Ab-initio based modeling of diffusion in dilute bcc FeNi and FeCralloys and implications for radiation induced segregationS. Choudhurya, L. Barnarda, J.D. Tuckerc,1, T.R. Allenb, B.D. Wirthd,e, M. Astaf, D. Morgana,aUniversity of Wisconsin Madison, 1509 University Avenue, Madison, WI 53706, USAbUniversity of Wisconsin Madison, 1500 Engineering Drive, Madison, WI 53706, USAcKnolls Atomic Power Laboratory, P.O. Box 1072, Schenectady, NY 12301, USAdUniversity of California Berkeley, 4165 Etcheverry Hall, Berkeley, CA 94720, USAeDepartment of Nuclear Engineering, University of Tennessee, Knoxville, TN 37996-2300, USAfDepartment of Materials Science and Engineering, University of California, Berkeley, CA 94720, USAarti cle i nfoArticle history:Received 3 September 2009Accepted 20 December 2010Available online 12 January 2011abstractA combination of ab-initio calculations and statistical mechanical models has been used to study diffusionbehavior in dilute ferritic FeCr and FeNi alloys. A full set of Onsager matrix coefcients (Lij) and tracerdiffusion coefcients (D+) were calculated both for vacancy and interstitial mediated diffusion. The keyresults are: (1) Cr is the fastest diffusing species by both vacancy and interstitial mediated transport fol-lowedbyNiforvacancyandFeforinterstitialmediateddiffusion, respectively;(2)weakinteractionsexist between Ni and Cr with vacancies as rst nearest neighbors; (3) the calculated D+ predict oppositetrends of radiation induced segregation (RIS) of Cr by vacancy (depletion) and interstitial (enrichment)diffusion mechanisms, perhaps explaining the lack of clear trends in experimentally determined Cr RISproles; (4) unlike the widely used Darken and Manning approaches, the calculated Lij can reliably pre-dict the direction of the vacancy defect ux, particularly at low temperatures; (5) the Lij calculated forinterstitial mediated transport indicate that the contribution of interstitial ux can be signicant in deter-mining RIS in these alloys; and (6) solute drag is unlikely to occur for vacancy mediated diffusion. In addi-tion, theLABoff-diagonaltermsforinterstitialtransport, whicharetypicallyassumedtobesmall, areshown to be as large as the diagonal elements LBB. These results provide a basis for understanding thecomplex radiation induced segregation behavior of Cr and Ni in ferritic/martensitic alloys. 2011 Elsevier B.V. All rights reserved.1. IntroductionFerritic/martensitic steels with Cr as the major alloying elementare likely to form the basis of many materials used in the next gen-eration of nuclear reactors. These alloys are known to exhibit bet-ter swelling and thermal shock resistance thanthe austeniticcounterparts and hence are considered as excellent candidatematerials for nuclear fuel cladding as well as rst wall and breederblanket structures for future fusion reactors [1,2]. Improvement inthe lifetime and performance of these alloys as structural materialsfor nuclear reactors requires a fundamental understanding of themicrostructural evolutioninawiderangeof temperatures andradiation conditions.Materialsusedintheradiationenvironments ofnuclearreac-torshavebeenknowntoformlargeconcentrationsofpointde-fects, e.g., vacanciesandinterstitials, aswell asmoreextendedstructural defects, e.g. dislocationsandvoids[1,3]. Formationofextended defects may degrade structural properties of these mate-rials. One radiation-induced phenomenon that potentially plays arole in microstructural changes and materials degradation is radi-ation induced segregation (RIS). RIS is the process by which the lo-cal composition of an alloy is altered due to preferentialparticipation of certain species with the vacancy and/or interstitialux tosinks [1,3]. Forexample,it iswell established that Cr de-pletesatthegrainboundaryinausteniticstainlesssteels, mostlikelyduetoaninverseKirkendall mechanism[4,5]. However,experimental observation of Cr RIS at the grain boundaries in fer-ritic/martensiticalloydoesnotshowanycleartrend, andinfacta much more complex segregation behavior as a function of irradi-ationdose. Bothenrichment anddepletionof Cr at the grainboundary has been observed [2,69]. Further, Ohnuki et al. [8] ob-served that the presence of other alloying elements also modify theRIS behavior of Cr. For example, Ohnuki et al. observed that Cr en-riches at grain boundaries in an Fe13Cr1Si alloy, but depletes inan Fe13Cr1Ti alloy when irradiated to 57 dpa with 200 keVC+ions. However, for ferritic steel witha similar Cr composition(Fe12Cr1Mo0.2C) Brimhall et al. [10] reported no measurablesegregation of Cr at the grain boundary when irradiated to about0022-3115/$ - see front matter 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.jnucmat.2010.12.231Corresponding author.E-mail address: ddmorgan@wisc.edu (D. Morgan).1The bulk of this work was done while at University of Wisconsin.Journal of Nuclear Materials 411 (2011) 114ContentslistsavailableatScienceDirectJournal of Nuclear Materialsj our nal homepage: www. el sevi er. com/ l ocat e/ j nucmat1 dpa with 5 MeV Ni++. It is not clear if these variations in RIS areduetointrinsicchangesinthetransportpropertiestothegrainboundary, orperhapsothermicrostructural differencesbetweendifferent alloys (e.g., it has been argued that formation of Cr precip-itates with other solutes at the grain boundary may alter the mea-sured RIS proles [1].)The inability to explain the RIS behavior in ferritic steels may beinpart duetoalackof completeunderstandingof thespeciesdependenceof theinterstitial ux. Interstitialsareproducedinequal quantities as vacancies under irradiation and the interactionbetween solutes and interstitial ux could play a signicant role indetermining the RIS proles of a solute. For example, it has beenargued[1] that inausteniticsteelsoversized(comparedtoFe)atoms like Cr tend to deplete at the grain boundary as the bindingenergy of Cr with Fe self-interstitials is negative. Such a hypothesismay help explain RIS behavior of Cr in austenitic steels, but doesnotresolvethevariationinexperimentallyobservedRISproleof Cr in ferritic steels. To help explain results in the ferritic steelsit has been suggested that the presence of alloying elements canchange the average lattice parameter of the Fe, and that this vari-ation can explain experimentally observed RIS prole based on thefact that Cr could change from an oversized to an undersized atomdepending on the lattice parameter of the alloy [1]. However, thesequalitativeargumentsarehighlyuncertainandrequireamorequantitative foundation.Among the minor alloying elements in ferritic steels the Ni con-tent plays an important role in determining the mechanical prop-ertiesof thesealloys[11]. SimilartoCr noclearRIStrendhasbeen observed in case of Ni. For example, in a ferritic steel alloysLittle and Stoter [12] observed that Ni enriches at the defect sinks,while Hosseni and Jones [13] reported depletion of Ni at the grainboundary in Fe3.6 wt.% Ni alloy. Similar to the case of Cr-rich pre-cipitates, formationofNi3Feprecipitatesfurthercomplicatesthemeasured RIS prole [13]. While it is realistic to expect that forma-tion of precipitates, the presence of various alloying elements, radi-ation dose and temperature may alter the evolution ofconcentration of Cr and Ni in ferritic/martensitic steels, experimen-tally it is difcult to separate these complicated interaction effects.In particular, it is difcult to determine what one might considerthe intrinsic RIS behavior in FeCr and FeNi alloys, which is duetothefundamentalspeciesdependenceofthebulkvacancyandinterstitial diffusioninthealloy. Thegoalsofthiswork ittouseab-initio methods to extract information about the bulk transportand intrinsic RIS behavior of Cr and Ni in these alloys.TheintrinsicRIStendenciesinFeCrandFeNialloyscanbeestablishedthroughthedeterminationof diffusionpropertiesofsolutes and the solvent. Such diffusion properties should take intoaccount the complex solutesolvent interactions as well as interac-tions between solutes with the defect uxes. For the sake of sim-plicityinthis paper, we choose dilute alloys of Cr andNi inbody-centered cubic Fe, as modeling the diffusion in more concen-tratedalloysrequiresasignicantlydifferent approach[14,15].Thispaperwill focusondiffusionbehaviorinferriticbccalloys,andresultswillbecomparedwithpreviouslyreportedmodelingof fcc alloys [16]. Results will also be compared to a recent study[17] on bcc FeCr that used a similar approach tothat discussedhereforvacancydiffusionandempirical potentialsforstudyingtheinterstitials(thecurrentworkwill useab-initiomethodsforboththevacanciesandtheinterstitials, andmodeltheirkineticsusing similar levels of approximation in their statistical treatment).Weadopt theframeworkof what arecommonlyreferredtoasmulti-frequency models to treat the statistical mechanics of diffu-sion of vacancies [18] and interstitials [19]. These models yield ki-netic parameters from hopping frequencies, which in turn can becalculatedfromab-initiomethods. Theresultswill becomparedwithknownexperimental data. Themainresultsof this paperare calculated values for tracer diffusion coefcients, D+i, and trans-port or Onsager coefcients, Lij (sometimes called phenomenolog-ical coefcients). The Lij are less commonly used and to understandthe utility of these parameters we here briey review their role intransport modeling.Transport of species inalloys is commonlydescribedusingFicks rst law:Ji =

Nj=1Dijnj(1)where Ji is the ux for species i, Dij is an element in the chemical dif-fusion coefcient matrix, and ni is the number of atoms of species iper unit volume [20]. Eq. (1) is a useful form of the ux equation be-cause ni are available experimentally. However, in this expressionthe kinetic and thermodynamic information about the system areinseparablycombinedinthediffusioncoefcient matrixD. Thethermodynamicdrivingforcefordiffusionisinfactthegradientinchemical potential, l, andnot thegradient inconcentration,which leads to the following, alternative expression for ux:Ji =

Nj=1Lijlj(2)where the kinetics are contained in the quantity Lij, the transport orphenomenological coefcients which are more commonly known asthe Onsager matrix [20].TheLijarethefundamentalkineticquantities andveryusefulfor bothunderstandingandquantitativepredictionof RISandothertransport behavior. Amongthevariablesdescribedabove,ofparticularinterestaretheoff-diagonaltermsLij(ij)containingkinetic information about coupled uxes between species i and j.Thesecrosstermsareessential foraquantitativeunderstandingofRISastheyincludeinformationaboutsolutedefectcoupling,e.g., vacancydrag, whichisnotavailablefromjustthediagonalterms. The off-diagonal terms will be used below to demonstratethe absence of vacancy drag mechanisms. Lij are generally not di-rectly measurable, making them difcult to obtain fromexperiment.One common path to obtaining Lij values is using either the Dar-ken [21] or Manning [22] relations to determine Lij indirectly fromexperimentallymeasuredtracerdiffusioncoefcients(D+). How-ever, such an approach has serious drawbacks. First, tracer diffu-sion coefcients (D+) are often available only over a limited rangeof temperature and composition. Furthermore species dependentexperimental diffusion data is typically available only for vacancymediateddiffusion[23,24], andhencethetransportcoefcientsinvolving interstitials cannot be determined [25]. In addition, whileDarkensapproachprovidesasimpleexpressiontocalculatethediagonal terms from measured D+, it does not supply Lij(ij). Hence,critical information about the defect-solute interactions as well asthe interaction between different atoms is absent. While it is pos-sible to calculate the off-diagonal term using Mannings approach,this only provides an approximate estimate of Lij, since it assumesthatthedefectjumpsareindependentof theenvironmentsur-rounding the defect. Moreover, it is not possible to predict negativeLij within Mannings framework, which can occur at least in theory.We will use our calculated Lij to assess the validity of the Darkenand Manning approximationsfor theLijs in the FeCr and FeNisystems.Lijs are particularly interesting to determine for interstitialtransport. Under radiation, interstitial dumbbellsareformedinmuchhigher concentrationsthanthermallyformedinterstitialsand transport of atoms through interstitial mechanisms could playa signicant role in determining RIS. Due to the lack of experimen-tal D+ for interstitial mediated diffusion it is almost impossible to2 S. Choudhury et al. / Journal of Nuclear Materials 411 (2011) 114assess their Lijfromexperiments. Afewtheoretical attempts[17,19] have been made to understand diffusion of atoms throughinterstitial dumbbell mechanisminferriticsteelsandanalyticalexpressions for Lij in terms of atomic energies have been developed[19]. However, no numerical values of interstitial Lij have been re-ported. One objective of this paper is therefore to determine a fullset of Lij Onsager coefcients for both vacancy and interstitial dif-fusionusing the energies obtainedfromab-initio calculations.These values will then be used to determine the more usual tracerdiffusioncoefcients, exploresolutecouplinganddragmecha-nism, and assess Darken and Manning approaches for calculatingL values. The implications for RIS will also be discussed.2. Ab-initio methodologyAllbulkcrystal, vacancyandinterstitialpropertieshavebeencalculatedusingtheViennaab-initioSimulationPackage(VASP)[26,27]. Inthesecalculations, weusedtheprojectoraugmentedwave(PAW) [28] methodwithaplanewavecut off of 350 eVand300 eVforvacancyandinterstitialcalculations, respectively.The calculations were spin-polarized and the PerdewBurkeErn-zerhof [29] parameterization of the generalized gradient approxi-mation(GGA) wasusedfor theexchangecorrelationpotential.The PAW potentials were generated using the following electroniccongurations: 3p6 3d5 4s1 for Cr, 3p6 3d7 4s1 for Fe, 3p6 3d8 4s2for Ni. We note that these PAW potentials include the p-electronsin the valence. PAW potentials without the p-electrons producedsignicant changes inthe interstitial energetics, oftenof over100 meV/interstitial for interstitial binding energies. Defect calcu-lations were performed with 54(1) atoms within a periodic3 3 3supercell of thebccconventional cell. Thecell shapeand volume are kept xed to that of pure Fe with a bcc structurebut internal ionic relaxations are allowed. Brillouin-zone samplingwas conductedusingtheMonkhorst andPackscheme[30]. A6 6 6k-pointmeshwasusedforvacancycalculations, while3 3 3 k-point mesh was employed for interstitial calculations.Theab-initiocalculationsforinterstitialsusealowercutoffandk-points(comparedtovacancies)asthecalculationsforintersti-tials were more computationally intensive and a much larger num-ber were required than for the vacancies. For vacancy calculationsthe errors associated with k-point mesh and energy cutoff are esti-mated to be less than 5 meV/defect for migration barriers, defectformation energies and binding energies. To understand the effectof cell size we calculated the vacancy formation energy and a fewrepresentative vacancy migration barriers with a 4 4 4 super-cell. The error for vacancy migration barriers and defect formationenergies associatedwithcell sizeis estimatedtobeless than20 meV and 40 meV, respectively. We did not calculate the vacancybindingenergyasafunctionof cell sizebut usingab-initoap-proaches Vincent et al. [31] reported that the change in rst near-est neighbor binding energy is less than 10 meV when the cell sizeis increased from 3 3 3 to 4 4 4 supercell. For interstitialswerequireonlymigrationenergiesasthesearewhatinuencethe kinetic predictions. The migration energies are converged withrespect to k-points and cutoff to within 10 meV. The errors associ-ated with cell size on interstitial migration are more complex to as-sess. However, comparing the values for FeCr interstitialmigration energies obtained in our 54 atom cell to those obtainedby Olsson et al. [32] in a 128 atom cell we nd changes of 60 meV.Although this is not as small as might be desired, it would likelycorrespond to a change of less than factor of three in diffusion coef-cientatevenarelativelylowtemperatureforRISof 400 C. Athoroughassessment of errorsassociatedwithab-initiosimula-tions for vacancy and interstitial energies are given in Refs.[33,34]. Migrationbarriers for vacancyandinterstitial hoppingwere calculated using the nudged elastic band method (NEB) withseven and three intermediate migration images, respectively. A cu-bic spline was tted to the migration energy prole and migrationbarriers were calculated by taking the energy difference betweenthesaddlepointwithrespecttotheenergyatthelatticepoint.Inthisregard, asaddlepoint representsthecongurationwherethe energy is maximum in the migration energy prole.3. Results3.1. Ab-initio calculations of vacancy diffusionThe approach to calculating diffusion constants is to nd hop-ping rates from ab-initio barriers and then use those in multi-fre-quencystatistical diffusionmodels. Inorder tounderstandtheeffect of different vacancy exchange mechanisms on solute diffu-sion, inthis workweadoptedthemulti-frequencyframeworkdeveloped by LeClaire [18] as shown in Fig. 1. It is assumed thatthe solute species is dilute enough that any solutesolute interac-tions may be neglected. The w0 is the rate of hopping in the pure Fe(not shown). We calculated the activation energies of each of thejump events,wi, for a vacancy in close proximity to a solute andthe results are presented in Table 1. The calculated migration bar-riers matchwell withmigrationbarriers availableinliterature[17,33,35]. The jumpfrequencies (wi) are calculatedfromthemigration barriers using the expressionwi = mi exp DHi;migvkT_ _: (3)We assumed a constant attempt frequency (mi) of 5 1012s1for all jumps and DHi;migvis the migration barrier for the ith vacancyjump. It is clear from Table 1 that the migration barriers are similarw2 w4 w3 w4w3w4w3FeCr/NiVacancy 1 2 3 4 Fig. 1. Illustration of multi-frequency model with rst nearest neighbor interactionfor vacancy jump in a dilute alloy with bcc structure. The numbers in the guresindicates the nearest-neighbor positions to solute.Table 1Migration barriers of vacancies obtained from ab-initio calculations.Jump type Calculated migration energy (eV)FeNi FeCrw00.67 0.67w20.68 0.58w30.55 0.69w40.69 0.65w/30.70 0.67w/40.67 0.63w//30.62 0.64w//40.59 0.62S. Choudhury et al. / Journal of Nuclear Materials 411 (2011) 114 3for FeCr and FeNi systems. In the ab-initio calculations we trea-ted the associative and dissociative jumps to and from the 2nd, 3rdand 4th nearest-neighbor positions explicitly (Table 1). However,inordertottheab-initiobasedratesintoLeClairesframeworkwe replaced the rate of the dissociative and associative jumps fromor to the rst nearest neighbor site of the solute with an effectiverate, denotedbyweff3andweff4respectively[16,36]. Theeffectiverate is determined by an average weighted by the number of path-ways to each nearest-neighbor distance as shown below.7weff3= 3w3 3w/3 w//3(4)and7weff4= 3w4 3w/4 w//4(5)ThejumprateswereusedtocalculatethevacancyOnsagercoefcients as a function of temperature in FeNi and FeCr alloyusingtherelationspresentedinRef. [37](seeAppendixA)andtheresultsareplottedinFig. 2. InthegureA = Fe, B = CrorNi.The dotted and the solid lines in the gure represent the off-diag-onal and the diagonal Onsager coefcients, respectively. It shouldbe pointed out that based on the expressions of Lij in Appendix Athe value of LAA is different for FeCr and FeNi alloys, as Cr andNi have different solute enhancement factor (b). We observed thatthe difference in value of LAA in FeCr and FeNi alloys is less than5% within the range of temperature described in the gure. Hencefor the sake of clarity of the gure we only present LAA in FeCr al-loy in Fig. 2. For the calculation of vacancy Onsager coefcients weuse a value for mole fraction of solute cB = 0.01, which correspondsto nB = 8.5 1020atoms/cm3. It can be observed that as the jumprates are weakly species dependent the Onsager coefcients of NiandCraresimilarwithintherangeoftemperaturedescribedinthe gure. In the gure a change of slope in Lijoccurs atT = 1043 Kbecauseof theferromagneticphasetransition, asex-plained later in this section.The interaction between the solute and the vacancy may affectthe ability of the solute to diffuse through vacancy defect ux. For adilute alloy, if the vacancy solute interaction is negligible then thesolute diffuses though vacancysolute exchange (the inverse Kir-kendall mechanism). However, if a strong attractive interaction ex-istsbetweenthesolute andthevacancythen thesoluteandthevacancy can migrate as a complex species, or in other words, thesolute is dragged in the direction of vacancy defect ux. In orderto understand how a vacancy binds with a solute atom (Cr, Ni) asa function of separation between solute and vacancy we calculatedthesolutevacancybindingenergyasafunctionof thenearestneighbor locations.Therstnearestneighborbindingenergybetweenthesoluteand the vacancy in a bcc iron matrix comprised of N atomic sitesis obtained from Eq. (6),DHbindv=E(N2; 1Cr;1V; 1nn) E(N1; 1V) E(N1; 1Cr) E(N);(6)where E(N 2; 1Cr;1V; 1nn) is the energy of the supercell contain-ing (N 2) Fe atom and one Cr atom with a vacancy as a rst near-est neighbor [34]. Similarly, the second, third and the fourth nearestneighbor binding energies were calculated and the results are pre-sentedinFig. 3a. Withthesignconventionusedhereanegativebinding enthalpy means attractive interaction and a positive valueindicates repulsion. The gure shows that a weak attractive interac-tion exists between the vacancy and the solute in almost all cases.The one exception is for the binding energy of Ni with the vacancywhen they are second nearest neighbors to each other, which showsa quite strong attraction. The source of this attraction is not clearbut it is possibly due to a combined effect of size as well as chemicalinteraction between the vacancy and the Ni atoms, as reported pre-viously[31]. Thecalculatedbindingenergiesagreewell (within25 meV) with previous ab-initio calculation for the rst and secondnearestneighborsintheFeNibinarysystem[31]. Thesebindingenergies suggest that the vacancydrag mechanismis unlikely,although the effect of the second neighbor Ni interaction could leadto vacancy drag.Tomakeamorequantitativeassessmentofthevacancydragthe Lij can be used to determine the vacancy wind (G), which is of-ten used to understand the effect of vacancies on the diffusion ofsolute atoms [38]. The parameterG is related to the off-diagonalOnsagercoefcient (LAB) throughtherelationG = LAB/LBB[39]. Itshould be noted that the Onsager coefcient LvB in a binary alloycanbeexpressedasLvB = LBB(G + 1). ItisclearthatforG < 1,LvB is positive, i.e. the vacancy and the solute diffuse in the samedirectionasacomplexspecies. Fig. 3bshowsthevacancywindasafunctionoftemperatureforFeCrandFeNisystem. Itcanbe seen that G > 1 at all temperatures of interest for both the al-loys. However, it has been observed that G < 1 for T < 150 K (notshown). Weconclude that at all temperatures of interest the va-cancydragof substitutional soluteis unlikelytooccur for theFeCr and FeNi systems,and the diffusion occurs through a va-cancysolute exchange mechanism. Such a conclusion is consistentwith thegenerally weakbinding between thesolute and theva-cancy (Fig. 3a). We note that If Ls are derived by applying DarkensapproachtoD+valuesthenLAB = 0andthevacancywind(G) isstrictly equal to zero.Fig. 3bdemonstratesthattheLFeCr/LCrCrratiochangessignataround 550 K for the FeCr system. Our calculation shows that LCrCrremains positive below this temperature as it must by the secondlawofthermodynamics[20]. Therefore, itisLFeCrwhichchangessignasthetemperatureisdecreasedbelow550 K. Wenotethatsuch change in sign of LAB is not predicted by applying ManningsapproachtoobtainLfromtheD+values, whichgivesLABP0(anegative value is also not possible to predict in Darkens approxi-mation, whichgivesLAB = 0). Moreover, verylittleexperimentalD+dataareavailableatlowtemperaturestocalculateeventhediagonal terms reliablyusingeither Darkens or Mannings ap-proach. Thus, themulti-frequencyapproachprovidesapowerfultool to determine Lij, including cross terms, over a wide tempera-ture range.Experimental validation of the energies calculated from ab-ini-tio calculations can be done by comparing the ab-initio calculatedvacancytracer diffusioncoefcients (D+) of Fe, Cr andNi withexperimentalmeasurements. Thevacancytracerdiffusioncoef-cient of Fe and the solutes as a function of temperature can be cal-culated using the expressions given below [18].14 0.830.880.930.981.031.08 LFeFe in Fe-Cr - ab initio LFeCr in Fe-Cr - ab initio LCrCr in Fe-Cr - ab initio LFeNi in Fe-Ni - ab initio LNiNi in Fe-Ni - ab initio 1/T ( x 10-3 K-1) LFeFe in Fe-Cr ab-initio LFeCr in Fe-Cr ab-initioLCrCr in Fe-Cr ab-initioLFeNi in Fe-Ni ab-initioLNiNi in Fe-Ni ab-initio12 10 8 6 4 Log (Lij(1/meVs)) Fig. 2. Vacancy Onsager coefcients obtained from multi-frequency model.4 S. Choudhury et al. / Journal of Nuclear Materials 411 (2011) 114D+Fe(0) = a2f0w0 expDHfvkBT_ _; (7)D+Fe(cB) = D+Fe(0)(1 bcB); (8)D+Cr=Ni(cB) = a2f2w2 exp(DHfv DHbindv)kBT_ _; (9)whereaisthelatticeparameter, Tisthetemperature, DHfvandDHbindvare the formation energy of the vacancy in pure Fe and therstnearestneighborbindingenergybetweenasoluteandava-cancy, respectively. D+Fe(0)isthetracerdiffusioncoefcientofFein pure Fe, while D+Fe(cB) and D+Cr=Ni (cB) represent the tracer diffusioncoefcientof Feand solutes inthedilute alloy,respectively. Notethat in the expression for D+ in Eqs. (7) and (9) the entropy termsare set to zero, which is a simplifying approximation. We will dis-cuss the effect of entropy terms on D+ later in this paper. Assumingthe alloys to be dilute in solute, the pure Fe lattice parameters areusedfor all tracer diffusioncalculations. Thecalculatedvacancyformation energy(2.23 eV)inpure Fematches wellwith vacancyformationenergiesavailableinliterature[17,33,40]. ForpurebccFe, we used correlation factor f0 = 0.7272 [18]. The correlation factorf2 is obtained as f2 = v/(2(w2/weff3) + v) where, v = 7 (1 + 0.512(w0/weff4))1 2(1 + 1.536(w0/weff4))1 (1 + 3.584(w0/weff4))1. [41] Be-foretheD+valuescanbedeterminedwemustaddresstheissueof magnetic changes in the Fe.It is well knownthat pureironwhencooledbelow1043 Kundergoes a phase transition from a paramagnetic to a ferromag-neticstate[42]. Thisphasetransformationaffectstheactivationenergy of vacancy diffusion in pure iron and its alloys. It has beenobserved that in the paramagnetic state D+ for pure iron follows anArrheniusrelationship, but belowtheCurietemperaturetheD+deviates downwards from the Arrhenius type relationship extrap-olated from the paramagnetic state. This deviation has been attrib-uted to temperature dependence of the activation energy forvacancydiffusionarisingduetothechangeofmagnetizationinpureFeanditsalloys[43]. Inotherwords, intheferromagneticstate the correct activation energy for vacancy diffusion is a func-tion of the spontaneous magnetization. We have modied the acti-vationenergiescalculatedwithab-initioapproachtoaccountforthemagneticchangeswithtemperatureusinganempiricalrela-tionship. The details of this approach are given in Appendix B.The calculated tracer diffusion coefcients for pure Fe (Eq. (7)),and for Cr and Ni (Eq. (9)) solutes in Fe, are shown by the solid linesin Fig. 4a. In the gure D+(cB) (see Eq. (8)) for Fe has not been shownseparatelyasthesoluteenhancementfactor(b)isweakathightemperatures. InFig. 4ascatteredsymbolsshowtheaverageofthe experimentally measured diffusion coefcients ([4448], refer-ences within [23,24]). It should be pointed out that among the listofreferences;weignoredreferencesinwhichdiffusiondataarepresented in Arrhenius form through the ferromagnetic transitiontemperatures. It is clear from the gure that the slope of the ab-ini-tio based diffusion coefcients as a function of inverse temperature(i.e. the activation barrier) matches well with experimental mea-surements. Further, therelativebehaviorof thediffusioncoef-cientsbetweenpureFe, Cr andNi predictedfromtheab-initiobased approach agrees well with experimentally observed behav-ior, i.e. both experimentally and ab-initio calculated D+ show thatCr is the fastest diffusing species by a vacancy mechanism while-1.5 -1 -0.5 0 0.5 40070010001300 Fe-Cr Fe-Ni -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 CrNi Repulsion Binding Binding Energy (eV) Nearest Neighbor Position (a) G, Vacancy Wind T (K) (b) 1st2nd3rd4thFig. 3. (a) Solutevacancy binding energy vs. nearest neighbor position; (b) vacancy wind, G, for FeCr and FeNi.0 2 4 6 8 10 12 40010001300 D*(Cr)/D*Fe - ab initio D*(Ni)/D*(Fe) - ab inito D*(Cr)/D*(Fe)- exps. D*(Ni)/D*((Fe) - exps. 0.850.90.9511.051.1 Fe -ab initio Ni - ab initio Cr - ab initio Fe - 10 exps. Ni - 3 exps. Cr - 4 exps. D*(Cr, Ni) (cB) /D*(Fe) (cB ) T (K)Ferromagnetic Tc=1043( K) ParamagneticParamagnetic Ferromagnetic Tc=1043( K) 1/T ( x 10-3 K-1) (a)(b) -14 -16 -18 -20 -22 Log (D*(m2/s)) 700Fig. 4. (a) Vacancy tracer diffusion coefcients of pure Fe, Ni and Cr in alloy. In the gure the solid and the dotted lines shows the tracer diffusion coefcients obtained fromab-initiocalculationsandaverageofexperimentalmeasurements, respectively;(b)ratioofCrandNivacancytracerdiffusioncoefcientsrelativetoFetracerdiffusioncoefcients.S. Choudhury et al. / Journal of Nuclear Materials 411 (2011) 114 5the diffusivity of Fe and Ni are quite close to each other. This resultcan be explained based on the migration energy (see Table 1) forthe jump involving direct exchange between the vacancy and thespecies, w2. Themigrationenergyfor w2jumpis 0.58 eVand0.68 eVfor Cr andNi, respectively, whilethemigrationbarrierfor Fe is 0.67 eV for the w0 jump. In Fig. 4b, we present the ratioof calculated and measured tracer diffusion coefcients of solutesto that of Fe as a function of temperature. We should point out thatin plotting calculatedD+ for Fe we have taken into considerationthesoluteenhancementfactorb(Eq. (8)), asitwasobservedbyWong et al. [17]. that b is signicant at low temperatures. The ratioof D+ is particularly important in predicting RIS proles in an irra-diated environment as the concentration prole of atoms dependson the relative value of the tracer diffusion coefcients rather thantheabsolutetracerdiffusioncoefcients. Itcanbeobservedthatwithinthetemperaturerange inwhichexperimental data areavailablethecalculatedtracer diffusioncoefcientsmatchwellwith experimental measurements except for the FeCr system be-low the ferromagnetic transition temperature. Although the abso-lute ratios are quite close, the experimental D+(Cr)/D+(Fe) does notseem to be rising with decreasing temperature, as seen in the ab-initio based model. Such error may arise from ab-initio calculatedenergies. Forexample, anerrorof20 meVinab-initiocalculatedmigration barrier (see Section 2) may explain the discrepancy be-tweencalculatedandmeasuredratioof D+inFeCr systematT = 900 K. Interestingly, for the FeNi system, within the same tem-perature ranges D+(Fe) is slightly greater than D+(Ni). However, atlow temperature D+(Ni) is greater than D+(Fe).We should point out that in similar ab-initio based modeling forFeCr in Ref. [17], the diffusion coefcients were calculated basedon nine frequencies. In the present calculations, although eight dif-ferent jumps (see Table 1) were considered, only four effective fre-quencies were used in calculating tracer diffusion coefcients andOnsagercoefcients. TounderstandtheerrorincorporatedusingfoureffectivejumpfrequenciesonRISpredictionwecalculatedtheratioof tracerdiffusioncoefcientof Crwithrespect toFeusing the jump barriers presented in Ref. [17], rst with all the ninefrequencies as presented in Ref. [17] and latter with four effectivefrequencies calculated from the jump barriers in Ref. [17]. We ob-served that independent of temperature the D+(Cr)/D+(Fe) only var-ies withina factor of 3betweenthe two approaches. As theanalytical expressions to calculate the Onsager coefcients for va-cancy transport provided in reference [37] only includes a limitednumber of frequencies, we used four effective frequencies through-outthisworktoensure theconsistencyofthecalculationsofD+and Lij throughout this work.It can be observed in Fig. 4a that ab-inito sets of data under pre-dict the experimental diffusion coefcients quite signicantly. Forexample, at T = 1193 K for pure iron, the ratio (R) of the average D+measured experimentally to ab-initio calculated D+ is R = 684.2. Asanempirical correctiontotheab-initiobasedvalueswesimplymultiplytheab-initiocalculatedD+forpureironandsolutesforall temperatures with the ratio R. We observed that the calculatednew D+ now lies within the error bar of the measured D+ over therange of temperaturespresented in Fig. 4a for Fe, Cr and Ni (notshown). Thediscrepancybetweenthemeasuredandcalculatedtracer diffusion coefcientscould arise from several factors (or acombinationof factors) that havebeenneglectedincalculatingthe tracer diffusion coefcients and/or fromerrors in the calculatedab-initiovalues. Theneglectedfactorsincludetheentropyofva-cancy formation and migration, both of which can have complexcontributionsfromvibrations, magnetismandelectronicsources[16]. For example, using molecular dynamics simulations the en-tropyofvacancyformationinferromagneticironhasbeenesti-matedtobe 1.78kB, [49] whichwouldenhance the D+valuesabout a factor of 6. Using frozen-phonon calculations, Huanget al. [50] recently calculated the various entropy terms for diffu-sion of impurities like Ta and Zr in pure Fe. It was found that theinclusionof these entropy terms yields goodagreement withexperimentaldatafortheirdiffusionconstants. Accurateestima-tionof all thefactorsneglectedincalculatingtheD+, orpreciseassessment of error originatingfromab-initiovaluesis beyondthe scope of this paper, because the primary focus of this researchistodevelop parameters forRIS modeling inFeCrNi alloys. Asmentioned above, RIS depends on the ratio of D+ of the solute withrespect to the solvent, rather than the absolute D+.Insummary, our ab-initiocalculationsaccuratelypredict therelativerelationshipof tracer diffusioncoefcientsbetweenFe,Cr and Ni, but the calculated tracer diffusion coefcients are signif-icantlylowerthattheexperimentalvalues. CalculatedLijforva-cancymediateddiffusionshowsthatvacancydragisunlikelytooccurbothinFeNi andFeCrbccalloysatall temperaturesofinterest.3.2. Ab-initio calculations of interstitial diffusionIt is well known that in metals the formation energy of intersti-tial dumbbells (-4 eV) is much higher than the vacancy formationenergy (-12 eV), which generally results in very low concentra-tion of interstitials at thermal equilibrium. However, under irradi-ation, interstitials and vacancies are produced in equal numbers, soasignicantpartofdiffusionofatomicspeciesunderirradiationcanoccurthroughinterstitial mediateddiffusion. Asmentionedin the introduction it is often difcult experimentally to obtain dif-fusiondataforinterstitialsinmulti-componentalloys. Recently,suchdata are increasingly obtainedusing ab-initio techniques[16,32,51]. Inthis paper, we adopt the multi-frequency Barbeand Nastar [19] model based on self-consistent mean eld theoryto calculate diffusion properties in FeCr and FeNi alloys throughinterstitial mechanisms using ab-initio techniques.Interstitials dumbbells in dilute bcc alloys can assume a numberof orientations, typically categorized by their crystallographicalignment. Ab-initio calculations have shown that the Fe self-inter-stitial in pure bcc Fe is most stable in the 1 1 0) dumbbell orienta-tion [33,52]. However, in case of self-interstitials in pure bcc Cr the1 1 1) is the most stable conguration [53]. It is to be noted that analternative 2 2 1) stable conguration has also been proposed re-cently for self-interstitials in pure bcc Cr [32]. Formation energiescalculatedelsewhere[34,54,55] reveal thatthe 110 ) orientationis more stable than the 1 1 1) orientationfor alltheinterstitialsand interstitial-solute complexes that are included in our calcula-tionof the interstitial phenomenological coefcients for diluteFeCr and FeNi alloys. It is notable that the FeNi 1 1 0) mixedinterstitial is unstable relative to an FeFe 1 1 0) interstitial withaNinearestneighbor, suggestingqualitativelythatNimigrationthrough interstitials could be unfavorable [54].We used the model developed by Barbe and Nastar [19] for dif-fusion of 1 1 0) interstitial dumbbells in dilute bcc metals to calcu-lateinterstitial transportcoefcientsfordiluteFeCrandFeNialloys. Within the framework of this model the hopping event ofan 1 1 0) ABinterstitial dumbbell involvesthedisplacement ofthe AB atom pair towards a target atom C to form a new dumbbellBC (ifB is the atom that jumps), while atom A now occupies thesubstitutional latticesiteof theinitial ABdumbbell. AsACcanbedifferent typesof atoms; suchahoppingmodemaychangethecompositionof thedumbbell. Followingtheworkof BarbeandNastarweonlyconsideredtherstnearestneighborjumpsand allowed displacements of atoms by simple translationofatoms or a combination of translation and rotation by 60. In thiscontext, atargetsiteisdenedasthenearestneighborsitethatanatominthedumbbellcanhoptodirectly,whileanon-targetsiteisanearestneighborsitethatanatominthedumbbellcan6 S. Choudhury et al. / Journal of Nuclear Materials 411 (2011) 114only hop to if the interstitial rst undergoes an onsite rotation. Itshouldbenotedthatdisplacementthroughtranslationrotationmechanism results in an orientation of the BC dumbbell differentthantheinitial ABdumbbell. Further, basedonRef. [19], eighttypes of dumbbell congurations are considered. Each of thedumbbell congurations can displace through a translationalmigration (denoted sixi) and a translation combined with a rota-tion (denoted xi), for a total of 16 jump types. These jump typesaredepictedschematicallyinFig5. Jumptypes3, 5, and7arenotpresentedbecausethesearesymmetriccounterpartsto2, 4,and6, respectively(i.e., jumptype3isidenticaltojumptype2inreverse). Usingab-initiomethodswecalculatedthemigrationbarriers of all the 16 types of hopping events in FeCr and FeNialloys and the results are shown in Table 2. The migration barrierof thejumptypei isdenedastheenergydifferencebetweenthe initial conguration for jump type i and peak transition energyfor jump type i. The results are compared with calculated migra-tionbarriersobtainedfromliterature. Unlike thevacancymigra-tion barriers it is clear fromTable 2 that the some of theinterstitial migrationbarriersarestronglyspeciesdependent. Inparticular, jump barriers involving dumbbells in which the startingconguration is mixed (x1, x2, s1x1 and s2x2 differ signicantlybetween the species.Asanaside, wenotethatweobservedtwotransitionstates,withaminimuminthemiddle, inthemigrationenergyprolefor s2x2 and s3x3 type of displacement in FeCr alloy (not shown).Thesemultiplemaximaandminimawerefoundevenwhenthenumber of images was increasedto7intheNEBcalculations.The local minimum at the middle corresponds to the 1 1 1) orien-tation of the FeCr dumbbell. From the local energy minimum thedumbbellcan, ifweignore anypossiblenewhoppingpaths thatmight be available from the intermediate minimum, proceed fur-ther and complete the jump or can jump back to its original cong-uration. Totakethispossibilityintoaccountweassumethatnosignicanttimeisspentintheintermediatelocalminimumanduse an effectivejump frequency equal toone half ofthe originaljumpfrequency. However, thisapproximatetreatmentisinade-quateasitispossibleforthedumbbell tomovefromthenewintermediate 1 1 1) state to other dumbbell orientations, includ-ing the possibility of a 1 1 1) translation, without completing thes2x2 and s3x3 type hops. To include these effects would requiresignicantmodicationofthebasicmodel, whichisbeyondthescope of this paper. Therefore, to determine the impact of ignoringthese effects, we assess how changes in s2x2 and s3x3 type hopsaffect the interstitial D+ values. The s2x2 and s3x3 are both trans-lation hops(see Fig. 5), without anyrotation. Inboth casestheirassociatedtranslationrotationhops, x2andx3, involvemuchlowerbarriers. Thisleadstoaveryweakdependenceof D+onthe exact values of s2x2 and s3x3 (for example, a change of a factorof 10 in s2x2 and s3x3 changes D+ by only 2%). Given that the s2x2and s3x3 type hops have relatively little impact on the D+ valuesfor the cases modeled here, the simple approximation of reducingtheir hop rates by two is used in this work.Similar to the vacancy case, interaction between the solute andthe interstitial affects the interstitial defect ux. It has been pro-posedthat undersizedsolutestendtobindmorestronglywithan interstitials than vacancies. Moreover, a soluteinterstitial com-plex is often much easier to transport than a solutevacancy com-plex [56]. It has been reported both experimentally [57] and fromab-initio calculations [55] that strong binding of Cr with self-inter-stitials in a concentrated FeCr alloys can reduce themobility ofthe interstitial dumbbell.The migration barriers presented in Table 2 are used to calcu-late the jump frequencies using Eq. (3). In all the calculations weusedaxedattemptfrequency(m) of 5 1012s1, thesameasfor the vacancy. The jump frequency is later used to calculate thejump probability using the following relation:Wi = cixi(10)where ci is probability of nding an interstitial in the initial cong-uration corresponding to jump type i. The expressions for calculat-ing ci are listed in Appendix C. The calculated jump probabilities are0 01 12 24 46 601246(a)(b) Fig. 5. Schematic diagrams of the interstitial dumbbells hopping events. (a) By translation; (b) by translationrotation mechanisms. In the gures, the dark and grey circlesrepresent the solvent and the solute atoms respectively. The migration barriers of the individual jumps are presented in Table 2. It is to be noted that in the gure jump types3, 5, and 7 are not presented separately as these jumps are symmetric counterparts to 2, 4, and 6, respectively (i.e., jump type 3 is identical to jump type 2 in reverse).S. Choudhury et al. / Journal of Nuclear Materials 411 (2011) 114 7utilized tocalculate the interstitial Onsager coefcients(Lij)usingtheexpressionsprovidedinAppendixC. ThecalculatedOnsagercoefcientsasafunctionof temperaturearepresentedinFig. 6.Similar to the case of vacancy we used A = Fe, B = Cr or Ni. It is nota-blefromFig. 6thattheoff-diagonal termintheLmatrix, LAB, isslightlyhigherthanthemaindiagonal termLBBforbothCrandNi. In most instances the off-diagonal terms are small with respecttothemaindiagonalterms, andconsequentlytheyareoftenne-glected. In this case however neglecting the off-diagonal terms willresult in signicant error in estimating diffusion ux [19]. The phys-ical consequence of this phenomenon is that a gradient in the chem-ical potential of the solute (either Cr or Ni) will result in a greaterux of Fe than of the solute species. It is not clear to what extentthis surprisinglylarge contributionfromtheoff-diagonal termswouldpersistintheconcentratedalloyastheLijcanbestronglyconcentration dependent. For vacancy mediated diffusion the effectof the ferromagnetic transition on the activation energy of diffusionwas estimated from the experimental measurement of D+ throughtheferromagnetictransitiontemperature(seeAppendixB). How-ever, similar measurement of D+ for interstitial mediated diffusionare not available. Hence, for interstitial mediated diffusion we havenot taken into account the effect of ferromagnetic transition on ourcalculated Onsager and tracer diffusion coefcients.In case of vacancy mediated diffusion we observed thatLvB = LBB(LAB=LBB 1), implyingthatthesignandmagnitudeofLAB plays a key role in assessing whether solute transport by vacan-cies occurs with the vacancy ux (vacancy drag) or against it (in-verse Kirkendall). While a formally equivalent analysis canperformedforinterstitialstoyieldLIB = LBB(LAB=LBB 1)it isnotclear what physical mechanism would allow solute to ow oppo-sitetheinterstitial uxdirection(LIB < 0). Infact, inspectionofthe expressions for LAB and LBB (Eq. (C.2) and (C.3) in Appendix C)reveals that both the coefcients are positive at all temperaturesand irrespective of the magnitude of the jump frequencies, as thefactor hw always less than unity. Thus the interstitial Onsager coef-cient LIB is always greater than zero and interstitial mediated sol-ute ux is in the same direction as the overall interstitial ux.After the Lij transport coefcients are determined, tracer diffu-sioncoefcientscanbecalculatedasafunctionof temperatureusing the following expression [20], thereby making a connectionbetween atomistic jump rates and macroscopic transportcharacteristics:D+B = LBBkBTnB(11)where D+Bis the tracer diffusioncoefcients via the interstitialmechanismfor speciesB, respectively, andnBisthenumber ofatoms of species Bper unit volume. Theresults areplottedinFig. 7a. In the case of Fe, the tracer diffusion coefcient is calculatedby evaluating the transportcoefcientLBBbyEq.(C.3)with everymigrationfrequencysetequal tothevalueforpureFe. Inotherwords, every xi and sixi are replaced by x0 and s0x0, respectively.It can be seen from Fig. 7a that although Cr and Fe have similarinterstitial tracer diffusion coefcients, D+(Ni) is at least two ordersof magnitude lower than for both Cr and Fe. To better understandthe implications of the calculated D+ on RIS, we plotted in Fig. 7btheratioofinterstitialtracerdiffusioncoefcientsofthesolutesto that of Fe as a function of temperature. The gure shows thatCr is the fastest diffuser followed by Fe and Ni. Moreover, relativeto Fe the diffusivity of Cr increases with decreasing temperature.However, the interstitial-mediated diffusivity of Ni shows theopposite trend as temperature decreases. Wong et al. [17] also cal-culated D+(Cr)/D+(Fe) for interstitial mediated diffusion usingmolecular dynamics simulations. Using two different semi-empir-icalpotentialstheypredictedthatD+(Cr)/D+(Fe)isclosetounitybetween 400 and 1100 K (more specically, Wong et al. found a ra-tio of approximately 1.1 within this temperature range for the po-tential that best agreed with the ab-initio calculated bindingenergies). Thus our predicted value of D+(Cr)/D+(Fe) differsTable 2Migration barriers of interstitial dumbbells obtained from ab-initio calculations.Jump type Description Calculated migration energy(eV), FeNiPublished value (eV),FeNiCalculated migration energy (eV),FeCrPublished value (eV), FeCrs0x0AA ?AA 0.84 0.84 0.80 [54], 0.78 [52]s1x1AB ?BA 0.69 0.46 [54] 0.48 0.42 [32]s2x2BA ?AA|B 0.47 0.58 s3x3AA|B ?AB 0.71 0.68 s4x4AA|B ?AA 0.80 0.79 s5x5AA ?AA|B 0.71 0.79 s6x6AA\B ?AA 0.40 0.32 s7x7AA ?AA\B 0.41 0.35 x0AA ?AA 0.36 0.35 0.37 [54], 0.34 [54], 0.33 [76]x1AB ?BA 0.41 0.45 [54] 0.25 0.23 [32]x2BA ?AA|B 0.09 0.13 [54] 0.33 0.33 [32]x3AA|B ?AB 0.33 0.45 [54] 0.42 x4AA|B ?AA 0.33 0.39 x5AA ?AA|B 0.28 0.26 0.22 [32]x6AA\B ?AA 0.36 0.36 x7AA ?AA\B 0.34 0.37 Symbols: AA|B means an AA dumbbell with a B on a target site. AA\B means an AA dumbbell with a B on a non-target site. AB means a mixed AB dumbbell. AA means anunmixed AA dumbbell without a B on a nearest neighbor site.0.830.880.930.981.031.08 Log (Lij (1/meVs)) 1/T ( 10-3 K-1) LAB, Cr - ab initio LAB, Ni - ab initio LBB, Cr - ab initio LBB, Ni - ab initio LAA, Fe - ab initio LFeCr ab-initioLFeNi ab-initioLCrCr ab-initioLNiNi ab-initioLFeFe ab-initio6 4 2 0 -2 -4 Fig. 6. Calculated values of interstitial Lijs based on ab-initio energetics.8 S. Choudhury et al. / Journal of Nuclear Materials 411 (2011) 114signicantlywithpreviouslyreportedvalues, particularlyatlowtemperatures. Inthenextsectionwediscusstheimplicationsofthe interstitial and vacancy relative tracer diffusivities in predict-ing RIS as a function of temperature.To summarize, we calculated the Lij and D+ for interstitial med-iated diffusion in FeCr and FeNi bcc alloys. Calculated D+ showsCristhefastestdiffusingspeciesfollowedbyFeandNi. Fortheinterstitialsitwasobservedthatoff-diagonal terms(LAB), whichare typically assumed to be small, are as large as the diagonal ele-ments LBB.4. Discussion of implications for radiation induced segregationIn this section we present the implications of the calculated tra-cerdiffusioncoefcientsinpredictingthespatial distributionofatoms near sinks in an irradiated environment. Under a radiationenvironment the local chemistry of the species near a sink dependson the relative diffusivity of the solute compared to the solvent byvacancyandinterstitial diffusionmechanismsandtheabsoluteuxesoftheinterstitialandvacancydefectspecies. Itshouldbepointed out that although the full set of Onsager coefcients werederived for vacancy and interstitial mediated diffusion, in this sec-tionweutilizethecalculatedD+todiscussRISbehavior. Thisisessentially equivalent to ignoring the cross terms in the Lij matrix.Ignoring Lij for vacancy mediated diffusion is acceptable for quali-tative analysis provided the cross terms do not cause any drag ef-fects or dominate the problem, which we have previously shown tobe the case. For interstitial mediated diffusion ignoring the LAB maycause an error in RIS prediction as LAB is of the same order of mag-nitude as LBB. In order to estimate the error caused by ignoring theLAB term we calculated ux ratios (JB/JA) and JI for a delta functionpeakof interstitialsforthecasewithandwithouttheLABterm(see Appendix D). It was found that by ignoring the LAB term, JB/JAchangesapproximatelybyafactorof2whilethechangeintheabsolute interstitial defect ux is insignicant ( D+(Fe),at all temperatures.Appendix C. Calculating interstitial Onsager coefcients basedon jump rates.By a process similar to that described in Appendix A, it is possi-ble to determine Onsager coefcients for interstitial diffusion fromatomistic migration barriers. Utilizing the model developed byV.Barbe and M. Nastar [19] for interstitial diffusion in dilute bcc met-als, we have calculated transport coefcients for a dilute binary al-loy of Cr and Ni in Fe for the interstitial mechanism. The primaryexpressions for calculating transport coefcients are:4kbTna2LAA = WA 6((2 s3)W3 (2 s4)W4)2(2 s3)W3 3(2 s4)W48w2(2 s4)W4((2 s3)W3 (2 s4)W4)(3 2s1)W1 (2 s2 hw)W2(C:1)4kbTna2LAB =4(2 s2)2W1(4wW4)(3 2s1)W1 (2 s2 hw)W2= 4kbTna2LBA(C:2)4kbTna2LBB = (2 s1)W1((1 2s1)W1 3(2 s2 hw)W2(3 s1)W1 (2 s2 hw)W2(C:3)where the quantities WA, w, and h are dened as:WA=3(2s0)W03(2s2)W218(2s5)c/BW524(2s7)c/BW7(C:4)w =(2 s3)W33(2 s4)W4 (2 s3)W3(C:5)h = (1 s2)221 s1(C:6)The central quantities in these expressions are the Wi, which aredened as the jump probability of the atomistic jump type i. Thismodel considerseightinterstitial congurations, eachwithbothatranslational migration(denotedsixi) andatranslationcom-bined with a rotation (xi), for a total of 16 jump types. The jumpprobability Wi is dened as follows:Wi = cixi(C:7)where xi is the jump frequency of jump type i, and ci is probabilityof nding an interstitial in the initial conguration corresponding tojump type i.Inspection of Fig. 5 reveals that there are in fact only four dis-tinct initial congurations: an unmixed dumbbell without a soluteon a nearest neighbor site (denoted cAA), a mixed dumbbell (cAB), anunmixeddumbbell withasoluteatomonatarget-typenearestneighbor site (cAA||B), and an unmixed dumbbell with a solute atomonanon-target-typenearestneighborsite(cAA\B). Thesecoef-cients are calculated to the rst order in c/B as follows:cAA = (1 c/BP) cI6(C:8)cAB = c/BPAB cI6(C:9)cAA[[B = c/BPAA[[B cI6(C:10)cAAlB = c/BPAAlB cI6(C:11)where c/B is the concentration of solute B farther than 1 nn distancefrom an interstitial dumbbell and cI is the concentration of intersti-tials. In a real alloy, where cB cI, the simplication that c/B = cB canbemade. ThequantitiesPAB, PAA||B, andPAA\BareBoltzmann-typeprobability weights calculated as follows:PAB = exp(DH3 DH2) (DH5 DH4)kbT_ _(C:12)PAA[[B = exp(DH5 DH4)kbT_ _(C:13)PAAlB = exp(DH7 DH6)kbT_ _(C:14)ThequantitiesDHiarethemigrationbarriersfortranslationrotation jump type i. The quantity P is a global probability weightintroduced for convenience, dened asP = 2PAB 4PAA|B 4PAAlB: (C:15)The jump frequencies xi are calculated with the expression inEq. (3). Finally, thecoefcients siaredenedsimplyastheratioof the translation frequency of jump type i to the translationrota-tion frequency:si = sixixi(C:16)12 S. Choudhury et al. / Journal of Nuclear Materials 411 (2011) 114Appendix D. Qualitative relation of RIS and kinetic parameters.Simple intuitive arguments based on tracer or chemical diffu-sion coefcients are often used to qualitatively estimate RIS trends.For example, if species B diffuses faster than species A by intersti-tial ux then interstitial ux is likely to cause enrichment of B at asink. Here we put these intuitive arguments on more quantitativefooting in order to assess the roleof off-diagonal Onsager coef-cients in the arguments. This more quantitative analysis is neces-saryastheoff-diagonal Onsager coefcientsfor theinterstitialuxwerefoundtobeaslargeasthediagonal elementsinthisstudy. For concreteness and simplicity, we focus only on interstitialux, although a similar approach could incorporate vacancy diffu-sion as well.For an interstitial mediated diffusion the eventual steady-stateconcentration prole at a sink of species under irradiation dependson (a) the absolute interstitial defect ux; (b) the ux ratio of sol-ute compared to the solvent by interstitial diffusion mechanisms.Topredicttheprolerigorouslyrequiresafullsimulationofthecoupleduxequations. Toprovideanestimateof thetrendsinRIS prole, in this appendix we present the change in the factors(a) and (b) immediately after a radiation damage cascade forma-tiononaplanefarawayfromthegrainboundary. Weconsideronly a one-dimensional problem. For the sake of simplicity we as-sume the vacancies are frozen and thus the diffusion is dominantlythrough interstitial mechanism. We particularly focus on the effectof includingor ignoringcross-termsLABonthecontributionofinterstitial ux.The ux expression (Eq. (2)) can be rewritten asJi =

Nj=1Lij@lj@x =

Nj=1LijHjkni(D:1)where l is the chemical potential, ni is the number of atoms of spe-cies i per unit volume, Hjk =@lj@nk, is the thermodynamic factor. For anideal solution, the chemical potential is expressed asli = l0i (T; P) kBT ln(ci) and thermodynamic factor is expressed asHij =kBTci= NkBTnifor i = j= 0 for i j(D:2)Combining Eqs. (D.1) and (D.2), the explicit ux for interstitialmediated diffusion can be written asJA = NkBT LAAnAnA LABnBnB LAInInI_ _(D:3)JB = NkBT LABnAnA LBBnBnB LBInInI_ _(D:4)JI = NkBT LAInAnA LBInBnB LIInInI_ _(D:5)Using the relation JA + JB = JI it is possible to showLAI = LAA LABand LBI = LAB LBBand LII = LAI LBI= LAA LAB LBA LBB(D:6)Weconsideraplane(p)farawayfromthegrainboundaryattime immediately after a radiation damage cascade creates a localchange in interstitial concentration. We assume that a total of Dninterstitials are formed, comprised of A and B atoms in the same ra-tio as the local concentration around plane (p). We write the com-positionrelationshipinplane p as npA = mnpB(andwe assumenearby planes have the same composition). For example, forcB = 0.01wegetm = 99. ThegradientofAbetweentheplanespand p-1 can be expressed asnA =npAnp1ADx=mnpBmnp1BDx= mnB. Thus for the plane p just afterthe cascade formation,nAnA=nBnB(D:7)Using the relations between the Onsager coefcients as well asEqs. (D.3), (D.4), and (D.7) the ratio of JB/JA is expressed asJB=JA = LABnAnA LBBnBnB (LAB LBB)nInI_ _ _LAAnAnA LABnBnB_(LAA LAB)nInI_= (LAB LBB)nInInBnB_ _ _ _ _nBnBnInI_ _(LAA LAB)_ _= (LAB LBB)=(LAA LAB)For the case when LAB LAA, LBB the ux ratio JB/JA can be rewrit-ten asJB=JA = LBBLAA= nBnAf D+BD+A(D:8)To introduce the tracer diffusion coefcients we have used thefact that the diagonal Onsager coefcients are related to the tracerdiffusion coefcients through LBB =nBkBTD+Band LAA =nAfkBTD+A, where fisthecorrelationcoefcient[20]. ItisclearfromEq. (D.8)thattheratioof uxJB/JAisproportional totheconcentrationratiotimes the ratio of tracer diffusion coefcients. RIS will occur whenthe ux ratio deviates from the nB/nA ratio. If we assume f ~ 1 thentheextentofRISwilldependontheratioofD+B/D+A. Thusforthecase when solute and solvent uxes are dominated by the diagonalOnsager coefcients, D+B/D+Acan be used for qualitative RIS predic-tion. However, itcanbeseenfromFig. 6thatbothfordiluteNiand Cr the off-diagonal Onsager coefcient (LAB) is of the same or-der of magnitude as diagonal coefcient (LBB). Thus, for the Osagercoefcientvaluesdetermined here, itisclearfromEq. (D.8)thatignoring LAB by predicting RIS purely based on the ratio of tracerdiffusioncoefcientswill causeerrorinRISpredictionindiluteFeCr and FeNi alloys. In order to estimate the error introducedbyignoringLABinourRISpredictionwecalculateuxratios(JB/JA)andJIfortheplanepfarawayfromthegrainboundary, forthe case with and without the LAB term. We assume LAB = LBBandLAA = kLBB where k - 100, which are similar values to what we ndwith A = Fe and B = Cr. The effect of LAB is estimated from(JB=JA)WithoutLAB=(JB=JA)WithLAB =1k_ _2k1_ _ =k 1 ( )2k=12 12k_ _For k 1,it is clear, that neglecting theLAB term changes theux ratio of solute and solvent by a factor of 2.Similarly, fromEqs. (D.5)and(D.7)itcanbeshownthatforlarge k there is an insignicant change in the total interstitial de-fect ux by ignoring the LAB term as given below.(JI)WithoutLAB =(JI)WithLAB = (k 1)(k 3) =- 1References[1] Z. Lu, R.G. Faulkner, G. Was, B.D. Wirth, Scr. Mater. 58 (2008) 878.[2] G. Gupta, Z. Jiao, A.N. Ham, J.T. Busby, G.S. Was, J. Nucl. Mater. 351 (2006) 162.[3] G.S. Was, FundamentalsofRadiationMaterialsScience, Springer, NewYork,2007.[4] T.R. Allen, J.T. Busby, G.S. Was, E.A. Kenik, J. Nucl. Mater. 255 (1998) 44.[5] T.R. Allen, G.S. Was, Acta Mater. 46 (1998) 3679.[6] R.E. Clausing, L. Heatherly, R.G. Faulkner, A.F. Rowcliffe, K. Farrell, J. Nucl.Mater. 141 (1986) 978.[7] Z. Lu, R.G. Faulkner, N. Sakaguchi, H. Kinoshita, H. Takahashi, P.E.J. Flewitt, J.Nucl. Mater. 351 (2006) 155.S. Choudhury et al. / Journal of Nuclear Materials 411 (2011) 114 13[8] S. Ohnuki, H. Takahashi, T. Takeyama, J. Nucl. Mater. 103 (1981) 1121.[9] H. Takahashi, S. Ohnuki, T. Takeyama, H. Kayano, J. Nucl. Mater. 96 (1981) 233.[10] J.L. Brimhall, D.R. Baer, R.H. Jones, Nucl. Mater. 122 (1984) 196.[11] R.L. Klueh,D.S. Gelles, S. Jitsukawa, A. Kimura, G.R. Odette, Nucl. Mater. 307(2002) 455.[12] E.A. Little, L.P. Stoter, Special Tech. Publ. 782 (1982) 207.[13] A.A. Hosseini, I.P. Jones,Phys. StatusSolidiA113(1989)57 (andreferencestherein).[14] A. Van der Ven, G. Ceder, Phys. Rev. Lett. 94 (2005) 045901.[15] A. Van der Ven, J.C. Thomas, Q.C. Xu, B. Swoboda, D. Morgan, Phys. Rev. B 78(2008) 104306.[16] J.D. Tucker, R. Najafabadi, T.R. Allen, D. Morgan, J. Nucl. Mater. 405 (2009) 216.[17] K.L. Wong, H.J. Lee, J.Y. Shim, B. Sadigh, B.D. Wirth, J. Nucl. Mater. 386388(2009) 227.[18] A.D. Leclaire, J. Nucl. Mater. 6967 (1978) 70.[19] V. Barbe, M. Nastar, Philos. Mag. 87 (2007) 1649.[20] A.R. Allnatt, A.B. Lidiard, AtomicTransport inSolids, CambridgeUniversityPress, Cambridge, 2003.[21] L.S. Darken, Trans. Am. Inst. Min. (Metall.) Eng. 175 (1948) 184.[22] J.R. Manning, Metal. Trans. 1 (1970) 499.[23] W.F. Gale, T.C. Totemeier, Smithells Metals Reference Book, Elsevier, SanFrancisco, 2004.[24] J. Askill, Tracer DiffusionData for Metals, Alloys andSimple Oxides, IFI/Plenum, New York, 1970.[25] M. Nastar, V. Barbe, A Self-Consistent Mean Field Theory for Diffusion in Alloys,MeetingonAtomicTransport andDefect PhenomenainSolids, Guildford,England, 2006.[26] G. Kresse, J. Furthmuller, Phys. Rev. B 54 (1996) 11169.[27] G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) 558.[28] P.E. Blochl, Phys. Rev. B 50 (1994) 17953.[29] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865.[30] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188.[31] E. Vincent, C.S. Becquart, C. Domain, Instr. Meth. Phys. Res. B 228 (2005) 137.[32] P. Olsson, J. Nucl. Mater. 386 (2009) 86.[33] C. Domain, C.S. Becquart, Phys. Rev. B 65 (2002) 024103.[34] P. Olsson, C. Domain, J. Wallenius, Phys. Rev. B 75 (2007) 014110.[35] C.C. Fu, J. Dalla Torre, F. Willaime, J.L. Bocquet, A. Barbu, Nat. Mater. 4 (2005)68.[36] J.D. Tucker, Thesis, University of Wisconsin, Madison, 2008.[37] Y. Serruys, G. Brebec, Philos. Mag. A 46 (1982) 661.[38] J.R. Manning, Phys. Rev. 139 (1965) A126.[39] J. Bocquet, G. Brebec, Y. Limoge, Diffusion in Metals and Alloys, Elsevier ScienceBV, Amsterdam, The Netherlands, 1996.[40] J. Wallenius, P. Olsson, L. Malerba, D. Terentyev, Inst. Meth. Phys. Res. Sec. B255 (2007) 68.[41] M.J. Jones, A.D. Leclaire, Philos. Mag. 26 (1972) 1191.[42] J. Crangle, G.M. Goodman, Roy. Soc. London Ser. a-Math. Phys. Sci. 321 (1971)477.[43] H. Nitta, Y. Iijima, Philos. Mag. Lett. 85 (2005) 543.[44] R.J. Borg, D.Y.F. Lai, Acta Metall. 11 (1963) 861.[45] S. Takemoto, H. Nitta, Y. Iijima, Y. Yamazaki, Philos. Mag. 87 (2007) 1619.[46] G. Hettich, H. Mehrer, K. Maier, Scr. Metall. 11 (1977) 795.[47] Y. Iijima, K. Kimura, K. Hirano, Acta Metall. 36 (1988) 2811.[48] C.G. Lee, Y. Iijima, T. Hiratani, K. Hirano, Mater. Trans. JIM 31 (1990) 255.[49] J. Wallenius, P. Olsson, C. Lagerstedt, N. Sandberg, R. Chakarova, V. Pontikis,Phys. Rev. B 69 (2004) 094103.[50] S. Huang, D.L. Worthington, M. Asta, V. Ozolins, G. Ghosh, P.K. Liaw, ActaMater. 58 (2009) 1982.[51] C. Domain, J. Nucl. Mater. 351 (2006) 1.[52] C.C. Fu, F. Willaime, P. Ordejon, Phys. Rev. Lett. 92 (2004) 175503.[53] D. Nguyen-Manh, A.P. Horseld, S.L. Dudarev, Phys. Rev. B 73 (2006) 020101.[54] E. Vincent, C.S. Becquart, C. Domain, J. Nucl. Mater. 359 (2006) 227.[55] T.P.C. Klaver, P. Olsson, M.W. Finnis, Phys. Rev. B 76 (2007) 214110.[56] R.G. Faulkner, S.H. Song, P.E.J. Flewitt, M. Victoria, P. Marmy, J. Nucl. Mater. 255(1998) 189.[57] A.L. Nikolaev, V.L. Arbuzov, A.E. Davletshin, J. Phys.: Condens. Matter 9 (1997)4385.[58] D. Terentyev, P. Olsson, L. Malerba, J. Nucl. Mater. 386 (2009) 140.[59] G.R. Odette, M.J. Alinger, B.D. Wirth, Ann. Rev. Mater. Res. 38 (2008) 471.[60] W.G. Wolfer, L.K. Mansur, J. Nucl. Mater. 91 (1980) 265.[61] T. Okita, W.G. Wolfer, J. Nucl. Mater. 327 (2004) 130.[62] C.H. Zhang, J. Jang, H.D. Cho, Y.T. Yang, J. Nucl. Mater. 386 (2009) 457.[63] T.R. Allen, L. Tan, J. Gan, G. Gupta, G.S. Was, E.A. Kenik, S. Shutthanandan, S.Thevuthasan, J. Nucl. Mater. 351 (2006) 174.[64] W.J. Phythian, R.E. Stoller, A.J.E. Foreman, A.F. Calder, D.J. Bacon, J. Nucl. Mater.223 (1995) 245.[65] R.E. Stoller, G.R. Odette, B.D. Wirth, J. Nucl. Mater. 251 (1997) 49.[66] R.E. Stoller, J. Nucl. Mater. 276 (2000) 22.[67] B.D. Wirth, G.R. Odette, D. Maroudas, G.E. Lucas, J. Nucl. Mater. 276(2000)33.[68] N. Soneda, T.D. De la Rubia, Philos. Mag. A 81 (2001) 331.[69] Y.N. Osetsky, D.J. Bacon, A. Serra, B.N. Singh, S.I. Golubov, Philos. Mag. 83(2003) 61.[70] Y.N. Osetsky, D.J. Bacon, B.N. Singh, B. Wirth, J. Nucl. Mater. 307 (2002) 852.[71] D.A. Terentyev, T.P.C. Klaver, P. Olsson, M.C. Marinica, F. Willaime, C. Domain,L. Malerba, Phys. Rev. Lett. 100 (2008) 14.[72] K.L. Wong, J.H. Shim, B.D. Wirth, J. Nucl. Mater. 367 (2007) 276.[73] L. Ruch, D.R. Sain, H.L. Yeh, L.A. Girifalco, J. Phys. Chem. Sol. 37 (1976) 649.[74] B. Drittler, N. Stefanou, S. Blugel, R. Zeller, P.H. Dederichs, Phys. Rev. B40(1989) 8203.[75] K. Hirano, M. Cohen, B.L. Averbach, Acta Metall. 9 (1961) 440.[76] R.A. Johnson, Phys. Rev. Lett. 134 (1964) A1329.14 S. Choudhury et al. / Journal of Nuclear Materials 411 (2011) 114