point V normalized with respect to the
sad
initial volume
Vsad−Vini V. OXIDE-IRON INTERFACES
∆V = (5)
rel
V
ini
In the previous section properties of He defects in ox-
aswellasthe relativechangeinthe distance betweenHe
ideshavebeenextensivelycharacterized,whichledtothe
and its nearest neighbor, ∆r , defined analogously.
rel
conclusion that He predominantly occurs in the form of
Figure 5 shows the migration barriers to scale fairly
interstitials. Furthermore it was found that formation
well with both of these measures, especially the relative
energies of these defects scale with the free volume at
He–nearestneighbordistance,butneithermeasureyields
the interstitial site almost independent of the chemistry
as convincing a scaling relation as in the case of the for-
ofthe hostoxide. Similarthoughlesspronouncedtrends
mation energies. The figure also contains the migration
barrier for Fe,5 which at 60meV is substantially lower couldalsobedemonstratedformigrationbarriersandHe
interstitial cluster binding energies.
than any of the migration barriers found in the oxides.
The oxide particles in ODS steels vary widely in com-
OnecanthusexpectHeinterstitialsinoxidestobemuch
positionandstructureresultinginarichvarietyofoxide-
less mobile than in iron.
matrixinterfacesofseeminglyarbitrarycomplexity.15Al-
though some orientation relationships have been exper-
imentally established (e.g., Fe|YAM, see Ref. 15), mod-
C. Helium interstitial clustering
eling these interfaces remains a computationally almost
intractabletask. Theresultsoftheprevioussection,how-
It has been established by previous first-principles
ever,suggestthat itshouldbe possible to rationalizethe
calculations5 that He interstitials in Fe exhibit a strong
behavior of He at oxide-iron interfaces primarily based
tendencytobindbothtootherHeinterstitialsandvacan-
on the geometry of the interface and the associated free
cies. TheresultingdefectcomplexesrepresentnascentHe
volume. Based on this rationale we decided to focus on
bubbles and therefore play a key role in understanding
twomodelsystems,Fe|MgOandFe|FeO|MgO.Thesein-
He embrittlement in iron and steels.
terfacesareparticularsuitableforacomputationalstudy
Complementing this information with regard to ODS
sincethe latticemismatchbetweenFeandMgOissmall,
steels, Fig.6 showsbinding energiesof smallHe intersti-
which means that for the present purpose interface dis-
tialclustersinoxides. ThelefthandpanelofFig.6shows locations can be neglected.40
the dependence of the binding energy on the number of
He atoms in the cluster for severaloxidesaswellas iron.
AccordingtothesedataHe–Heinteractionsinoxidesare
A. Ideal interfaces
typicallymuchweakerthaninironwith the exceptionof
MgO. In some cases the interaction is even repulsive.
The trends displayed in Fig. 6(a) can be readily ex- Theinterfacebetweentwomaterialswithrocksaltand
plained, atleastqualitatively,by consideringthe density body-centered cubic structure, respectively, is described
7
((ab)) OMg Registry along [010] 00000.....12345 FFee oonn ttoopp ooff MOg (c) 123 Energy difference (eV) ormation energy (eV) 345 (aM)gO Fe MgO (bM)gO FeO Fe FeO MgO
0.0 F
0
0.0 0.1 0.2 0.3 0.4 0.5
Fe Registry along [100] 9
3)
Å
FIG. 7. (a,b) Illustration of the Baker-Nutting orientation me ( 8
relationship. Projection of (a) rocksalt MgO and (b) body-
u
centeredcubicFealong[001]. Thelatterisrotated by45deg vol
about the [001] axis such that the [100]rocksalt and [110]bcc noi 7
directionsareparalleltoeachother. (c)Variationofinterface oro
energy as a function of the lateral displacement of the two V
6
crystals with respect to each other.
−10 −5 0 5 10 −10 −5 0 5 10
Position along [001] (Å) Position along [001] (Å)
Å) 30
e/ (a) (c) FeO FeO FIG. 9. Helium interstitial formation energies and their
ensity ( 2205 MgO Fe MgO MgO Fe MgO Frees|pFeecOti|vMegOVoirnotnerofiacveosluasmainfuancfotrionthoefp(oas)itiFoen|MpegrOpenanddicu(lbar)
d
ge 15 to the interface. Helium atoms placed in therange indicated
ar bythe horizontal bars relax into theinterface.
h 10
c
e
ag 5
er tionoftheFe|MgOinterfacewasestablishedbyscanning
Av 0
the energy landscape as a function of in-plane displace-
10
(b) (d) ment, which yields the Fe-on-top-of-O configuration as
%)
01] ( 5 tthaienmedosfrtosmtabthleeoFnee|.MTghOemFeo|dFeelOb|My ginOsemrtoindgelOisatthoemnsobin-
0 0
g [ the outermost Fe layers such that the Fe and O atoms
n
o −5 formasquarelatticeparalleltotheinterface. Bothmod-
al
n els were subsequently fully relaxed allowing both ionic
ai −10
Str motion as well as cell shape and volume changes.
−15 After relaxation of the Fe|MgO interface model the
−10 −5 0 5 10 −10 −5 0 5 10
Fe slab is under a compressive in-plane strain of 4.8%
Position along [001] (Å) Position along [001] (Å)
while the MgO half is under a tensile in-plane strain of
−1.2%. The strain along [001] is of the respective op-
FIG.8. Plane-averagedchargedensityandout-of-planestrain
posite sign near the interface as shown in Fig. 8(b) and
for the (a,b) Fe|MgO and (c,d) Fe|FeO|MgO interfaces as a
quicklydecaystothe bulkvalue withincreasingdistance
functionofpositionperpendiculartotheinterfaceplane. The
colored spheres in the bottom panel indicate the atomic po- to the interface. For the Fe|FeO|MgO geometry the in-
sitions. plane strains for the Fe and MgO part are 5.2% and ap-
proximatelyzero,respectively. Distinctlynon-zerostrain
along[001]isonlyobservedforthe Feslab,forwhichthe
by the Baker-Nutting orientation relationship.34 It is il- strain is tensile directly next to the FeO layer but com-
lustratedinFig.7,whichshowsthatthe(001) and pressive anywhere else. The different magnitudes of the
rocksalt
(001) planesaswellasthe[100] and[110] di- strain in the Fe and MgO parts reflect the fact that the
bcc rocksalt bcc
rections are parallel to each other. Among the earth al- tetragonal shear modulus is almost 1.5 times higher for
kalineoxidesMgOhasthesmallestlatticemismatchwith MgO than for Fe.
Fe (4% based onthe experimentallattice constants) and
wasthereforeselected forthe presentstudy. Incidentally
Fe–MgO interfaces have recently attracted a lot of at- B. Helium at interfaces
tention since they exhibit a strong transverse magnetic
resistance effect.41–45 Startingfromthefullyrelaxedinterfacemodels,Hein-
InthepresentworkbothFe|MgOandFe|FeO|MgOin- terstitialswereinsertedsamplingalldistinctknownbulk
terfacemodelswereincluded,thegeometriesofwhichare sites as well as sites in the interface. The calculated
illustratedinFig.8. Firsttheminimumenergyconfigura- formation energies for these configurations are shown in
8
Fig. 9. fect formation energies for vacancies and substitutional
Forbothinterfaceconfigurationsthe formationenergy He, the results are qualitatively similar. Furthermore,
for He interstitials is by far the lowest if the latter are for He interstitials conventional and hybrid functionals
located at the interface. For example in the case of the provide formation energies that are also quantitatively
Fe|MgO interface the formation energy is 2.7eV to be similar. Since the latter are computationally much more
compared with values of 3.3eV and 3.9eV near the cen- demanding, further calculations employed conventional
ter of the MgO and Fe slabs, respectively. In the MgO functionals only.
part the formation energy is thus almost identical to the Comparedto ironHe interstitial formationenergies in
bulk value at the equilibrium lattice constant, whereas oxides are significantly lower and exhibit a considerable
the corresponding value for Fe is noticeably lower than materials dependence. Using data for a wide range of
its unstrained bulk counterpart. This behavior is caused differentoxidesandvolumesitwasshownthatthe latter
by two effects: Firstly, the tetragonal shear modulus is dependence can be described with good accuracy by a
softerforironthanfortheoxide,asaresultofwhichthe scaling relation based on the Voronoi volume of the He
Fe slab is more severely strained. Secondly, the Poisson site. This finding demonstrates that the behavior of He
ratio of iron is less than 0.29, which implies that the av- can be largely rationalized in terms of free volume. It
erage volume per atom changes with strain. In the case thereby greatly simplifies the task of understanding the
at hand, one observes an increase in the free volume at behaviorofHeinODSsteelssinceitallowsustoseparate
theHeinterstitialsitefrom7.4˚A3 toabout8.0˚A3,which structure and chemistry.
in combination with Fig. 4 readily explains the observed Heinterstitialmigrationbarrierswerefoundtobesys-
decrease in the formation energy.46 tematically higher than in Fe. While the migration en-
Comparison of formation energies and Voronoi volu- ergies do not obey a scaling relation as cleanly as for
mina shows that inside both the Fe and MgO slabs the the formation energies, as a general trend the migration
formation energy scales with the free volume. At the barriersdecreaseifthe relativechangeinthe He–nearest
Fe|MgO interface the lowest formation energy also cor- neighbor distance from the initial configuration to the
responds to the largest free volume. In the case of the saddle point increases.
Fe|FeO|MgO interface the formation energy is lower at In iron He interstitials bind strongly to each other,
theFeO|MgOinterfacethanattheFeO|Feinterfaceeven which facilitates the formation of He bubble nuclei.5 In
thoughthefreevolumessuggesttheoppositetrend. This contrastin most oxides the binding energies between He
effectisrelatedtotheearlierobservationthatFedoesnot
interstitialsaremuchsmallerindicatingaweakerpropen-
fallonthesamescalingrelationastheoxides(seeFig.4). sity for bubble formation. Binding between He intersti-
While this implies thatone cannotpredictformationen-
tials was observed to scale with the volume density of
ergies at interfaces based exclusively on free volume, it Heinterstitialsites,withhigherdensities(e.g.,MgO,Fe)
nonethelessdemonstratesclearlythatHeinterstitialsare
leading to stronger binding.
more strongly attracted to interfaces than to either one
The variabilityand complexityof oxide-ironinterfaces
of the bulk phases. For the two systems considered here
in ODS steels prevents a direct first-principles study of
as well as e.g., Fe|Y O interfaces,47 one observes an in-
2 3 Hesequestrationattheseinterfaces. Inthepresentstudy
creaseinthefreevolumeattheinterfacecomparedtothe
itwasthereforedecidedtostudytwoparticularsimplein-
bulk phases. This behavior can be attributed to a com-
terfaces,Fe|MgOandFe|FeO|MgO.Inviewofthescaling
parably weak adhesion between the oxide and the iron
relationsdescribedabove,onecanexpecttheseinterfaces
matrix, which is related to the transition from mixed
toactasprototypesforthetypesofinterfacesthatoccur
ionic-covalent to metallic bonding across the interface.
in real materials. The calculations revealed that in both
interfaces the formation energies of He interstitials are
significantlylowerthanineitherofthe bulk phases. The
VI. DISCUSSION AND CONCLUSIONS low formation energies could again be correlated with a
larger free volume at the interfaces. While in the MgO
Inthissectiontheimplicationsoftheforegoinginvesti- slabs the He interstitial formation energies reached al-
gationforthe understandingofHesequestrationinODS mostexactlythe valuethatwasobtainedearlierforbulk
steels will be discussed. To simplify the discussion, first MgO,formationenergiesforpositionsinsidethe Feslabs
the major results of this work will be recapped. deviated significantly from the bulk value. This behav-
A detailedexaminationofthree differentoxides ofdif- ior is directly related to strain fields that distort the Fe
ferentchemicalcomposition,stoichiometryandstructure lattice andthereby affectthe free volumeavailableatin-
using both conventional as well as hybrid XC function- terstitial sites. The strain effect is much larger in the Fe
als showed that intrinsic defects limit the range within matrix because its tetragonalshear modulus is consider-
which the electron chemical potential can vary. Within ably lower than the one of MgO.
thethermodynamicallyallowedrangeHeinterstitialsare Now one can combine all this information to develop
themoststableformofhelium,yettheywillbindtoexist- a schematic energylandscape for He interstitials in ODS
ingvacancies. Whileconventionalandhybridfunctionals steels. In the Fe matrix He interstitial formation ener-
yielddifferentvaluesforbandgapsaswellasabsolutede- giesarehigh butmigrationbarriersare low. Incontrast,
9
directionalbonding and a more localizedelectroncharge
iron matrix interface oxide particle
density comparedto iron. As a result oxides adoptmore
low solubility highest higher solubility
fast migration solubility slower migration openstructureswithlargerinterstitialsites(bothinionic
andelectronicterms),anddefectinducedstrainfieldsare
less extended than in metals. These qualitative features
y
g
er are also found in amorphous oxide particles and small
n
E oxide inclusions. (As shown in Refs. 10 and 12, already
verysmalloxideclusterscontainingjustafewatomsfea-
ture pronounced directionalbonding). It is therefore ex-
pected thatthe resultsobtainedinthe presentworkalso
transpire to these more general situations.
In summary, in this paper it has been demonstrated
FIG. 10. Schematic energy landscape for He interstitial mi- that oxideparticles in ODS steels havea higher He solu-
gration inan ODSsteel. IntheFematrix formation energies bilitythantheFematrix,whichisprimarilytheresultof
are high but migration barriers are low, while the opposite largerinterstitialsites. Thesolubilityatoxide-ironinter-
applies for the oxide particles. The smallest formation en-
facesis evenlargerthaninthe bulk oxides. Strainfields,
ergies and thus the highest solubilities are predicted in the
which can affect in particular the iron matrix, lead to
interface region. Strain fields can lead to gradients near the
solubility gradients near oxide-iron interfaces. The data
interface that depending on the sign of the strain field can
obtained in this study not only provides valuable insight
either increase or decrease toward the interface.
intothe behaviorofHe inODSsteels butcanbe usedto
deriveparametersforrateequationmodels ofHeseques-
tration in ODS steels.14
formation energies in the oxide particles are lower while
migration barriers are higher than in Fe. The lowest
formation energies are observed at the interface. Since
ACKNOWLEDGMENTS
iron is elastically softer than the majority of oxides con-
sidered in this study, the iron matrix is more likely to
be strained. This affects the free volume available to I would like to thank J. Marian for his continuous in-
He interstitials and accordingly their formation energies terestinthisworkandmanyfruitfuldiscussions. Helpful
(and solubilities). Combining these data, one can ob- discussions of experimental findings with L. Hsiung and
tain a schematic energy landscape as the one sketched M. Fluss are gratefully acknowledged. This work has
in Fig. 10. In this particular plot the free volume in been performed under the auspices of the U.S. Depart-
the Fe matrix is assumed to increase toward the inter- mentofEnergybyLawrenceLivermoreNationalLabora-
faceleadingtoagradualslopingofthe landscapetoward tory under ContractDE-AC52-07NA27344with support
the oxide particle. It is, however, equally possible that fromtheLaboratoryDirectedResearchandDevelopment
the free volume decreases toward the interface, leading Program.
to a landscape that rises toward the interface. In real-
ity the character of the strain field sensitively depends
Appendix: Bulk properties of oxides
on the structure of the interface, which can include e.g.,
interface dislocations, amorphous regions, and chemical
gradients (in so far as they translate to strain). In fact, This appendix summarizes results for groundstate
experimentally He bubble formation is observed not to propertiesoftheoxidesincludedinthisstudyasobtained
occur around all precipitates,15 for which strain effects using different computational methods. Based on these
could be a reasonable explanation. results the parameters for the defect calculations were
Inthe introduction, itwas pointedoutthat oxidepar- chosen, in particular the XC functionals.
ticles in ODS steels exhibit a broad spectrum of struc-
tural and chemical variations, including amorphous re-
1. Alumina
gions and extremely small nanometer-sized inclusions.
While these structures were not explicitly studied here,
thepresentworkhasestablishedcleartrendsthatholdfor The ground state structure of alumina (Al O ) is
2 3
avarietyofdifferentlocalenvironmentsbothstructurally corundum(StrukturberichtsymbolD5 ,spacegroupR¯3c,
1
and chemically. It was also demonstrated that He in- number 167), which has a primitive unit cell of rhombo-
corporationin oxides is qualitatively different from iron. hedral symmetry containing ten atoms with Al and O
ThisismostclearlyvisibleintheHeinterstitialformation atoms occupying Wyckoff sites 4c and 6e, respectively.
energies (Fig. 4), for which oxides display systematically The structure can also be described using a hexagonal
lower values than iron for the same free volumes. This setting, in which case the unit cell compromises three
behavior can be readily understood in terms of mixed times as many atoms. The relation between the two set-
covalent–ionic bonding in the oxides vs metallic bonding tings is discussed in detail in Ref. 55. Following com-
iniron. Thecovalentcharacterofoxidesleadstostronger mon procedure the structural parameters in Table III
10