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:burr.patrick@gmail.com(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 composition, 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 considered. 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 total 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

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

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