From solid solution to cluster formation of Fe and Cr in $\alpha$-Zr

4.1 MB·English

by  P. A. Burr

From solid solution to cluster formation of Fe and Cr in α-Zr
P.A.Burra,b,∗,M.R.Wenmana,B.Gaultc,M.P.Moodyc,M.Ivermarkd,e,M.J.D.Rushtona,M.Preusse,L.Edwardsb,
R.W.Grimesa
aCentreforNuclearEngineeringandDepartmentofMaterials,ImperialCollegeLondon,London,SW72AZ,UK.
bInstituteofMaterialsEngineering,AustralianNuclearScience&TechnologyOrganisation,Menai,NewSouthWales2234,Australia.
cDepartmentofMaterials,UniversityofOxford,ParksRoad,Oxford,OX13PH,UK.
dHighTemperatureMaterials,SandvikMaterialsTechnology,73427Hallstahammar,Sweden.
eUniversityofManchester,SchoolofMaterials,M139PL,UK.
5
1
0
2

t
c Abstract
O
Tounderstandthemechanismsbywhichthere-solutionofFeandCradditionsincreasethecorrosionrateofirradiated

1 Zralloys,thesolubilityandclusteringofFeandCrinmodelbinaryZralloyswasinvestigatedusingacombination
1 ofexperimentalandmodellingtechniques—atomprobetomography(APT),x-raydiffraction(XRD),thermoelectric

power (TEP) and density functional theory (DFT). Cr occupies both interstitial and substitutional sites in the α-Zr
]
i lattice;Fefavoursinterstitialsites,andalow-symmetrysitethatwasnotpreviouslymodelledisfoundtobethemost
c
s favourable for Fe. Lattice expansion as a function of Fe and Cr content in the α-Zr matrix deviates from Vegard’s
- law and is strongly anisotropic for Fe additions, expanding the c-axis while contracting the a-axis. Matrix content
l
r ofsolutescannotbereliablyestimatedfromlatticeparametermeasurements,insteadacombinationofTEPandAPT
t
m wasemployed. Defectclustersformathighersolutionconcentrations,whichinduceasmallerlatticestraincompared
. to the dilute defects. In the presence of a Zr vacancy, all two-atom clusters are more soluble than individual point
t
a defectsandasmanyasfourFeorthreeCratomscouldbeaccommodatedinasingleZrvacancy. TheZrvacancyis
m criticalfortheincreasedapparentsolubilityofdefectclusters;theimplicationsforirradiationinducedmicrostructure
- changesinZralloysarediscussed.
d
n
o
1. Introduction SPPdissolution[1,3,20–26].Itisimportanttolimithy-
c
[ drogenuptakeduringreactoroperationbecausehydro-

Zr alloys are widely used in the nuclear industry as gen causes dimensional changes to the cladding [24],
3
v nuclear fuel cladding and other structural components. reduces its ductility [24] and reduces integrity perfor-
2 FeandCrarecommonalloyingelements,addedtoim- mance in hypothetical accident scenarios [27, 28], and
3 prove corrosion resistance [1–3]. These elements are potentially in the storage conditions relevant to spent
7 known to exhibit near-negligible solid solubility in α- nuclearfuel[27,29].
6
0 Zr, and therefore segregate to form second phase par- Recent advanced transmission electron microscopy
. ticles (SPPs) [4–6]. One of the key aspects of the mi- (TEM) [30] and atom probe tomography (APT) stud-
1
0 crostructural evolution of Zr alloys under irradiation is ies [31] have shown that clusters of Fe and Cr form at
5 the amorphisation and subsequent dissolution of SPPs, (cid:104)a(cid:105) and (cid:104)c(cid:105) dislocation loops following the re-solution
1 which leads to the re-solution of the alloying elements process. ThiswaspreviouslysuggestedbyTEMinves-
v: intheZrmatrixabovetheirsolubilitylimits[7–19]. In tigation[9–11,17,18,32,33]butnotobserveddirectly.
i turn, this has an impact on the physical and corrosion It has been suggested that irradiation induced defects
X
properties of the alloy. In particular, surface oxidation mayalsoactastrappingsitesforhydrogen,therebyin-
ar andhydrogenpick-upfractionareknowntobestrongly creasingtheterminalsolidsolubilityofhydrogeninα-
affectedbyalloycompositionandthepresenceanddis- Zr[34,35].
tributionofSPPsandexperienceamarkedincreaseafter
The solubility of Fe in Zr — and to a lesser extent
also that of Cr in Zr — has also been investigated us-
∗Correspondingauthor. ing atomic scale simulations, but so far, the clustering
Emailaddress:[e-mail](P.A.Burr) behaviour of alloying elements has hardly been con-
PreprintsubmittedtoActaMaterialia October13,2015
sidered using such methods. Early work by Pere`z and α-ZrlatticeandthatZrvacanciesarecrucialforthefor-
Weissmann[36]andofPasianotetal.[37]investigated mationandgrowthoftheclusters. Finally, theworkis
the possible mechanism for Fe accommodation in the summarisedandtheimplicationsforirradiatedZralloys
α-Zrlattice,buttheirDFTcalculationswerelimitedto discussed.
smallsupercellscontaining36and48Zratomsrespec-
tively. In particular, Pasianot et al. [37] observed that
2. Methodology
whenFesubstitutesforZr,itoccupiesalowsymmetry
configuration that is displaced slightly from the lattice
2.1. Materials
site. More recent studies have employed, in one case,
Binary Zr-alloys were melted in an arc furnace in
slightlylargersupercells(54ZratomsbyLumleyetal.
a water cooled copper crucible under an argon atmo-
[6]and48ZratomsbyChristensenetal.[38,39]),but
sphereatWesternZirconium,USA.TheZrstartingma-
onlythemoreconventionalinterstitialsites(tetrahedral
terial was in the form of chips while Cr and Fe were
and octahedral) were considered. Furthermore, recent
small beads. All alloying elements were standard ma-
publications[40–46]haveshownthatevenlargersuper-
terials used by Western Zirconium for their production
cells(∼300atomsifnofinitesizecorrectiontermisap-
ofzirconiumalloys. The125gbuttonswerere-melted
plied)arerequiredtoavoidcomputationalartefactsthat
three times in order to ensure chemical homogeneity.
maysignificantlyaffecttheapparentstabilityofdefects
Furtherdetailsofthematerialsprocessingcanbefound
inα-Zr. Itisevidentthatastate-of-the-artevaluationof
in[49,50]. ThebuttonswereanalysedatWesternZir-
theextrinsicdefectsinZrisneeded.
coniumusinginducedcoupledplasma-atomicemission
Inpreviouspapers,theauthorsconsideredtheforma-
spectroscopy and combustion analysis (oxygen and ni-
tionofSPPs[6,47]andtheirinteractionwithH[26,48].
trogen) to determine the chemical constituent of each
Here,theauthorsareconcernedwiththeconditionsrel-
sample, presented in Table 1. It is acknowledged that
evanttoirradiatedZralloys,inwhichtheSPPsarepar-
sample Zr-0.05Cr contains notable amounts of Fe and
tially dissolved. The current work employs a suite of
Sn contaminations and the results from this alloy are
experimental and theoretical approaches to investigate
highlightedinsubsequentsection. Allotheralloyswere
the solubility of Fe and Cr in pristine and defective Zr
producedwithahighdegreeofpurity.
andtheformationofclusterscontainingFe, Crandin-
trinsic defects. First, DFT simulations reveal that Cr
Table1: Chemicalcompositionofthebinaryalloysin
may occupy both interstitial and substitutional sites in
wt.ppm. HfandNbwereconsistentlylessthen23and
the α-Zr lattice, and the results were corroborated by
20ppmrespectively. Siandanyotherpotentialimpuri-
spatialdistributionmapsproducedwithAPT.Thesim-
tieswerealwaysbelowthedetectionlimit.
ulationsalsoindicatedthattheFe-Zrbinarysystemsex-
hibitsalargedeviationfromVegard’slaw,therebyindi-
Samplename Cr Fe Cu N O Sn
catingthatlatticeparametermeasurementsbyXRDdo
not provide a suitable estimate of solute concentration Zr-0.1Fe <20 1049 10 NA NA <8
intheα-Zrmatrix. Thematrixcontentoffastquenched Zr-0.2Fe <20 1927 11 NA NA <8
binarysampleswasthenmeasuredusingTEPandAPT, Zr-0.4Fe <20 4298 <10 44 810 <8
Zr-0.6Fe <20 6226 22 NA NA <8
showing that an increasing amount of Fe and Cr was
Zr-0.8Fe <20 8943 19 NA NA <8
trapped in solution with increasing nominal composi-
tion, despite the formation of SPPs. The lattice ex- Zr-0.05Cr 475 217 11 NA NA 1155
pansionduetoalloyingadditionsofthebinarysamples Zr-0.15Cr 1608 37 <10 NA NA <8
was then measured by XRD, and the predicted devia- Zr-0.30Cr 2869 41 <10 43 849 <8
tionfromVegard’slawwasobserved. Complementary
DFT simulations also highlight that the preference for The as-cast buttons were cross rolled at 540°C with
interstitialoversubstitutionalaccommodationisdepen- an intermediate recrystallisation anneal at 600°C to a
dentontheatomicstrainenvironmentandanargument final thickness of 3 mm. Subsequently, 3 × 3 × 40
isputforwardforclusteringofdilutedefectsasameans mm3 matchstick samples were cut and β heat-treated
to reduce overall lattice strain. A first nearest neigh- for 10 minutes at 1000°C in a vertical furnace flushed
bour analysis carried out with APT provides evidence withargon,followedbywaterquenching,inanattempt
that these clustering tendencies occur in oversaturated to maintain most of the Fe and Cr into α-Zr solution.
Cr-Zr alloys. Further simulations show that larger de- Scanning and transmission electron microscopy inves-
fect clusters may be favourably accommodated in the tigation showed that complete solid solutions were not
2
obtainedevenattheseveryhighcoolingrates. Instead to stabilise the thermo-electricity, each specimen was
asignificantnumberofsmallSPPshadformed[50]. leftfor1minaftermountingbeforemeasurement. Each
measurementhadadurationof20sinwhichtheinitial
2.2. X-raydiffraction value and the variation from this value were recorded.
Eachsurfaceofthematchstickspecimenwasmeasured
Thematchstickswerecutinto3×3×2mm3 cubes,
twicetogiveanaveragevaluefromeightmeasurements
mountedina5×5gridbeforegrindingandpolishingto
peralloyconcentration.
produce an XRD-sample with an approximate surface
dimensionof15×15mm2. Thex-rayanalysiswascar-
2.4. Atomprobetomography
riedoutonaPhilipsPW3710diffractometerusingCu-
APTwasperformedonaCamecaLEAP3000XSi,
K radiation in Bragg-Brentano geometry. Diffraction
α withaflightpathof90mm. Theexperimentswerecon-
profiles were recorded ranging from the {101¯0} to the
ducted at a base temperature of −213±5◦C, in laser-
{303¯2} reflections with a step size of 2θ = 0.02° and a
pulsing mode (∼10ps, 532nm, spot size <10µm di-
recordingtimeof20sperstep. Theprofileswereanal-
ameter). Throughout the analysis, the DC voltage was
ysed using Rietveld analysis to determine the a and c
increased to keep a detection rate of 5 ions per 1000
latticeparameters.
pulses. Specimens were prepared by means of a FIB
lift-outprocedure,fromamechanicallypolishedsample
2.3. Thermoelectricpowermeasurements
oftheZr-alloys,usingaZeissAurigaandanelectropol-
TEP experiments measure the Seebeck coefficient ishedMogridassupport[62]. Thedatasetswererecon-
(S),whichistheelectricpotentialdifferencethatarises structedusingstate-of-the-artalgorithms[63],resulting
when two metals in tight contact form a thermocouple inthetypicaltomogramsshowninFigure1(a)forabi-
withtwojunctionsheldunderatemperaturedifference. naryCr-Zrspecimen containing0.26at%Cr. Therein,
The Seebeck effect consists of two parts: a chemical a2at%Crisoconcentrationsurfacehighlightsthepres-
gradient found at the junctions between the two met- ence of small regions enriched in Cr, up to 4–5 at%,
als (the Peltier effect) and a thermal gradient within whichappeartobealigned,maybealongatwinbound-
the same metal (the Thomson effect). In the present ary, similar to what was discussed in [49]. The signal
case, matchstick samples were clamped between cop- tobackgroundnoiseratioonthemajorpeakofCrwas
perblocks,whichweremaintainedattemperatureTand 120:1,andonthemajorpeakofFeitwasabove200:1,
T+∆T,respectively. ThemeasuredTEP,whichisrel- providing a high level of certainty when labelling the
ativetothereferencemetaltowhichitisclamped,can CrandFeatoms. Isotopicratioof56Fe/54Fewasfound
then be plotted as a function of concentration of solid tobe14.3,whichcompareswellwiththenaturalabun-
solutioninthematrix[51–54]. ThesignoftheTEPcan dance(15.81). Thisprovidesconfidencethatthesignal
bediscussedintermsofincompleted-bandsofelectrons was mostly generated from Fe ions and not molecular
intransitionmetals[55]andFermisurfaces[56].Foran impuritiessuchasCO.
elementwithunfilledd-orbitals,suchasZr,theaddition As evidenced in Figure 1(b), which contains a two-
of a solute atom into the matrix can either increase or dimensional density map computed from the same to-
decrease the TEP of the alloy. As all transition met- mogram, the data exhibits features that can be directly
als have fewer available d-orbitals than Zr, the energy related to the crystallography of the specimen [64]: a
difference decreases, which results in the TEP becom- so-calledpolecanbeseenatthebottomleftalongwith
ingmorenegativewithincreasingsoluteconcentrations aseriesofzonelinesexhibitingsix-foldsymmetry.This
[57]. Previous work has shown that the Seebeck co- polecanbeattributedtothe(0002)atomicplanesatthe
efficient of zirconium alloys is sensitive to solute con- specimensurfacecausingtrajectoryaberrations. These
centration,textureandcoldwork,butisnotaffectedby planes are also imaged within the APT data. Imag-
thepresenceofsmallvolumefractionofSPPs[58–61]. ing atomic planes is common in APT, and allows for
Allsampleswerepreparedusingthesameprocedureto calibration of the tomogram [65, 66] as well as site
minimise variations in texture and microstructure (see occupancy analyses [67–70], which are facilitated by
section2.1),thereforeanychangerecordedinTEPcan datatreatmentmethodssuchasspatialdistributionmaps
beattributedtovariationsinsoluteconcentrations. [67,71,72].Thelatteraresimilartosplitradialdistribu-
The measurements were conducted at INSA, Lyon, tionfunctionsinvestigatingthelocalneighbourhoodof
France using a TechMetal Promotion instrument and a eachatomalongaspecificdirection. Anothercommon
Cureference. Thetemperatureoftheclampingcopper datatreatmenttechnique,thecalculationandanalysisof
blockswasheldat15±0.2°Cand25±0.2°C. Inorder thedistancebetweenagivenspeciesanditsfirstnearest
3
Figure1: (a)Three-dimensionalreconstructionofthedatasetfromtheZr-Crsample. Thegreensurfaceencompasses
regionscontainingabove2at.%Cr.(b)Top-downprojectionofthedatasetshownin(a),withredrepresentingregions
ofhigherdensityandblueoflowerdensity.
neighbours[73–75],wasalsousedheretoestimatethe criteria were imposed for atomic relaxation within the
matrixcompositionfollowingtheprotocoldescribedin memoryconservativeBFGSalgorithm[81,82]: theen-
[75]asreportedin[49],butalsotoinvestigatetheclus- ergydifferencewaslessthan1×10−6eV,forcesonin-
teringtendencyofCrinZr. dividual atoms less than 0.1eVnm−1 and for constant
pressure calculations, stress components on the cell of
lessthan1MPa.
2.5. Computationalcalculations
Defect formation energies Ef were calculated using
DFT simulations were carried out using the castep equation1.
code[76]withthePBEexchange-correlationfunctional
(cid:88) 1
[77], ultra-soft pseudo potentials [78] and a consistent Ef = EDFT−EDFT± µ(i)+ E (1)
plane-wavecut-offenergyof450eV. d p i 2 int
Supercells containing 150 Zr atoms were modelled
whereEDFTandEDFTarethetotalenergiesfromthede-
using a 2 × 2 × 2 k-point sampling grid [79]. The d p
fectiveandperfectDFTsimulations,µ isthechemical
linear elastic theory correction term of Varvenne et al. i
potentialofallspeciesithatareaddedorremovedfrom
[45](aneto)wasemployedtoreducefinitesizeeffects.
theperfectcrystaltoformthedefect,andE isthecor-
The elastic constants of α-Zr, which feed into aneto, int
rectiontermfortheinteractionenergyofthedefectwith
were calculated by performing small lattice perturba-
its periodic images, calculated using aneto [45]. The
tions. The resulting stiffness constants are (in units
chemical potential µ is calculated as the DFT energy
of GPa): c = 141.97, c = 65.36, c = 68.02,
11 12 13 peratomofthemetallicelementsintheirgroundstate;
c = 148.71,c = 30.22andc = 38.30. Sincethese
33 44 66 forFethegroundstateistheferromagneticBCCphase,
systems are metallic, density mixing and Methfessel-
for Cr it is the anti-ferromagnetic BCC phase. Aneto
Paxton[80]coldsmearingofbandswereused(smear-
calculations employed a radial cutoff of 15A˚ with 20
ingwidth= 0.1eV).Testswerecarriedouttoensurea
divisionsoftheFouriergrid,whichyieldedenergyval-
convergenceof1×10−3eV/atomwithrespecttoallpa-
uesconvergeduptothe4thdecimalplace.
rameters. Nosymmetryoperationswereenforcedwhen
The relaxation volume (Ω) of a defect is defined as
calculating point defects and all simulations were spin
thedifferenceinvolumebetweenasupercellcontaining
polarised.
thedefectandtheperfectZrsupercell;seeequation2.
The energy convergence criterion for self-consistent
calculations was set to 1×10−8eV. Similarly robust Ω=V(Zr M )−V(Zr ) (2)
x y z
4
When calculating Ω, mass action is not taken into ac-
Cr at% Fe at%
count, in other words, the number and types of atoms
6.0
between the defective and perfect cell do not have to
Fe 0 0
bethesame. Infact,consideringthesubscriptofequa-
Cr
tion2, foraninterstitialdefect x = z, whilstforasub- 0.1
stitutional defect x + y = z. A related quantity often 5.5 0.1
0.2
found in the literature is the defect formation volume,
in which the number of atoms of each species is con- C) 0.2 0.3
served between perfect and defective cells. However, (cid:176)V/ 5.0 0.4
thedefectformationvolumeisonlyproperlydefinedfor mS (
D 0.3 0.5
intrinsicdefects[83],asthereferencevolumeofaniso-
latedextrinsicatomisnotastrictlydefinedquantity. In 4.5 0.6
0.4
the current work, only relaxationvolumes will be con-
sidered.
Configurational averaging of physical properties re- 4.0 0.5
lated to the presence of defects (such as Ω) was per-
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
formedfollowingequation3:
Total alloy content at%
(cid:80)
n(i)X(i)exp(−Q)
X¯ = i(cid:80) i (3) Figure2: TEPmeasurementsagainstnominalcompo-
n(i)exp(−Q)
i i sitionofthebinaryalloys. Right-handsideaxesarethe
where calibratedconcentrationsofCrandFedissolvedinsolu-
∆E (i)
Q = f (4) tion. Errorbarsexpressthestandarddeviationofeight
i k T samples per datum. The hollow symbol represent the
B
and X is the physical property of interest, i is a defect lowpurityalloy.
configuration,n(i)isitsmultiplicity,∆E (i)isthediffer-
f
enceinformationenergywithrespecttothemoststable
defect and all other symbols retain their conventional therefore, includesalltheFeandCratomsthatarenot
meaning. withinSPPs,irrespectiveoftheatomsbeinguniformly
dispersedinsolution,segregatedatthegrainboundaries
orclusteredinatmospheresofhigherCrorFedensity.
3. ResultsandDiscussion
These results were published previously [49], and the
3.1. Fe-ZrandCr-Zrbinaryseries relevant findings are summarised thus. The samples
containing1.30at.%Feand0.26at.%Crwerechosen
TEP measurements of the samples evaluate the See-
beckcoefficientoftheα-Zrmatrix,bypassingtheresis- for APT analysis. Fe and Cr are either found within
small particles or atmospheres along grain boundaries,
tancecontributionsoftheSPPs[51–54,59–61]. There-
fore,anychangeinSeebeckcoefficientobservedacross or are randomly distributed within the matrix. The to-
tal composition calculated using APT is in excellent
the binary alloys, compared to the unalloyed reference
agreement with the nominal composition of the alloys
sample, can be related to the amount of Fe or Cr in
(1.34±0.026 at.%Fe and 0.25±0.01 at.%Cr respec-
solution. TEP results are plotted against composition
in Figure 2. It is observed that the Seebeck coefficient tively). Selecting volumes that contained no SPPs or
grainboundaries,theconcentrationofalloyingelement
decreases with increasing content of Fe or Cr, indicat-
in solution was calculated to be 0.42±0.015 at.% Fe
ing that, despite the formation of SPPs, an increasing
and0.21±0.01at.%Cr.
amountofalloyingelementwastrappedinsolutionwith
increasingnominalcompositionofthesamples. ThechangeinSeebeckcoefficientmeasuredbyTEP
Toobtainquantitativeconcentrationsfortheα-Zrma- is reasonably linear with increased alloy concentration
trix,itisnecessarytoestablishadatumbycalibratingat (see Figure 2). Therefore, assigning the concentration
least one TEP point that has a known alloy concentra- values obtained by APT to the points at 1.30 at.% Fe
tioninsolution.Sinceallofthesamplesexhibitedsome and0.26at.%Cr,andextrapolatinglinearlysothatthe
segregationofFe/CrtoSPPs,APTwasemployedtocal- referencesamplehasanalloyconcentrationofzero,the
culatethematrixcontentofFeandCrinα-Zrbysam- solution concentration of the remaining samples was
plingavolumecontainingnoSPPs.Thematrixcontent, obtained. The resulting scales are plotted on the right-
5
hand y-axis in Figure 2. The deviation from the linear termed basal tetrahedral by some authors) and all the
regressionthenrepresentsanewestimateoftheuncer- dumbbellconfigurations. Thesubstitutionaldefectcon-
tainty, which is added to the uncertainty arising from sistently relaxed to the off-site substitutional position
ATP measurements and that of TEP measurements, to discussedin[36]unlesssymmetryconstraintswereim-
formthetotalerroraboutthesolutionconcentrationof posed. Thissuggeststhatthehighspinhigh-symmetry
eachsample(usedinlatergraphs).Inthefollowingsec- substitutional site observed by Christensen et al. [38]
tions, when referring to alloy composition, we refer to maybeduetoinsufficientlyhighdegreeofconvergence
these calculated concentrations of Cr or Fe in solution duringgeometryrelaxation.
ratherthanthenominalcompositionofthebuttons. Similarly, the tetrahedral site relaxed into the newly
observedcrowdionconfigurationifasuitablylargesim-
3.2. FeandCrsolubilityinα-Zr ulationcellisemployed.Thecurrentworkidentifiesan-
To calculate the solution energy of the alloying el- otherinterstitialsitethathasnotpreviouslybeensimu-
ement within the non-interacting regime, DFT simula- lated: theoff-siteoctahedral. Thisissignificantlymore
tions were performed with single Fe and Cr defects in stable than any other interstitial site for Fe, but is un-
thesupercellsdescribedpreviously,equivalenttoanal- stablefortheaccommodationofCr,whichisconsistent
loy concentration of 0.67at% Fe/Cr. A number of re- withthelargeratomicradiusofCr. Mo¨ssbauerstudies
centstudies[40–46]havehighlightedthatfinitecellsize ofFeinα-Zr[84]suggestedthat∼ 30%ofthetotalFe
effectsmaysignificantlyaffectthepredictedstabilityof insolutionislocatedinoff-centreinterstitialsitesofthis
Zrselfinterstitialsatoms(SIAs);simulationsperformed type.
using a small size of supercell, or without the use of Regarding the accommodation of Fe, all interstitial
finite-sizecorrectionmethods,yieldspuriousresults. In sitesprovidemorefavourablesolutionenergiesthanthe
particular,theworkbyVarvenneetal.[45]showedthat substitutionalsite.Thisisinagreementwithexperimen-
with an energy correction term calculated from linear tal diffusivity measurements [85–87], and all previous
elastic theory, as used in the current work (E ), su- DFTcalculations[6,36,37]withthesoleexceptionof
int
percells containing 200 Zr atoms accurately described [38]. ItisunclearfromtheliteraturewhetherCratoms
Zr SIAs, and supercells containing only 96 Zr atoms exhibitasimilarpreferenceforinterstitialaccommoda-
yieldeddifferencesofonly40–150meV.Allthedefects tion. Experimental diffusivity measurements indicated
considered in the current work cause significantly less that Cr diffuses 2–4 orders of magnitude slower than
lattice strain than Zr SIAs. This provides confidence Fe in α-Zr. However, Fe is reported to diffuse 6–9 or-
that the supercell employed in this study, which con- ders of magnitude faster than for Zr self-diffusion and
tained150Zratoms,issufficientlylargetoavoidspuri- 9–16ordersofmagnitudefasterthansubstitutionalso-
ous finite size effects. This is corroborated by the fact lutes[86–89]. ThissuggeststhatthetransportofCrin
thatEp wasintherangeof3–60meVforallpointde- the α-Zr lattice may be mediated by an interstitial so-
int
fects. As a further confirmation, all point defect simu- lute. The current work is in excellent agreement with
lationswererepeatedunderconstantvolumeconditions a previous DFT publication [6], which highlights that
(ε = 0) and constant pressure conditions (σ = 0), and whilstthepreferredsiteforCrsolutionissubstitutional,
the difference in energy between the two methods was the difference in energy between that and the intersti-
consistentlylessthen0.6%. tial octahedral site is very small, and therefore Cr is
Many interstitial positions, as well as the substi- expected to exhibit both substitutional and interstitial
tutional and Zr-Fe and Zr-Cr dumbbell configura- behaviour. Despite this, subsequent modelling studies
tions were considered; The resulting formation ener- havenotconsideredthepossibilityofinterstitialaccom-
gies (Eε=0), relaxation volumes (Ωε=0) and anisotropic modationforCr[38].
f
strainsonthesupercell(εσ=0,εσ=0)arereportedinTa- Experimental evidence of the dual nature of Cr ac-
11 33 commodationinZrisprovidedbyourAPTwork. The
ble2forallstabledefects.1 Someinterstitialpositions
dataset of the sample is shown in Figure 1(a). A
were found to be unstable, that is, the defects moved
10 × 10 nm2 subset of the data centred on the (0002)
toanothersiteuponrelaxation; theseincludethetetra-
poleindicatedinFigure1(b),andgoingdownthewhole
hedral positions (which appears as stable when simu-
lengthofthedataset,wasexported2. Advancedspecies-
latedinsmallsupercells),thehexahedralposition(also
specific spatial distribution maps were applied to this
1Eσ=0andΩσ=0werewithin0.07%and3.7%ofEε=0andΩε=0 2Duetothesmallsizeoftheregion,alargeportionofthoseatoms
f f
respectively. reside at the boundary and have a limited number of neighbours.
6
Table 2: Defect formation energy (Eε=0) and volumetric properties for all defects that may accommodate Fe or Cr
f
in bulk α-Zr. εσ=0 and εσ=0 are the strain in the a and c direction respectively. Full relaxation volume tensors are
11 33
presentedinAppendix A.
Eε=0(eV) Ωε=0(A˚3) εσ=0(%) εσ=0(%)
f 11 33
Fe off-sitesubstitution 1.388 −10.40 −0.15 −0.15
substitution 1.709 −16.25 −0.02 −0.40
octahedral 1.079 13.70 −0.02 0.27
off-siteoctahedral 0.941 13.53 −0.07 0.25
trigonal 1.212 13.42 0.18 −0.26
crowdion 1.172 13.52 0.10 0.09
Cr off-sitesubstitution 1.732 −11.31 −0.03 −0.21
substitution 1.892 −12.83 −0.11 −0.13
octahedral 1.882 15.20 −0.05 0.30
trigonal 1.968 13.65 0.16 −0.17
crowdion 2.061 15.41 0.15 0.08
subsetandtheresultingdataareplottedinFigure3.The 30
Zr-Zr distribution exhibits broad peaks corresponding 15
to the (0002) atomic planes. The slight and progres-
25
siveshiftawayfromtheexpectedlocationofthepeaks
cctaahaulroeensnsaibegnved-etbhraieanitstgtwr[eci9berd0yuei]snts.ettatdrThlilbhetoouegmtdirZoaairsinp-nthCooiprrfcteiCddaoikrinsrsstaertcoiiobntfimuottthhnsieoe,rneeZt,lxoarhmw-tZiihvboreiigctdrsthiaospmmtZer,iarebkaaausssttouidtohmriensass-t. 5Zr counts (x10) 10 1250 Cr counts (x10)
Tathoimssisbceoinngsilsotecnattewdiathtinatesirgstniitfiiaclasnittefsr.acTthioisniosfththeefiCrsrt Zr− 5 10 Zr−
evidenceofinterstitialsprovidedbyAPT.Asimilarpro-
5
cedurewasattemptedontheFe-containingsample,but
the large volume fraction of SPPs in the sample [49]
0 0
made it extremely challenging to achieve a sufficient
signal-to-noiseratiotogeneratedefinitiveresults. −1.0 −0.5 0.0 0.5 1.0
z offset (nm)
3.3. Latticeexpansionofbinaryalloys
AccommodationofFeandCrpointdefectsintheα-
Figure3: In-depthspatialdistributionmapshowingthe
Zr matrix is predicted to cause noticeable lattice strain
Zr-Zrdistribution(inpurple)andtheCr-Zrdistribution
(see Table 2). Therefore, the lattice parameters of the
(in blue). Dashed lines indicate d spacings from
(0002)
α-Zrmatrixweremeasuredforallbinarysamplesusing
XRDdata.
XRD.Thechangeinlatticeparameteraasafunctionof
alloyconcentrationisshowninFigure4,togetherwith
aprojectionofVegard’slawcalculatedfromthelattice
TheDFTpredictionsrelyonconfigurationalaverages
parameters of the pure elements (dashed line) and the
ofthedilutepointdefectsattemperaturesof25K,300K
DFT predictions of lattice expansion due to the pres-
and 600K. This averaging technique does not include
ence of defects (dotted lines). Note that the change in
other temperature effects such as thermal expansion or
compositionandthechangeinlatticeparameterarevery
phononscattering. Inotherwords,themodelrepresents
small,neartheaccuracylimitoftheXRDequipment,as
asolutionthathasbeenhomogenisedatthosetempera-
highlightedbythelargeerrorbars.
turesandsubsequentlyquenched.
Figure 4 shows that, for Cr-Zr solution, both Veg-
Therefore,theresultstakendonotbareaquantitativeweight,butcan
providequalitativeinformation. ard’s law and the DFT predictions are in good agree-
7
5.162 Cr Fe
A) 5.160
(cid:176)(
c
er 5.158
et
m
0 0
a 5.156
r
a 600(cid:176) C
p
e 300(cid:176) C
c 5.154
latti 25(cid:176) C
5.152
Cr Fe
3.235
0 0
)
(cid:176)(A 600(cid:176) C
meter a 33..223334 32050(cid:176)(cid:176) C C
0 0
a
par 3.232 600(cid:176) C
e
attic 3.231 300(cid:176) C
l
25(cid:176)

3.230 C
0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6
0 0
Cr at. % Fe at. %
Figure4: Measuredlatticeparameteraasafunctionofalloyingelementinsolution(points)andlatticeparametersa
andcfromtheory(lines). DashedlinerepresentsVegard’slaw;Dottedlinesrepresentthecurrentpredictionoflattice
expansionusingconfigurationalaveragingofDFTsimulationsofdilutedefects. Hollowsymbolrepresentlowpurity
sample. Verticalerror barsrepresent theuncertainty dueto machineerror, backgroundnoise and Rietveldanalysis;
horizontal error bars represent the compound uncertainty of TEP measurements, APT concentration measurements
andstandarddeviationfromlinearregressionofFigure2.
8
ment with experimental observations, if the low purity Applied pressure (GPa)
sample(hollowcircle)isdisregarded. However,forthe 5 2.5 0 −2.5
Fe-Zr solution DFT predictions differ greatly to Veg-
2.2
ard’slaw. Further,forthealatticeparameter,DFTpre-
dictionsareinbetteragreementwithexperimentaldata 2.0 CrZr
atlowconcentrations(nearthesolidsolubilityofFein
Crtri
Zr) but the agreement is somewhat lost at higher con- 1.8 Croct
centrations. Fortheclatticeparameter,DFTresultsare
in stark contrast to Vegard’s law in that an expansion eV) 1.6 Crcrow
is expected instead of a contraction. XRD data for the solE ( 1.4 FFeeZtrri
clatticeparameterwereinconclusiveasthelowmulti-
Feoffoct
plicityofthecdirectioncausedtoomuchscatterinthe
1.2 Feoct
data. The predicted deviation from Vegard’s law sug-
geststhatlatticeparametermeasurementsarenotasuit- 1.0 Fecrow
ablemeanstoestimatethesoluteconcentrationofalloy-
ingelements.
−5 −4 −3 −2 −1 0 1 2 3
When performing the configurational average, it is
Hydrostatic strain (%)
implicit that the defects are not interacting; therefore,
strictly, the average is only valid at the dilute limit. Figure5: EnergyofsolutionofFe(beige)andCr(blue)
Since the binary solid solutions investigated here are accommodated as interstitial species (hollow symbols)
abovetheirrespectivesolubilitylimits,itispertinentto andsubstitutionspecies(filledsquares)asafunctionof
assumethatthealloyingatomsareinteractingwitheach hydrostatic strain. The simulation cells were strained
other. Morespecifically,thecompressivestrainfieldof prior to adding the defect, by applying an external hy-
an interstitial defect is likely to increase the formation drostatic pressure, displayed in the secondary x-axis
energyofanotherinterstitialdefectinitsvicinity,whilst above(positive=compressive).
reducingthatofasubstitutionaldefect(which,duetoits
negativerelaxationvolume,hasatensilestrainfieldas-
sociated with it). This hypothesis was corroborated by
repeating the defect simulations in pre-strained super- definedbycombiningasubstitutionaldefect(M )with
Zr
cells,seeFigure5. Underacompressivestrain,thesta- and an octahedral or off-site octahedral interstitial de-
bilityofsubstitutionaldefects(filledsquares)increases fect (M ), since these are the most stable defects
i(oct)
while that of interstitial defects (hollow symbols) de- with opposing strain fields for Cr and Fe respectively
creases; and the opposite is true under a tensile strain. (from Section 3.3). All such configurations that could
This helps explain the lack of preference between in- fit in a 5 × 5 × 3 supercell of α-Zr (150 Zr atoms)
terstitial and substitutional accommodation that is ob- were investigated, leading to defect-defect separations
served in the APT spatial distribution map of Cr (Fig- that range from 2.30A˚ for the first nearest neighbour
ure3). (1nn)to6.97A˚ forthe7nnconfiguration.Whenconsid-
Aswellasaffectingtherelativesolutionenergies,the ering mixed Fe-Cr clusters, the Cr atoms were placed
strain fields of the point defects may provide a driv- in the substitutional sites and the Fe atoms in the in-
ing force for diffusion: defects with opposing strain terstitial sites, {CrZr : Fei(off-oct)}, owing to the smaller
fieldsmayattracteach-otheratdistancesofuptoafew atomic radius of Fe and its preference for interstitial
angstroms, whilst defects with same-sign strain fields sites (see section 3.3). The simulations of the clus-
will repel one another. When combined with the ex- ters were relaxed to a high level of force convergence
trememobilitiesofFeandCr[37,85,91],thismaylead (0.05eVA˚−1) and the atomic positions were perturbed
totheformationofdefectclusterswithareducedover- bysmallamountsinrandomdirections.Furthermore,to
alllatticestrain, henceareducedlatticeexpansionand provide greater degrees of freedom to the simulations,
morefavourablesolutionenergy. these were repeated under σ = 0 conditions as well as
ε = 0 conditions. This combination ensures that the
3.4. Clusterformation BFGS minimiser [81, 82] is unlikely to trap atoms in
To investigate the formation of Fe and Cr clusters, localenergyminima. Inotherwords,thestartingposi-
simulations containing two extrinsic species were first tionsarejustthat,andtheextrinsicatomswereexpected
considered. Thestartingpositionsfortheclusterswere toexploretheenergysurfaceuntillowestenergyconfig-
9
(a)Fe-Fedumbbell (b)Cr-Crdumbbell (c)Fe-Fe4nn
(d)Fe-Fe3nn (e)Cr-Cr3nn (f)Fe-Fe5nn
Figure 6: Brown spheres represent Fe atoms, dark blue spheres represent Cr atoms, turquoise spheres represent Zr
atoms,translucentspheresrepresenttheinitialpositionofselectedatoms.
urationswerefound.Figure6showstheinitialandfinal smaller than those of the single-atom dilute defects.
configurations of some clusters that have moved from Furthermore,thecombineddefectvolumeofdiluteFe
Zr
theiroriginallatticesites. Inallcases,theresultingde- andFe is3.13A˚3,whilethatofthebounddumbbellis
i
fectisanelongatedorextendeddefect,ofteninvolving only−2.49A˚3. SimilarlyforCr, thecombinedvolume
one or more Zr SIAs. Most notably the 1nn clusters ofdilutedefectsis3.81A˚3comparedwith−1.95A˚3for
moved into a split substitutional (or dumbbell) around the dumbbell. Finally, in the mixed case (in which Cr
a Zr lattice site. An in-depth analysis of the dumbbell takesthesubstitutionalsiteandFetakestheoff-octin-
configurations, including a comparison with the intrin- terstitialsite),thecombinedvolumeofdilutedefectsis
sic Zr dumbbells from previous work [46, 92], is pre- againgreaterthanthatofthedumbbell(2.50A˚3against
sentedinAppendix B. −0.42A˚3). Thissuggeststhatpartofthebindingenergy
Asummaryoftheformationenergies, bindingener- comesfromareductionoflatticestrain.
giesandrelaxationvolumesofallthe2-atomdefects— Notably,manyoftheFe-Fedefectpairsexhibitnega-
in their final relaxed positions — are presented in Ta- tiverelaxationvolumes,resultinginatensilestrainfield,
ble3.Inallcasesthedumbbelldefectisconsistentlythe despite the addition of one extra atom in the supercell.
moststableconfiguration,independentofthespeciesin- In particular, the most favourable configurations (1nn,
volved, and the relative preference for the dumbbell is 3nn&4nn)exhibittensilestrainfieldsarisingfromre-
ashighas0.5–0.6eVcomparedtothenextmoststable laxationvolumesof−2.49A˚3,−2.39A˚3and−3.59A˚3.
configuration. Importantly, allconfigurations up to the ToinvestigateifmorethantwoFeorCratomscould
5thnnareasinglelatticejumpawayfromthedumbbell be accommodated in or around a single Zr vacancy, a
configuration.Whilstthisprovidesincompleteinforma- third and then a fourth interstitial atom were added to
tion about kinetics of cluster formation, it does imply the relaxed dumbbell configurations, as these are the
thatmultiplepathsexistformigratingextrinsicspecies most stable 2-atom clusters (see Figure 7). The result-
toreach(andbetrappedin)thedumbbellconfiguration. ingsolutionenergies(normalisedperextrinsicatom,as
With regard to the lattice expansion, all two-atom definedin[48])anddefectvolumesarepresentedinFig-
clustersexhibitrelaxationvolumesthataresignificantly ure 8. Clusters containing 3 and 4 Fe atoms exhibit
10