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Ab initio explanation of disorder and off-stoichiometry in Fe-Mn-Al-C κ carbides Poulumi Dey,1 Roman Nazarov,2 Biswanath Dutta,1 Mengji Yao,1 Michael Herbig,1 Martin Fri´ak,3,4 Tilmann Hickel,1 Dierk Raabe,1 and J¨org Neugebauer1 1Max-Planck-Institut fu¨r Eisenforschung GmbH, D-40237 Du¨sseldorf, Germany 2Lawrence Livermore National Laboratory, Livermore, CA 94550, USA 3Institute of Physics of Materials, v.v.i., Academy of Sciences of the Czech Republic, CZ-61662 Brno, Czech Republic 4Central European Institute of Technology, CEITEC MU, Masaryk University, Kamenice 5, CZ-625 00 Brno, Czech Republic (Dated: January 25, 2017) Carbides play a central role for the strength and ductility in many materials. Simulating the 7 impact of these precipitates on the mechanical performance requires the knowledge about their 1 atomic configuration. In particular, the C content is often observed to substantially deviate from 0 the ideal stoichiometric composition. In the present work, we focus on Fe-Mn-Al-C steels, for 2 which we determined the composition of the nano-sized κ carbides (Fe,Mn) AlC by atom probe 3 n tomography (APT) in comparison to larger precipitates located in grain boundaries. Combining a densityfunctionaltheorywiththermodynamicconcepts,wefirstdeterminethecriticaltemperatures J for the presence of chemical and magentic disorder in these carbides. Secondly, the experimentally 4 observed reduction of the C content is explained as a compromise between the gain in chemical 2 energy during partitioning and the elastic strains emerging in coherent microstructures. ] PACSnumbers: 61.50.Nw,61.72.jd,71.15.Mb i c s - I. INTRODUCTION The arrangement of κ carbides within the microstruc- l r ture is to a large extent determined by the E2 crys- t 1 m tal structure of κ carbide. It resembles a perovskite- Fe-Mn-Al-C based steels have recently attracted close type cubic structure with Al atoms at the corners of the . attention because of their high strength and ductility1,2 at along with a high corrosion resistance and a compara- cube, Fe atoms at the face-centered sites (correspond- m ing to L1 ), and C atoms at the body-center octahedral bly low density3. This combination of their properties 2 site (also called L(cid:48)1 ). Therefore, the nominal composi- - makes them also attractive for automotive applications. 2 d tion is Fe AlC. Experiments for high-Mn alloys indicate The excellent mechanical performance of the Fe-Mn-Al- 3 n thattherecanbeasignificantmanganesecontentreplac- C steels is mainly attributed to microstructure features o ingtheFeatoms,yieldinga(Fe,Mn) AlCcomposition14. c thatcorrelatewithdeformationmechanismsandstrongly 3 While different ordered E2 structures for varying Mn [ depend on the amount of Al in the material. High-Mn 1 contents are shown in Fig. 1, it is still unclear which steelswithalowAlcontent(<5wt.%)typicallyundergo 1 Mncontentwouldcorrespondtothermodynamicequilib- amicrostructurerefinementbytheactivationofdeforma- v rium, how relevant the ordering is, and how this affects 0 tiontwinningintheaustenitephase, whichincreasesthe the mechanical properties. 1 strain-hardening rate4,5. When the Al content in these 9 steels(about30wt.%Mnand1.3wt.%C)ishigherthan The orientation relationship between the regularly ar- 6 6 wt.%, an annealing produces finely dispersed nano- ranged κ carbides and the γ matrix is reported to be 0 . sized κ carbides (Fe,Mn)3AlC in the austenitic matrix. (001)/(001) in experiments15. The interfaces are coher- 1 Experiments showed that these precipitates strengthen ent in case of κ nano-precipitates without indications 0 Fe-Mn-Al-C steels, thereby making them interesting for of misfit dislocations. Such a microstructure can, how- 7 applications. For example, the large age-hardenability ever, not be understood if a completely stoichiometric 1 : of these alloys is attributed to the homogeneous pre- composition is assumed. As the density functional the- v cipitation and dispersion of κ carbides in the austenitic ory (DFT) calculations performed in this paper show, Xi matrix6–10. a nominal E21 structure would have its elastically hard Using specific heat treatments, a γ/κ regular mi- axis in the (001) direction and the resulting misfit of 9% r a crostructure can be achieved, which strongly influences with respect to the lattice constant of the matrix mate- the ductility at ambient temperatures11. Further, κ car- rial would be too large, to ensure coherent interfaces. bides improve the creep resistance of Fe-based alloys at Indeed, electron microprobe experiments have shown high temperatures making them desirable materials for considerable deviations from the Fe AlC stoichiometry. 3 manufacturinghigh-temperaturecomponentssuchasgas In an experimental work done by Palm et al.16, the turbine blades and vanes in aircraft engines, aerospace off-stoichiometriccompositionobservedisFe Al C 3+y 1−y z andpowergeneratingplants. Thepresenceofnano-sized where y may vary between -0.2 and +0.2 and z between κ carbides therefore yields mechanical properties of Fe- 0.42 and 0.71. Other experimental works have proposed basedalloysthataresimilartoNi-basedsuperalloys12,13, an off-stoichiometric composition of Fe AlC 17–19. All 3 0.5 provided the desired microstructure is achieved. these observations demonstrate in particular a depletion 2 ofCinκcarbidesascomparedtothenominalE2 struc- perature applications the chemical and magnetic order 1 ture of Fe AlC. can break down. We thus studied the impact of such 3 In spite of these experimental evidences of off- magnetic and/or chemical disorder on the stability of κ stoichiometric C compositions of κ carbides, further carbides. measurements that can resolve the properties of nano- precipitates are desired. In the present study, we use atom probe tomography for this purpose, since II. METHODOLOGY it combines near-atomic resolution with ppm chemical sensitivity20. At the same time, the theoretical investi- We perform calculations using DFT27,28 as imple- gation of the C depletion is still limited. The main focus mented in the Vienna Ab Initio Simulation Package of previous theoretical studies has been on perfectly or- (VASP)29–31. Theelectron-ioninteractionisdescribedby dered κ carbides. In a previous investigation, DFT has using projector augmented-wave (PAW) potentials32,33. been employed to compare the properties of Fe Al-L1 3 2 The generalized-gradient approximation (GGA) func- and ordered Fe AlC-E2 structures and to underline the 3 1 tionalofPerdew,BurkeandErnzerhof(PBE)34 hasbeen roleplayedbyC21,22. Thesestudieshaveshownthatthe employed. The Methfessel-Paxton method35 has been addition of C atoms decreases the magnetic moment of used for the Fermi surface smearing with a 12×12×12 the neighboring Fe atoms and yields a heat capacity and Monkhorst-Pack grid36 in a 1×1×1 5-atom unitcell for elasticconstantsofFe AlC-κthatareappreciablydiffer- 3 the κ carbides shown in Fig. 1. A supercell (SC) size ent from Fe Al-L1 21. In similar spirits, the energetics 3 2 of 2×2×2 (40 atoms) has been considered for the dis- and magnetic properties of Fe Al and Fe AlX (where X 3 3 ordered and vacancy calculations with a corresponding =H,B,C,N,O)compoundsareinvestigatedusingDFT, Monkhorst-Pack grid of 6×6×6. The single-electron among which Fe AlC turns out to be most stable when 3 wavefunctionshavebeenexpandedbyusingplanewaves comparing cohesive energies23. The reduction in mag- up to an energy cut-off of 500 eV. The energies are con- netization of Fe Al due to the addition of C has been 3 verged to a precision of better than 1 meV/formula unit explained by relaxation effects induced by the C atom in (f.u.). the Fe Al structure23,24. 3 We first study the occupation of the metal sublattice The computation of the elastic constants of κ car- sites,sinceithasanimpactontheCsolvationenergiesin bides has revealed that these carbides are more rigid κ carbide. To determine the equilibrium Mn content, κ than the parental Fe Al-L1 structure24. In the same 3 2 carbides (Fe Mn )AlC with integer x are considered. work, the issue of different chemical configurations has 3−x x Fororderedconfigurations,asingle5-atomunitcell(refer been discussed for the Fe-Mn sublattice by considering to Fig. 1) is used, which is periodically repeated. The (Fe Mn )AlC with integral values for x from 0 to 3. 3−x x chemical disorder of the κ carbides has been simulated In a subsequent work by the same group25, low Mn con- by the special quasi-random structure (SQS) scheme37 centrations in κ carbides have been investigated which in a 2×2×2 SC. Two kinds of SQS are generated: one show absence of any kind of interaction between sub- withchemicaldisorderontheFe-Mnsub-latticeonly(i.e. stitutional Mn atoms thereby indicating a random alloy keeping the symmetry of L1 ) and the other one with a system. A previous work further indicates the relevance 2 random distribution of all metal atoms (i.e. using the of point defects such as C vacancies (treated in the di- symmetry of fcc). In order to generate these SQS, corre- lute limit) for the thermodynamic stability of the rele- lation functions of up to five-body figures are used. The vant phases22. While these studies provided important chosenSQSinourstudyhavethelowestcorrelationerror insight into the structure and thermodynamics, none of in terms of the error-function introduced in our previous themfullyexplainedtheabovementionedCreductionof work38. κ carbides. As we will show in the present study, this is Thestabilityoftheseκcarbidesisinvestigatedbycom- mainly due to the geometrical constraints of a coherent puting the Helmholtz free energy difference between the interface to the matrix material. precipitates and the surrounding solid solution assuming In our first ab initio study of κ carbides14, we have thatthetwophasesarethermodynamicallyequilibrated: already been able to reveal and explain the Al deple- tion in these carbides by coherency strains. We have pointed out that this effect alone is not sufficient, but ∆F(T,V,x,y,z) (1) that it occurs concurrently with a reduced C content in = ESC[(Fe Mn )Al C ]−TSκ(x,y,z) theseprecipitates. Inthepresentwork,wenowprovidea κ 3−x+y x 1−y z deeper theoretical understanding of the C concentration − (3−x+y)µ −xµ −(1−y)µ −zµ . Fe Mn Al C in off-stoichiometric κ carbides, which is benchmarked against experimental data. This investigation requires The first term is the ground state energy ESC deter- κ the careful application of various thermodynamic con- mined in a DFT supercell (SC) calculation. The sec- cepts. One of them is the application of a constrained ond term gives the entropy contribution. Neglecting the paraequilibrium26, which allows us to focus on C only. vibrational entropy, which is small compared to the con- Further, at operational conditions relevant for high tem- figurational contribution, we express the entropy solely 3 by the latter one: andPMstatecanbeexpressed40as∆E =N2 (cid:80) J mag i(cid:54)=j ij and the above equation transforms to (cid:16) 3−x x Sκ(x,y,z)=−k (3−x)ln +xln B 3 3 2 ∆E k T = . (4) +ylny+(1−y)ln(1−y) B C 3N mag (cid:17) +zlnz+(1−z)ln(1−z) , (2) It may be noted that the values of T obtained using C Eq.(3)typicallyoverestimatetheexperimentalvalues41, where k is the Boltzmann constant, x,y and z the con- B but provide correct qualitative trends. tent of Mn, Fe antisites on the Al sublattice and C in κ, Single-crystalline elastic constants of the disordered κ respectively. ThethirdterminEq.(1)balancesthether- carbides are determined using tetragonal and trigonal modynamicexchangeofatomsbetweentheκcarbideand (rhombohedral) cell-shape deformations38. Due to the the γ matrix. In the spirit of a grand-canonical ensem- fact that our SQS supercells in general do not possess ble, this exchange can be described by taking/removing cubic symmetry, strains have been applied along struc- atoms from the chemical reservoir, which is determined turally equivalent directions, the resulting stresses are by the free energy of the γ solution. used to calculate elastic constants and these have been In the present paper, the chemical reservoir is repre- then averaged (for details see, e.g., Ref. 38). sented by the chemical potentials µ of the involved el- X The theoretical investigations are supported by exper- ements X = Fe, Mn, Al, C. They depend on the (ex- imental investigations on the C content in κ carbide. perimentally given) composition, temperature, and vol- For this purpose, a high-Mn steel of the composition Fe- ume of the γ matrix and are computed by DFT (see ap- 29.8Mn-7.7Al-1.3C (wt.%) has been used, which is aged pendix A). An advantage of using chemical potentials is at 600◦C for 12 weeks. The material has undergone a that they provide a physically intuitive tool to describe solid solution treatment at 1100◦C for two hours and is continuous changes in the chemical composition of the subsequently oil quenched prior to ageing. A system- considered alloys without being limited to discrete sto- atic repetition of various aging treatments ensures that ichiometries imposed by finite size supercells. This is the present conditions yield a thermodynamically stable particularly useful for the constrained paraequilibrium26 partitioning of the chemical elements. Further details of discussed in the second part of the paper, where we en- alloy casting and thermo-mechanical processing are re- force an equality of chemical potentials between κ and γ ported elsewhere6,14. The sample is etched with 1% Ni- for the interstitial C atoms. We note that the ab initio talsolutionandcharacterizedusingafieldemissionscan- derivation of µ from DFT energies for a specific super- X ningelectronmicroscope(SEM)ZeissXB1540equipped cell ESC[Fe Mn Al ] implies that the absolute value of γ x y z withanelectronbackscatterdiffraction(EBSD)detector. thechemicalpotentialsinthematrixisdependentonthe Needle-likeatomprobetomography(APT)samplesfrom given pseudopotential (see appendix A for details). grain boundary and grain interior regions are prepared The energetically favoured magnetic phase in the via a standard FIB procedure by a dual-beam focused- κ carbides is determined by computing the free en- ion-beam(FIB)system(FEIHeliosNano-Lab600i)42. A ergy difference in Eq. (1) for ferromagnetic (FM), anti- LEAPTM3000XHRsystem(CamecaInstruments)isem- ferromagnetic double layer (AFMD), and non-magnetic ployedforAPTanalysiswithvoltage-pulsingat200kHz (NM) phases. Since the FM phase is found to be the pulse repetition rate, 0.005 atom/pulse detection rate, T =0Kgroundstate,itisusedinthecalculations,ifnot 15% pulse fraction at 70 K. stated otherwise. The γ matrix is consistently treated in an anti-ferromagnetic (AFM) state. Paramagnetic (PM) energies for κ carbides are again obtained by a 2×2×2 III. RESULTS AND DISCUSSION supercell using the SQS scheme, which mimics a random distribution of collinear local moments as closely as pos- sible for this SC. This procedure has been performed for A. Experiment the chemically ordered as well as the disordered κ car- bides. Asindicatedintheintroduction,itisthemainpurpose The Curie temperature, T , is estimated within our of the theoretical investigations in this paper to reveal C study from the mean field approximation of the Heisen- the reasons for the C off-stoichiometric compositions in berg model39 κ carbides. Previously, our own measurement14 for a κ-containing steel, namely an Fe-29.8Mn-7.7Al-1.3C (in k T = 2N (cid:88)J , (3) wt.%)alloy,hasgivenavaluez =0.61. Withthepresent B C 3 mag ij experimental evaluation, we employ a much longer aging i(cid:54)=j treatment to ensure thermodynamic equilibrium. whereN isthenumberofmagneticatomsintheunit- Figure 2 shows the microstructure of the same alloy mag cellandJ arethemagneticexchangecouplingconstants as used in Ref. 14 after the prolonged ageing. It clearly ij between sites i and j. Using mean field approximation, showstwodifferentmorphologiesofκcarbides,whichare the energy difference ∆E per unitcell between the FM the bright protruding phases after etching (Fig. 2(a) & 4 a) b) c) d) FIG. 1: (Color online) Crystal structures of (a) Fe AlC, (b) Fe MnAlC, (c) FeMn AlC, and (d) Mn AlC. Red, golden, green 3 2 2 3 and black balls represent Al, Fe, Mn and C atoms respectively. (b)). On the one hand, there are nanosized κ precipi- Feinthesupercell,weobtainy =0.125,whichiscloseto tates in the grain interior (GI) i.e. within the austen- the reported experimental composition16. We then ob- ite matrix γ, regularly aligned along specific directions tain at T =0 K an increase of the free energy difference (Fig. 2(b) & (c)), which are orthogonal (cid:104)001(cid:105) crystallo- betweenthecarbideandtheγ matrix(seeEq.(1))byap- graphic directions6,15. On the other hand, a µm-scale prox.1eVascomparedtothestoichiometriccomposition lamellarstructuremainlycomposedofalternativecoarse of Fe AlC. This energy increase enters the temperature 3 κ carbides and solute-depleted austenite γ is observed dependent antisite formation energy given by 0 0 at regions next to grain boundaries (GB) between γ grains (Fig. 2(a)). Also a small fraction (<1%) of fer- Ff (T)=ESC −ESC −µ (T)+µ (T). (5) rite α is detected in these regions by EBSD (not shown AS FeAl Fe3AlC Fe Al here). This κ0+γ0+α lamellar microstructure initiates theconfigurationalentropyintheκcarbideisconsidered at GBs and grows into GI region. The chemical com- if the antisite concentration is determined by position of GI κ carbide as measured by APT is found to be Fe1.99Mn1.10Al0.91C0.60 and that of GB κ0 carbide (cid:34) Ff (T)(cid:35) to be Fe Mn Al C . These chemical composi- c =exp − AS . (6) 1.69 1.35 0.95 0.87 FeAl k T tions confirm deviations from stoichiometric C concen- B trations in κ carbides. The nano-sized GI κ carbides seems to be stabilized by the coherence constrain, show- Neglecting again vibrational contributions, the tempera- ing almost the same composition after 24 hours14 and turedependenceofthedefectformationenergyoriginates 12 weeks. These GI precipitates are observed to barely solely from the one in the chemical potentials µFe(T) coarsen after prolonged ageing maintaining an average and µAl(T) imposed by the γ matrix (see appendix A sizeofapprox. 20nm. ThelargerGBκcarbides,incon- for details). It takes care of the fact that with increas- trast to this, represent a thermodynamicallymore stable ing temperature the chemical potential decreases due to state, since they grow on expense of the matrix phase enhanced configurational entropy. in the grain interior. The microstructure evolution upon UsingEq.(6), onecanexpect0.001%oftheAlatoms aging has been thoroughly studied and will be discussed tobereplacedbyFeat600◦C.Aselasticeffectsareinthe elsewhere. Full coherency of the GI κ/γ interface has focus of the present investigations, the lattice constant been observed by high-resolution transmission electron of the κ carbide has also been constrained to that of the microscopy and no indication for a segregation to this surrounding Fe matrix. Even the decrease of the antisite interface is found by APT measurements. For the pur- formation energy due to this strain (from 1 eV to 0.8 eV pose of the present study, however, most important is atT =0K)istoosmalltoyieldanoff-stoichiometriccon- the noticeable difference in the C concentrations in GI centration higher than 0.01 % at elevated temperatures. and GB κ carbides with the former (GI) showing more The situation is different in the case of Mn antisites on C reduction than the latter (GB). the Al sublattice if C vacancies are additionally present at neighboring sites (see Ref. 14 for details). In this case an Al reduction of up to 10 at.% can be observed. For the purpose of the present investigations this effect is B. Chemical and magnetic order still not decisive and it is justified to assume a filled Al sublattice,whichstabilizestheκcarbideandactsather- When investigating with ab initio simulations the C modynamicdrivingforceforthepartitioningofC.Using content(z)ofκcarbides(Fe Mn )Al C ,werep- this assumption implies that the volume fraction of κ 3−x+y x 1−y z resenttheMnandAlcontributions(xandy)ina2×2×2 vs. γ is fixed during the thermodynamic modeling and supercell. We first discuss the Al contribution (i.e. fix not subjected to an equalization of chemical potentials the values x = 0 and z = 1). Replacing one Al atom by (constrained paraequilibrium). 5 where also the configurational entropies in the γ matrix (via the T-dependence of µ (T)) and the κ carbide (via X Eq. (2)) are taken into account. We first note that the free energy difference at T = 0 K is negative in a large part of the plotted chemical potential and in particular for µ corresponding to the Mn experimental matrix composition. As can be seen from Eq.(1), anegativesignimpliesthattheformationofthe κcarbideisexothermic. ForT =600◦C(873K)thefree energy difference becomes at the experimental composi- tion positive for all phases except Fe MnAlC, for which 2 it is almost zero (-14 meV), implying that Fe MnAlC is 2 thermodynamically stable. The κ carbide formation out ofthesolutesolutionisonlyexothermicuptoapprox.625 ◦C, below this temperature the carbide will grow on the expense of the γ matrix, as indeed experimentally ob- servedfortheGBcarbides. However,fortheGIcarbides, the elastic coherency strain has an additional impact on C partitioning as discussed below. Regarding the Mn distribution, the results show that for the exact experimental composition (red dash-dotted line in Fig. 3), Mn-free Fe AlC and Fe MnAlC are en- 3 2 ergetically almost degenerate at T = 0 K, but that Fe MnAlC is energetically clearly preferred at 600 ◦C. 2 In the latter case, this is also true if one allows a slight variation of the composition (green shaded area). The result is in good agreement with the experimentally ob- servedMncontentinκcarbide(Fe Mn Al C ). 1.99 1.10 0.91 0.60 FeMn AlC and Mn AlC will only form if the Mn chem- 2 3 ical potential (Mn content) in the alloy is substantially increased. The stability of Fe MnAlC at T = 600 ◦C is mainly 2 causedbyconfigurationalentropyintheκcarbide,which lowers the energy of this phase with respect to Fe AlC 3 byapprox.0.15eV(comparetherelativepositionsofthe maroon (x = 0) and green (x = 1) lines for T = 0K FIG. 2: (Color online) Microstructure of a Fe-29.8Mn-7.7Al- (dotted) and T = 600 ◦C (solid) in Fig. 3). We have 1.3C (wt.%) alloy aged at 600◦C for 12 weeks: (a) SE image therefore also investigated the impact of (Fe-Mn) config- showing the grain boundary (GB) (κ +γ +α) phases and 0 0 uration in the Fe sub-lattice on the DFT supercell en- graininterior(GI)(κ+γ)phases. (b)zoomed-inSEimageat ergy ESC [Fe MnAlC] in Eq. (1). For this purpose the GI region highlighting the nano-sized GI κ-precipitates. (c) κ 2 APT analysis of GI (κ+γ) phases where κ-precipitates are results of a regular Mn arrangement (periodic repetition visualizedbyCiso-concentrationsurfaceatathresholdvalue oftheunitcell)andanSQSdisorderedstructurearecom- of 9 at.%. pared in Fig. 3 (thick and thin solid line for x = 1) and showanegligibledifference. Acomparisonoverthewhole volume range relevant for subsequent considerations is performed in Fig. 4, where also the impact of magnetic TheequilibriumconcentrationofMnintheκcarbideis disorder is taken into account. The differences of the or- determined via Eq. (1), setting y =0 and z =1. Chang- der of max. 25 meV/unitcell can be translated into an ingthechemicalpotentialchangestheamountofMnand order-disorder transition temperature T . The latter is the thermodynamically most stable carbide phase. The OD a resultof thecompetition between formation enthalpies correspondingphasediagramasfunctionofµ isshown Mn (at T = 0K) and configurational entropy given by the in Fig. 3. The obtained dependence allows us not only expression to connect to our experimental alloy composition (red dash-dotted line; see appendix A for details), but also ESC[SQS]−ESC[ordered]=T Sκ(1,0,1), (7) κ κ OD to investigate chemical and thermodynamic trends. On theonehand,weconstructedtheT=0Kphasediagram with the entropy Sκ defined in Eq. (2). Since the (dottedlinesinFig.3)toseethechemicaleffectonphase stoichiometry in an order/disorder transition remains stabilities. On the other hand, we generalized it to the unchanged, any contributions from chemical potentials annealing temperature of 600 ◦C (solid lines in Fig. 3), (compare Eq. (1)) cancel. Using this equation for a Mn 6 FIG. 3: (Color online) Free energy differences for the κ car- FIG.4: (Coloronline)Freeenergydifferenceofchemicallyor- bide formation according to Eq. (1) with varying Mn con- deredanddisorderedFe MnAlCfordifferentmagneticphases 2 tent x (given by the labels). The results for T = 0 K (dot- is shown as a function of volume at T = 0 K. The energies ted lines) and for the experimental annealing temperature of have been rescaled such that the ground state configuration 600◦C(solidlines)arecompared. Thecolorshadingindicates of Fe MnAlC is taken as a reference. 2 the phase stability at 600◦C as a function of the Mn chemi- calpotentialwithrespecttothereferencepotentialdescribed in appendix A.The chemical potentialscorrespondingto the orderedstructure(AFMD,yieldsavanishingnetmagne- compositionoftheexperimentalalloy(Fe-29.8Mn-7.7Al-1.3C tization) and a completely non-magnetic (NM, unrealis- in wt.%) are for both temperatures shown as a red dash- ticscenarioofvanishinglocalatomicmagneticmoments) dotted lines. For Fe MnAlC the ground state energy of an 2 configuration for comparison. ordered Mn arrangement (thick green line) and of an SQS The results for T = 0 K (Fig. 4) show that the FM disordered structure (thin green line) are compared. phase in chemically ordered κ carbide is energetically most favourable and therefore indeed the correct choice for ground state ab initio calculations. However, some concentration of x=1, one obtains a T of approx. 75 OD of the disordered structures are energetically very close K. Therefore, any chemical ordering will be lost at room to the ground state. In particular, the difference of the temperature, which is in agreement with observations in PM to the FM state is approx. 75 meV/unitcell, which experiment,buthasnotbeenconsideredinpreviousthe- is smaller than that of the AFMD and NM states. This oretical studies24,25. indicates,ontheonehand,alowCurietemperature,T . C To complete the considerations on chemical order and Using Eq. (4), the Curie temperature T for a transi- C to emphasize the crucial role played by the chemical or- tion from chemically disordered FM to the PM phase is dering in the Al sub-lattice for the formation of κ car- approx. 60 K. Even a combined magnetic and chemical bides, we discuss the free energies with chemical dis- disorderingofanoriginallyFMorderedstatewouldonly order in both Fe-Mn and Al sub-lattices (Fig. 4). We require 90 K. This number is only an estimate, because find that the additional chemical disorder in the Al sub- Eq. (3) is based on a mean-field approximation and does lattice makes the formation of κ carbides substantially notdistinguishbetweenFeandMnatoms. Nevertheless, less favourable. Using Eq. (7), the corresponding order- our study supports those experiments44 that do not ob- disorder transition temperature is ≈ 1400 K. Further serve any macroscopic magnetic order in κ carbides at calculations showed that these findings are qualitatively room temperature. On the other hand, we observe very similar for other compositions of κ carbide. little difference between structural properties (e.g. equi- In the following we extend the concept of disorder librium lattice constant) of a FM and a PM material in also to the magnetic degrees of freedom. Experimen- contrast to, e.g., a NM calculation (see also Sec. IIIC). tally, the relevance of magnetic disorder for this carbide This justifies the application of the FM approach, if a isinconclusive. Afewexperimentalworksindicateκcar- PM calculation is not feasible. bides to be ferromagnetic43, in agreement with theoreti- We can now investigate the stability of κ carbides as calcounterparts24. Ontheotherhand,someexperiments a function of its composition (as given by Eq. (1)) to ex- suggestκcarbidenottobemagnetic44. Inourtheoretical plain the experimentally observed C off-stoichiometry in approach,wecomparetheab initiofreeenergies,accord- κ carbides. Due to the computational effort, no thermo- ingtoEq.(1), correspondingtoordered(FM)anddisor- dynamic excitations such as lattice vibrations and mag- dered (PM) spin configurations. In order to evaluate the netic entropy are taken into account. Their impact on, energy difference, we further add another magnetically e.g.,vacancyformationenergiesistypicallysmallatroom 7 FIG. 5: (Color online) Area modulus48,49 of (a) a cubic-symmetry approximant of Fe MnAlC with disorder Fe-Mn sublattice 2 inFMstate,(b)Fe MnAlC ,i.e. withreducedCcontentinFMstate,(c)Fe AlC,i.e. withoutMninFMstate,(d)Fe AlC 2 0.625 3 3 in PM state. The calculation (values in GPa) is based on the determined elastic constants C , C and C summarized in 11 12 44 Table I (visualization by the SC-EMA software package50–52). temperature45. Due to the low order-disorder transition (001)/(001), this misfit is obtained from our DFT calcu- temperature,nochemicalsuperstructure/orderingonthe lations to be in the stoichiometric case as high as 9%. Fe-Mn sublattice can be expected. In principle, also the This value is too large to allow coherent interfaces with- magnetic disorder should be taken into account. Due to out misfit dislocations. the fluctuating moments in this phase, however, the nec- Synchrotron diffraction experiments (not shown here) essary relaxations, e.g., for a vacancy calculation would have indicated a reduction in the lattice misfit to 1.4% require sophisticated approaches such as the spin-space between off-stoichiometric κ carbide and γ matrix in the averaging (SSA) method46. This goes beyond what is grain interior. In order to enforce a completely coher- currently feasible for a complex alloy like the κ carbides. entinterfacewithoutmisfitdislocations,asitisobserved Having in addition the limited impact of magnetism on for GI carbides (at least for the small channels of γ structural properties in mind (Fig. 4), we restrict most material6), a compromise of the lattice constant of both of our calculations to the magnetic ground state (FM). phases is required. It will depend on the volume fraction of the phases and the elastic energy associated with a compression or elongation. C. Elastic properties Togetadeeperunderstandingoftheelasticproperties of κ carbides, we determined its elastic tensor. Accord- The κ carbides have so far been considered as an in- ing to the investigations of the previous section, we first dividual bulk phase. However, the experimental find- use the composition Fe MnAlC with chemical disorder 2 ings provided above clearly indicate that the C off- and ferromagnetic order for this purpose. The results stoichiometry is strongly related to the microstructure. aresummarizedinTableI.Acomparisonoftheseelastic The main difference between GI and GB precipitates is constants with those of a cubic elastic approximant38,47 the coherency to the matrix material. We argue that based on the values reported in Ref. 24 for an ordered, next to configurational entropy also the strain caused by ferromagnetic unitcell shows that the chemical disor- the degree of coherency drives the C out of the carbide. der has only limited impact on elastic properties of the The coherency is related to the lattice parameter mis- studied κ carbide. The directional dependence of the match between κ carbides and the γ matrix material. corresponding single-crystalline Youngs modulus yields For the experimentally observed orientation relationship a significant anisotropy of the ferromagnetic κ carbide. 8 The hard (cid:104)001(cid:105) direction has an almost twice as large TABLE I: Single-crystalline elastic constants (C , C , C , Youngs modulus (394 GPa) as the soft (cid:104)111(cid:105) direction 11 12 44 B) calculated for different chemical compositions and mag- (215 GPa). For our considerations, however, the area netic states of κ carbide. The selections are identical with modulus of elasticity48,49, which provides the amount of those shown in Fig. 5. In the cases (a) and (b), a disor- energyneededforcoherentplanarloadingwithinaplane dered configuration of Fe and Mn is considered. For com- normaltothevectorn,ismorerelevant. Thedirectional parison, elastic constants of a cubic elastic approximant38,47 dependence of these normal vectors, n, is visualized in basedonresultsobtainedfororderedFe MnAlCfromRef.24 2 Fig. 5a, which still shows an anisotropy. The Youngs are shown. All values are in GPa. modulus is highest for the {001} planes, i.e. the cor- responding energy required for epitaxial loadings within Composition Magn. C11 C12 C44 B theplanesthatarerelevantfortheκ/γcoherencyishigh- (a) (Fe2,Mn)AlC FM 418 77 82 191 est. Thisobservationtogetherwiththelargemisfitof9% (b) (Fe2,Mn)AlC5/8 FM 282 167 94 205 (c) Fe AlC FM 446 109 72 221 makes the stabilization of an (001)/(001) very unlikely, 3 (d) Fe AlC PM 439 90 96 206 in puzzling disagreement with experiment. 3 Ref. 24: Fe MnAlC FM 436 80 92 199 2 A reduction of the C content is expected to yield a smaller misfit. The question is, however, how it influ- encestheelasticproperties. Thechallengeofcorrespond- ing calculations of the elastic tensor is to ensure a cubic crystal structure of the 2×2×2 supercell. A tetragonal distortion would not only increase the numerical effort significantly,itisalsoinconflictwiththephysicalexpec- tationforaninfinitelylargesystem. Theonlyreasonable choice that fulfills this constraint is the presence of three C vacancies. The resulting area modulus of elasticity is shown in Fig. 5b. It reveals that some of the elastic constants are softer, as expected from the high vacancy concentration, whilethebulkmodulusishardlychanged (Tab. I). More important is the observation that (cid:104)001(cid:105) has now turned into the elastically soft direction, there- with resolving the before mentioned puzzle. Due to the central importance of the elastic proper- tiesfortheupcominginvestigations,wealsoinvestigated the impact of the assumptions formulated at the end of Sec. IIIB. Figure 5c allows a comparison of the area FIG.6: (Coloronline)Volumedependenceoftheenergycon- modulus for Fe MnAlC with the Mn-free version, while 2 tributions to the formation energies of C, Fe and Al vacan- Fig.5dshowstheresultsofafullyparamagneticcalcula- cies in FM κ carbide according to Eq. (8). The energies are tion. Inbothcasesaclosesimilaritytotheresultsforthe rescaled such that the perfect Fe AlC (filled symbols) at its 3 FM Mn-containing version shown in Fig. 5a is obtained. equilibrium lattice constant is taken as a reference. The ver- For the area modulus as well as the bulk modulus the ticaldashedlinemarkscompromisingvolumebetweenκcar- maximum changes are of the order of 10 %. This jus- bide and γ matrix, if both have a volume fraction of 50% tifies our choice for the chemical and magnetic degrees (compare with Fig. 8). of freedom. In addition, we note that the area modulus does not show a strong anisotropy, if the C content is re- for species X, according to the expression duced (Fig. 5b). We therefore consider in the upcoming calculations the bulk modulus instead of the area modu- Ef (V)=ESC (V)−ESC (V)+µ (T) (8) lus. X-Vac X-Vac Fe3AlC X analogous to Eq. (5) where µ (T) is obtained at 600◦C X asdefinedinappendixA.Thevolumedependenceofthe supercell energies entering Eq. (8) are shown in Fig. 6, D. Vacancy formation energy where the energies of the perfect carbide (filled symbols) shouldbecomparedwiththeenergiesofthedefectstruc- Due to the coherency strain, we expect a driving force tures (open symbols). Since these defect energies are for forCtoleavetheκcarbideanddissolveinthematrix. A FeandAlvacanciesformostvolumessubstantiallyhigher Cdepletion(asdescribeinSec.IIIE)isexpectediftheC thanthoseofFe AlC,theirvacancyformationisunlikely. 3 vacancy formation is exothermic, or if the energy loss is ThesituationisdifferentforthecaseofC.Attheequilib- small enough to be compensated by a gain in configura- rium volume of κ carbide, for example, the red symbols tional entropy at finite temperature. We have therefore are below the green symbols, i.e. the C vacancy forma- investigatedthecorrespondingvacancyformationenergy tionenergyisnegative(-290meV).Hence,theremovalof 9 asingleCatomfromanotherwiseperfectκcarbideisan exothermicprocess. Theoriginofthenegativeformation energy lies largely in the large configurational entropy in the γ matrix, where the C concentration is low. The consequences of this driving force will be discussed in the next subsection. Second, there is a remarkable volume dependence of the C vacancy formation energy, yielding a substantial reduction to even more negative values under volumet- ric compression. This reduction is a consequence of the largenegativevacancyformationvolumeofapprox. 7A˚3, which allows the system to efficiently release strain en- ergy by creating C vacancies. As a consequence, the for- mation of C vacancies is more feasible in κ carbides that areformedascoherentprecipitatesintheFematrixthan in incoherent particles as formed near grain boundaries. FIG. 7: (Color online) Schematic picture of C partitioning As discussed at the end of Sec. IIIB, the calculations betweenκcarbideandγ matrix: AssuminganequalCdistri- bution in the as-cast state (dashed line), there is a chemical are performed for Mn-free κ carbide. This is mainly drivingforceforCtoaccumulateinAl-orderedregions. Since due to the fact that we would otherwise need to treat thisimposesanelasticenergypenalty,thedecompositionwill the Fe-Mn sublattice as a disordered alloy, which results remain incomplete. in a huge increase in the number of configurations to be considered for the calculation of (multiple) vacancies. WhileweshowedinSec.IIICthattheeffectontheelas- formκcarbide(whichisanexothermicprocess). TheAl tic energy is small, we have also tested the impact for ordering,whichisusedinthisworktodefinetheregionof the chemical part of the vacancy formation. We realize κcarbide, isvolumeconserving, whiletheCpartitioning that the difference in formation energies of a C vacancy isnot. Sincethecoherencyconditionpreventsanyrelease in Mn free (Fe AlC) and Mn containing (Fe MnAlC) κ 3 2 of elastic energy by plastic relaxation via misfit disloca- carbides can be up to 0.24 eV. We have further consid- tions at the interface, partitioning unavoidably increases ered the impact of magnetism on the vacancy formation the misfit and thus the elastic energy. This mechanism energies, by performing a fully paramagnetic calculation prevents a complete filling of the Al-ordered region (i.e. for a single chemical configuration. These calculations the κ carbide) with C. are extremely challenging and prone to errors, but the These considerations show that the required energy obtained deviations from the FM calculation are in the minimization also needs to take the chemical and elas- same order of magnitude as the chemical difference. It tic energy of the γ matrix into account. In principle, we is therefore clear that the upcoming calculations cannot shoulddeterminethevolumedependentCsolubilityina aim at a quantitative reproduction of the experimental disorderedFe-Al-Mnmatrix. AsmentionedinSec.IIIB, results,sincethenumericalefforttoachievethisaccuracy Mn has been removed from the considerations, but even would be enormous. However, the general mechanisms the treatment of Al disorder in the γ matrix would re- for the C partitioning discussed in the following are not sult into a large configuration space. Two limiting cases affected by these approximations. canbeconsideredinstead: (i)theγ matrixconsistsofFe and C only, or(ii) the γ matrix is itself an ordered Fe-Al phase. For reasons that will be discussed below, we only E. Partitioning between κ and γ use the first scenario. The optimization of the composite consisting of κ pre- Given that the combination of configurational entropy cipitate and γ matrix does not only affect the C concen- and the coherency constraint results in negative vacancy trations, but also the coherent lattice parameters. We formationenergies,itisclearthatthecommonlyapplied expressthelatterbyVκ+γ,thecoherentvolumeperunit- concepts of dilute point defects cannot be used for the cell, which is an intensive thermodynamic variable. It present study. Rather, since the concentration changes capturesboththehydrostaticchangeoflatticeconstants are well above a few percent, thermodynamic concepts of cubic nano-precipitates and the volume change in a developed for alloy decomposition become appropriate. tetragonal distortion, if the coherency is only assumed In this sense we discuss the problem as an incomplete C for the in-plane lattice constant (biaxial strain) and the partitioning between κ carbide and γ matrix as shown normal component is relaxed. Therefore, the Helmholtz in Fig. 7, i.e. we have in upcoming considerations the freeenergyofthecompositeofκprecipitateandγmatrix following physical picture in mind: After casting, C and is given by Al are homogeneously distributed in the sample. During annealing the onset of Al ordering occurs along with a Ftot(T,V,cκ,cγ) = vκFκ(T,V,cκ) chemical driving force for C to enter these regions and + v Fγ(T,V,c ), (9) γ γ 10 where v and v are volume fractions of κ and γ respec- determines a concentration c and a corresponding con- κ γ κ tively and c and c are their corresponding C concen- centration c as given by Eq. (10). Since c cannot be κ γ γ γ trations. It can be split into an elastic, a chemical and a represented by a 2 × 2 × 2 supercell and since Veg- a configurational part. In the case of the κ carbide (ex- ardslawisfulfilled,alinearinterpolationisemployedfor pressions for γ matrix are similar) the definition of the eachV inordertodetermineFγ(T,V,c ). Subsequently, γ first two terms is given by theequilibriumcoherentvolumeV isobtainedbythe κ+γ minimizationofthetotalfreeenergy(Eq.(9))oftheκ-γ Eelas(V,cκ) = Eκ(V,cκ)−Eκ(Vκ(cκ),cκ) composite with respect to the volume V which is com- Echem(Vκ(cκ),cκ) = Eκ(Vκ(cκ),cκ) mon for both the phases. The results for Vκ+γ are again linearly interpolated. − Eκ(V (c ),c ). κ exp exp Theprocedureisrepeatedforvariousvolumefractions The chemical part covers the change of the concentra- of the phases which enter Eq. (10). Fig. 8 shows the tion at equilibrium cubic volume as obtained from the resulting Vκ+γ together with the unstrained equilibrium Murnaghan equation of state53,54. The elastic part cov- volumes of the individual phases. The results of Vγ indi- ersthevolumedeformation(hydrostaticorbiaxial). The catesonceagainthattheCconcentrationintheγ matrix reference is the homogeneous C distribution (see Fig. 7) dependsonthevolumefractionvκ foragivenconcentra- with a concentration determined by experiment cexp at tion cκ due to Eq. (10). The lower the C concentration its respective equilibrium volume V (c ). inκcarbide,themoresimilarthelatticeconstantsofthe κ exp The C concentrations are not independent, but are unstrained phases get. coupledduetothefactthatthetotalnumberofCatoms Assuming coherency of the carbide in all three dimen- during the partitioning must be conserved: sions, the common volume per unitcell of the composite V will be closer to that of the κ phase than of the κ+γ c v +c v =c . (10) γ phase, because the former is stiffer and has the larger κ κ γ γ exp bulkmodulus. Nevertheless,theκcarbideshowsasignif- The C concentrations c and c are both defined with icant adaptation of its lattice constant, too. The impact κ γ respect to the octahedral sublattice that corresponds to ofthevolumefractiononV issmallandcanbesafely κ+γ the body-centered positions in Fig. 7, i.e. one per four neglected. A larger volume fraction v (larger impact γ metal atoms. If the C concentration in this sublattice is on V ) is compensated by a higher C concentration κ+γ 100 at.% (complete filling of this sublattice), then the C (i.e. increasingV )oftheγ matrix. Further,V shows γ κ+γ concentration per unitcell would be 20 at.% (the other hardly any concentration dependence, since the effects 80 at.% are metal atoms). Since the experimentally de- of V and V cancel each other. Therefore, the value of κ γ termined C concentration per unitcell (averaged over κ V ≈ 49˚A3/unitcell can be safely used as a universal κ+γ and γ) is only 9 at.%, the sublattice C concentration is parameter of the system. cexp =9/20=45 at.%. After the volume optimization, we now perform the Thepossibilitytooccupyonlyonesublatticelimitsthe energy minimization with respect to C concentration. In numberofconfigurationsandhasthusastrongimpacton principle,therearetwoprocedurespossibleandbothare the configurational entropy. This is taken into account compared for a volume fraction v = 0.5 and the an- κ by including the number of available sublattices (sγ=4 nealing temperature of T = 600 ◦C. First, one can in- and sκ=1). Therefore, the overall expression for the free troduce temperature dependent chemical potentials for energyoftheindividualphasesσ (=κorγ)inEq.(9)is C (see Eq. (A.11) of appendix B), which we now treat as formally independent in both phases and which have Fσ(T,V,c )=E (V,c )+E (V (c ),c ) σ elas σ chem σ σ σ both been plotted in Fig. 9d with a common x axis (c κ (cid:20)c c (cid:18) c (cid:19) (cid:18) c (cid:19)(cid:21) and c are coupled by Eq. (10)). The thermodynamic +k Ts σ ln σ + 1− σ ln 1− σ . (11) γ B σ s s s s equilibrium is then determined by the intersection point σ σ σ σ of these two lines. We note that exactly the same result The particle conservation (10) enables us to express is obtained, if the particle conservation (10) is used and c in terms of c . Under these circumstances, Eq. (9) the Helmholtz free energy is directly minimized. γ κ simplifies, i.e., Ftot(T,V,c ,c ) = Ftot(T,V,c ) imply- The dependence of the free energies on c is shown in κ γ κ κ ing that we have to perform the minimization only over Figs. 9b and 9c. Similarly to Fig. 8, DFT data points a single concentration c . Before doing so, we consider can only be provided for the individual phases, while an κ the minimization with respect to the volume V in order interpolation (polynomial fit) is used for the composite. to obtain V for different values of concentration c . Thefreeenergy Ftot(T,V,c )inFig.9c(solidline)illus- κ+γ κ κ Forthispurpose, theMurnaghanequationofstateisap- trates that starting from the homogeneous distribution, plied to the energy-volume curve for an integer number the partitioning of C atoms yields first a gain in energy ofCatomsin2×2×2supercells. Thisprocedureisper- before it increases again when too many C atoms are formedforbothphasesseparately. IfseveralCconfigura- transferred into the carbide. The minimum energy is tions are possible, an averaging of the energies has been achieved at an equilibrium sublattice C concentration in performed. EachCatomremovedfromthesupercellofκ the κ carbide of approx. 55 at.%.

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Ab initio explanation of disorder and off-stoichiometry in Fe-Mn-Al-C kappa carbides PDF

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Ab initio explanation of disorder and off-stoichiometry in Fe-Mn-Al-C κ carbides Poulumi Dey,1 Roman Nazarov,2 Biswanath Dutta,1 Mengji Yao,1 Michael Herbig,1 Martin Fri´ak,3,4 Tilmann Hickel,1 Dierk Raabe,1 and J¨org Neugebauer1 1Max-Planck-Institut fu¨r Eisenforschung GmbH, D-40237 Du¨sseldorf, Germany 2Lawrence Livermore National Laboratory, Livermore, CA 94550, USA 3Institute of Physics of Materials, v.v.i., Academy of Sciences of the Czech Republic, CZ-61662 Brno, Czech Republic 4Central European Institute of Technology, CEITEC MU, Masaryk University, Kamenice 5, CZ-625 00 Brno, Czech Republic (Dated: January 25, 2017) Carbides play a central role for the strength and ductility in many materials. Simulating the 7 impact of these precipitates on the mechanical performance requires the knowledge about their 1 atomic configuration. In particular, the C content is often observed to substantially deviate from 0 the ideal stoichiometric composition. In the present work, we focus on Fe-Mn-Al-C steels, for 2 which we determined the composition of the nano-sized κ carbides (Fe,Mn) AlC by atom probe 3 n tomography (APT) in comparison to larger precipitates located in grain boundaries. Combining a densityfunctionaltheorywiththermodynamicconcepts,wefirstdeterminethecriticaltemperatures J for the presence of chemical and magentic disorder in these carbides. Secondly, the experimentally 4 observed reduction of the C content is explained as a compromise between the gain in chemical 2 energy during partitioning and the elastic strains emerging in coherent microstructures. ] PACSnumbers: 61.50.Nw,61.72.jd,71.15.Mb i c s - I. INTRODUCTION The arrangement of κ carbides within the microstruc- l r ture is to a large extent determined by the E2 crys- t 1 m tal structure of κ carbide. It resembles a perovskite- Fe-Mn-Al-C based steels have recently attracted close type cubic structure with Al atoms at the corners of the . attention because of their high strength and ductility1,2 at along with a high corrosion resistance and a compara- cube, Fe atoms at the face-centered sites (correspond- m ing to L1 ), and C atoms at the body-center octahedral bly low density3. This combination of their properties 2 site (also called L(cid:48)1 ). Therefore, the nominal composi- - makes them also attractive for automotive applications. 2 d tion is Fe AlC. Experiments for high-Mn alloys indicate The excellent mechanical performance of the Fe-Mn-Al- 3 n thattherecanbeasignificantmanganesecontentreplac- C steels is mainly attributed to microstructure features o ingtheFeatoms,yieldinga(Fe,Mn) AlCcomposition14. c thatcorrelatewithdeformationmechanismsandstrongly 3 While different ordered E2 structures for varying Mn [ depend on the amount of Al in the material. High-Mn 1 contents are shown in Fig. 1, it is still unclear which steelswithalowAlcontent(<5wt.%)typicallyundergo 1 Mncontentwouldcorrespondtothermodynamicequilib- amicrostructurerefinementbytheactivationofdeforma- v rium, how relevant the ordering is, and how this affects 0 tiontwinningintheaustenitephase, whichincreasesthe the mechanical properties. 1 strain-hardening rate4,5. When the Al content in these 9 steels(about30wt.%Mnand1.3wt.%C)ishigherthan The orientation relationship between the regularly ar- 6 6 wt.%, an annealing produces finely dispersed nano- ranged κ carbides and the γ matrix is reported to be 0 . sized κ carbides (Fe,Mn)3AlC in the austenitic matrix. (001)/(001) in experiments15. The interfaces are coher- 1 Experiments showed that these precipitates strengthen ent in case of κ nano-precipitates without indications 0 Fe-Mn-Al-C steels, thereby making them interesting for of misfit dislocations. Such a microstructure can, how- 7 applications. For example, the large age-hardenability ever, not be understood if a completely stoichiometric 1 : of these alloys is attributed to the homogeneous pre- composition is assumed. As the density functional the- v cipitation and dispersion of κ carbides in the austenitic ory (DFT) calculations performed in this paper show, Xi matrix6–10. a nominal E21 structure would have its elastically hard Using specific heat treatments, a γ/κ regular mi- axis in the (001) direction and the resulting misfit of 9% r a crostructure can be achieved, which strongly influences with respect to the lattice constant of the matrix mate- the ductility at ambient temperatures11. Further, κ car- rial would be too large, to ensure coherent interfaces. bides improve the creep resistance of Fe-based alloys at Indeed, electron microprobe experiments have shown high temperatures making them desirable materials for considerable deviations from the Fe AlC stoichiometry. 3 manufacturinghigh-temperaturecomponentssuchasgas In an experimental work done by Palm et al.16, the turbine blades and vanes in aircraft engines, aerospace off-stoichiometriccompositionobservedisFe Al C 3+y 1−y z andpowergeneratingplants. Thepresenceofnano-sized where y may vary between -0.2 and +0.2 and z between κ carbides therefore yields mechanical properties of Fe- 0.42 and 0.71. Other experimental works have proposed basedalloysthataresimilartoNi-basedsuperalloys12,13, an off-stoichiometric composition of Fe AlC 17–19. All 3 0.5 provided the desired microstructure is achieved. these observations demonstrate in particular a depletion 2 ofCinκcarbidesascomparedtothenominalE2 struc- perature applications the chemical and magnetic order 1 ture of Fe AlC. can break down. We thus studied the impact of such 3 In spite of these experimental evidences of off- magnetic and/or chemical disorder on the stability of κ stoichiometric C compositions of κ carbides, further carbides. measurements that can resolve the properties of nano- precipitates are desired. In the present study, we use atom probe tomography for this purpose, since II. METHODOLOGY it combines near-atomic resolution with ppm chemical sensitivity20. At the same time, the theoretical investi- We perform calculations using DFT27,28 as imple- gation of the C depletion is still limited. The main focus mented in the Vienna Ab Initio Simulation Package of previous theoretical studies has been on perfectly or- (VASP)29–31. Theelectron-ioninteractionisdescribedby dered κ carbides. In a previous investigation, DFT has using projector augmented-wave (PAW) potentials32,33. been employed to compare the properties of Fe Al-L1 3 2 The generalized-gradient approximation (GGA) func- and ordered Fe AlC-E2 structures and to underline the 3 1 tionalofPerdew,BurkeandErnzerhof(PBE)34 hasbeen roleplayedbyC21,22. Thesestudieshaveshownthatthe employed. The Methfessel-Paxton method35 has been addition of C atoms decreases the magnetic moment of used for the Fermi surface smearing with a 12×12×12 the neighboring Fe atoms and yields a heat capacity and Monkhorst-Pack grid36 in a 1×1×1 5-atom unitcell for elasticconstantsofFe AlC-κthatareappreciablydiffer- 3 the κ carbides shown in Fig. 1. A supercell (SC) size ent from Fe Al-L1 21. In similar spirits, the energetics 3 2 of 2×2×2 (40 atoms) has been considered for the dis- and magnetic properties of Fe Al and Fe AlX (where X 3 3 ordered and vacancy calculations with a corresponding =H,B,C,N,O)compoundsareinvestigatedusingDFT, Monkhorst-Pack grid of 6×6×6. The single-electron among which Fe AlC turns out to be most stable when 3 wavefunctionshavebeenexpandedbyusingplanewaves comparing cohesive energies23. The reduction in mag- up to an energy cut-off of 500 eV. The energies are con- netization of Fe Al due to the addition of C has been 3 verged to a precision of better than 1 meV/formula unit explained by relaxation effects induced by the C atom in (f.u.). the Fe Al structure23,24. 3 We first study the occupation of the metal sublattice The computation of the elastic constants of κ car- sites,sinceithasanimpactontheCsolvationenergiesin bides has revealed that these carbides are more rigid κ carbide. To determine the equilibrium Mn content, κ than the parental Fe Al-L1 structure24. In the same 3 2 carbides (Fe Mn )AlC with integer x are considered. work, the issue of different chemical configurations has 3−x x Fororderedconfigurations,asingle5-atomunitcell(refer been discussed for the Fe-Mn sublattice by considering to Fig. 1) is used, which is periodically repeated. The (Fe Mn )AlC with integral values for x from 0 to 3. 3−x x chemical disorder of the κ carbides has been simulated In a subsequent work by the same group25, low Mn con- by the special quasi-random structure (SQS) scheme37 centrations in κ carbides have been investigated which in a 2×2×2 SC. Two kinds of SQS are generated: one show absence of any kind of interaction between sub- withchemicaldisorderontheFe-Mnsub-latticeonly(i.e. stitutional Mn atoms thereby indicating a random alloy keeping the symmetry of L1 ) and the other one with a system. A previous work further indicates the relevance 2 random distribution of all metal atoms (i.e. using the of point defects such as C vacancies (treated in the di- symmetry of fcc). In order to generate these SQS, corre- lute limit) for the thermodynamic stability of the rele- lation functions of up to five-body figures are used. The vant phases22. While these studies provided important chosenSQSinourstudyhavethelowestcorrelationerror insight into the structure and thermodynamics, none of in terms of the error-function introduced in our previous themfullyexplainedtheabovementionedCreductionof work38. κ carbides. As we will show in the present study, this is Thestabilityoftheseκcarbidesisinvestigatedbycom- mainly due to the geometrical constraints of a coherent puting the Helmholtz free energy difference between the interface to the matrix material. precipitates and the surrounding solid solution assuming In our first ab initio study of κ carbides14, we have thatthetwophasesarethermodynamicallyequilibrated: already been able to reveal and explain the Al deple- tion in these carbides by coherency strains. We have pointed out that this effect alone is not sufficient, but ∆F(T,V,x,y,z) (1) that it occurs concurrently with a reduced C content in = ESC[(Fe Mn )Al C ]−TSκ(x,y,z) theseprecipitates. Inthepresentwork,wenowprovidea κ 3−x+y x 1−y z deeper theoretical understanding of the C concentration − (3−x+y)µ −xµ −(1−y)µ −zµ . Fe Mn Al C in off-stoichiometric κ carbides, which is benchmarked against experimental data. This investigation requires The first term is the ground state energy ESC deter- κ the careful application of various thermodynamic con- mined in a DFT supercell (SC) calculation. The sec- cepts. One of them is the application of a constrained ond term gives the entropy contribution. Neglecting the paraequilibrium26, which allows us to focus on C only. vibrational entropy, which is small compared to the con- Further, at operational conditions relevant for high tem- figurational contribution, we express the entropy solely 3 by the latter one: andPMstatecanbeexpressed40as∆E =N2 (cid:80) J mag i(cid:54)=j ij and the above equation transforms to (cid:16) 3−x x Sκ(x,y,z)=−k (3−x)ln +xln B 3 3 2 ∆E k T = . (4) +ylny+(1−y)ln(1−y) B C 3N mag (cid:17) +zlnz+(1−z)ln(1−z) , (2) It may be noted that the values of T obtained using C Eq.(3)typicallyoverestimatetheexperimentalvalues41, where k is the Boltzmann constant, x,y and z the con- B but provide correct qualitative trends. tent of Mn, Fe antisites on the Al sublattice and C in κ, Single-crystalline elastic constants of the disordered κ respectively. ThethirdterminEq.(1)balancesthether- carbides are determined using tetragonal and trigonal modynamicexchangeofatomsbetweentheκcarbideand (rhombohedral) cell-shape deformations38. Due to the the γ matrix. In the spirit of a grand-canonical ensem- fact that our SQS supercells in general do not possess ble, this exchange can be described by taking/removing cubic symmetry, strains have been applied along struc- atoms from the chemical reservoir, which is determined turally equivalent directions, the resulting stresses are by the free energy of the γ solution. used to calculate elastic constants and these have been In the present paper, the chemical reservoir is repre- then averaged (for details see, e.g., Ref. 38). sented by the chemical potentials µ of the involved el- X The theoretical investigations are supported by exper- ements X = Fe, Mn, Al, C. They depend on the (ex- imental investigations on the C content in κ carbide. perimentally given) composition, temperature, and vol- For this purpose, a high-Mn steel of the composition Fe- ume of the γ matrix and are computed by DFT (see ap- 29.8Mn-7.7Al-1.3C (wt.%) has been used, which is aged pendix A). An advantage of using chemical potentials is at 600◦C for 12 weeks. The material has undergone a that they provide a physically intuitive tool to describe solid solution treatment at 1100◦C for two hours and is continuous changes in the chemical composition of the subsequently oil quenched prior to ageing. A system- considered alloys without being limited to discrete sto- atic repetition of various aging treatments ensures that ichiometries imposed by finite size supercells. This is the present conditions yield a thermodynamically stable particularly useful for the constrained paraequilibrium26 partitioning of the chemical elements. Further details of discussed in the second part of the paper, where we en- alloy casting and thermo-mechanical processing are re- force an equality of chemical potentials between κ and γ ported elsewhere6,14. The sample is etched with 1% Ni- for the interstitial C atoms. We note that the ab initio talsolutionandcharacterizedusingafieldemissionscan- derivation of µ from DFT energies for a specific super- X ningelectronmicroscope(SEM)ZeissXB1540equipped cell ESC[Fe Mn Al ] implies that the absolute value of γ x y z withanelectronbackscatterdiffraction(EBSD)detector. thechemicalpotentialsinthematrixisdependentonthe Needle-likeatomprobetomography(APT)samplesfrom given pseudopotential (see appendix A for details). grain boundary and grain interior regions are prepared The energetically favoured magnetic phase in the via a standard FIB procedure by a dual-beam focused- κ carbides is determined by computing the free en- ion-beam(FIB)system(FEIHeliosNano-Lab600i)42. A ergy difference in Eq. (1) for ferromagnetic (FM), anti- LEAPTM3000XHRsystem(CamecaInstruments)isem- ferromagnetic double layer (AFMD), and non-magnetic ployedforAPTanalysiswithvoltage-pulsingat200kHz (NM) phases. Since the FM phase is found to be the pulse repetition rate, 0.005 atom/pulse detection rate, T =0Kgroundstate,itisusedinthecalculations,ifnot 15% pulse fraction at 70 K. stated otherwise. The γ matrix is consistently treated in an anti-ferromagnetic (AFM) state. Paramagnetic (PM) energies for κ carbides are again obtained by a 2×2×2 III. RESULTS AND DISCUSSION supercell using the SQS scheme, which mimics a random distribution of collinear local moments as closely as pos- sible for this SC. This procedure has been performed for A. Experiment the chemically ordered as well as the disordered κ car- bides. Asindicatedintheintroduction,itisthemainpurpose The Curie temperature, T , is estimated within our of the theoretical investigations in this paper to reveal C study from the mean field approximation of the Heisen- the reasons for the C off-stoichiometric compositions in berg model39 κ carbides. Previously, our own measurement14 for a κ-containing steel, namely an Fe-29.8Mn-7.7Al-1.3C (in k T = 2N (cid:88)J , (3) wt.%)alloy,hasgivenavaluez =0.61. Withthepresent B C 3 mag ij experimental evaluation, we employ a much longer aging i(cid:54)=j treatment to ensure thermodynamic equilibrium. whereN isthenumberofmagneticatomsintheunit- Figure 2 shows the microstructure of the same alloy mag cellandJ arethemagneticexchangecouplingconstants as used in Ref. 14 after the prolonged ageing. It clearly ij between sites i and j. Using mean field approximation, showstwodifferentmorphologiesofκcarbides,whichare the energy difference ∆E per unitcell between the FM the bright protruding phases after etching (Fig. 2(a) & 4 a) b) c) d) FIG. 1: (Color online) Crystal structures of (a) Fe AlC, (b) Fe MnAlC, (c) FeMn AlC, and (d) Mn AlC. Red, golden, green 3 2 2 3 and black balls represent Al, Fe, Mn and C atoms respectively. (b)). On the one hand, there are nanosized κ precipi- Feinthesupercell,weobtainy =0.125,whichiscloseto tates in the grain interior (GI) i.e. within the austen- the reported experimental composition16. We then ob- ite matrix γ, regularly aligned along specific directions tain at T =0 K an increase of the free energy difference (Fig. 2(b) & (c)), which are orthogonal (cid:104)001(cid:105) crystallo- betweenthecarbideandtheγ matrix(seeEq.(1))byap- graphic directions6,15. On the other hand, a µm-scale prox.1eVascomparedtothestoichiometriccomposition lamellarstructuremainlycomposedofalternativecoarse of Fe AlC. This energy increase enters the temperature 3 κ carbides and solute-depleted austenite γ is observed dependent antisite formation energy given by 0 0 at regions next to grain boundaries (GB) between γ grains (Fig. 2(a)). Also a small fraction (<1%) of fer- Ff (T)=ESC −ESC −µ (T)+µ (T). (5) rite α is detected in these regions by EBSD (not shown AS FeAl Fe3AlC Fe Al here). This κ0+γ0+α lamellar microstructure initiates theconfigurationalentropyintheκcarbideisconsidered at GBs and grows into GI region. The chemical com- if the antisite concentration is determined by position of GI κ carbide as measured by APT is found to be Fe1.99Mn1.10Al0.91C0.60 and that of GB κ0 carbide (cid:34) Ff (T)(cid:35) to be Fe Mn Al C . These chemical composi- c =exp − AS . (6) 1.69 1.35 0.95 0.87 FeAl k T tions confirm deviations from stoichiometric C concen- B trations in κ carbides. The nano-sized GI κ carbides seems to be stabilized by the coherence constrain, show- Neglecting again vibrational contributions, the tempera- ing almost the same composition after 24 hours14 and turedependenceofthedefectformationenergyoriginates 12 weeks. These GI precipitates are observed to barely solely from the one in the chemical potentials µFe(T) coarsen after prolonged ageing maintaining an average and µAl(T) imposed by the γ matrix (see appendix A sizeofapprox. 20nm. ThelargerGBκcarbides,incon- for details). It takes care of the fact that with increas- trast to this, represent a thermodynamicallymore stable ing temperature the chemical potential decreases due to state, since they grow on expense of the matrix phase enhanced configurational entropy. in the grain interior. The microstructure evolution upon UsingEq.(6), onecanexpect0.001%oftheAlatoms aging has been thoroughly studied and will be discussed tobereplacedbyFeat600◦C.Aselasticeffectsareinthe elsewhere. Full coherency of the GI κ/γ interface has focus of the present investigations, the lattice constant been observed by high-resolution transmission electron of the κ carbide has also been constrained to that of the microscopy and no indication for a segregation to this surrounding Fe matrix. Even the decrease of the antisite interface is found by APT measurements. For the pur- formation energy due to this strain (from 1 eV to 0.8 eV pose of the present study, however, most important is atT =0K)istoosmalltoyieldanoff-stoichiometriccon- the noticeable difference in the C concentrations in GI centration higher than 0.01 % at elevated temperatures. and GB κ carbides with the former (GI) showing more The situation is different in the case of Mn antisites on C reduction than the latter (GB). the Al sublattice if C vacancies are additionally present at neighboring sites (see Ref. 14 for details). In this case an Al reduction of up to 10 at.% can be observed. For the purpose of the present investigations this effect is B. Chemical and magnetic order still not decisive and it is justified to assume a filled Al sublattice,whichstabilizestheκcarbideandactsather- When investigating with ab initio simulations the C modynamicdrivingforceforthepartitioningofC.Using content(z)ofκcarbides(Fe Mn )Al C ,werep- this assumption implies that the volume fraction of κ 3−x+y x 1−y z resenttheMnandAlcontributions(xandy)ina2×2×2 vs. γ is fixed during the thermodynamic modeling and supercell. We first discuss the Al contribution (i.e. fix not subjected to an equalization of chemical potentials the values x = 0 and z = 1). Replacing one Al atom by (constrained paraequilibrium). 5 where also the configurational entropies in the γ matrix (via the T-dependence of µ (T)) and the κ carbide (via X Eq. (2)) are taken into account. We first note that the free energy difference at T = 0 K is negative in a large part of the plotted chemical potential and in particular for µ corresponding to the Mn experimental matrix composition. As can be seen from Eq.(1), anegativesignimpliesthattheformationofthe κcarbideisexothermic. ForT =600◦C(873K)thefree energy difference becomes at the experimental composi- tion positive for all phases except Fe MnAlC, for which 2 it is almost zero (-14 meV), implying that Fe MnAlC is 2 thermodynamically stable. The κ carbide formation out ofthesolutesolutionisonlyexothermicuptoapprox.625 ◦C, below this temperature the carbide will grow on the expense of the γ matrix, as indeed experimentally ob- servedfortheGBcarbides. However,fortheGIcarbides, the elastic coherency strain has an additional impact on C partitioning as discussed below. Regarding the Mn distribution, the results show that for the exact experimental composition (red dash-dotted line in Fig. 3), Mn-free Fe AlC and Fe MnAlC are en- 3 2 ergetically almost degenerate at T = 0 K, but that Fe MnAlC is energetically clearly preferred at 600 ◦C. 2 In the latter case, this is also true if one allows a slight variation of the composition (green shaded area). The result is in good agreement with the experimentally ob- servedMncontentinκcarbide(Fe Mn Al C ). 1.99 1.10 0.91 0.60 FeMn AlC and Mn AlC will only form if the Mn chem- 2 3 ical potential (Mn content) in the alloy is substantially increased. The stability of Fe MnAlC at T = 600 ◦C is mainly 2 causedbyconfigurationalentropyintheκcarbide,which lowers the energy of this phase with respect to Fe AlC 3 byapprox.0.15eV(comparetherelativepositionsofthe maroon (x = 0) and green (x = 1) lines for T = 0K FIG. 2: (Color online) Microstructure of a Fe-29.8Mn-7.7Al- (dotted) and T = 600 ◦C (solid) in Fig. 3). We have 1.3C (wt.%) alloy aged at 600◦C for 12 weeks: (a) SE image therefore also investigated the impact of (Fe-Mn) config- showing the grain boundary (GB) (κ +γ +α) phases and 0 0 uration in the Fe sub-lattice on the DFT supercell en- graininterior(GI)(κ+γ)phases. (b)zoomed-inSEimageat ergy ESC [Fe MnAlC] in Eq. (1). For this purpose the GI region highlighting the nano-sized GI κ-precipitates. (c) κ 2 APT analysis of GI (κ+γ) phases where κ-precipitates are results of a regular Mn arrangement (periodic repetition visualizedbyCiso-concentrationsurfaceatathresholdvalue oftheunitcell)andanSQSdisorderedstructurearecom- of 9 at.%. pared in Fig. 3 (thick and thin solid line for x = 1) and showanegligibledifference. Acomparisonoverthewhole volume range relevant for subsequent considerations is performed in Fig. 4, where also the impact of magnetic TheequilibriumconcentrationofMnintheκcarbideis disorder is taken into account. The differences of the or- determined via Eq. (1), setting y =0 and z =1. Chang- der of max. 25 meV/unitcell can be translated into an ingthechemicalpotentialchangestheamountofMnand order-disorder transition temperature T . The latter is the thermodynamically most stable carbide phase. The OD a resultof thecompetition between formation enthalpies correspondingphasediagramasfunctionofµ isshown Mn (at T = 0K) and configurational entropy given by the in Fig. 3. The obtained dependence allows us not only expression to connect to our experimental alloy composition (red dash-dotted line; see appendix A for details), but also ESC[SQS]−ESC[ordered]=T Sκ(1,0,1), (7) κ κ OD to investigate chemical and thermodynamic trends. On theonehand,weconstructedtheT=0Kphasediagram with the entropy Sκ defined in Eq. (2). Since the (dottedlinesinFig.3)toseethechemicaleffectonphase stoichiometry in an order/disorder transition remains stabilities. On the other hand, we generalized it to the unchanged, any contributions from chemical potentials annealing temperature of 600 ◦C (solid lines in Fig. 3), (compare Eq. (1)) cancel. Using this equation for a Mn 6 FIG. 3: (Color online) Free energy differences for the κ car- FIG.4: (Coloronline)Freeenergydifferenceofchemicallyor- bide formation according to Eq. (1) with varying Mn con- deredanddisorderedFe MnAlCfordifferentmagneticphases 2 tent x (given by the labels). The results for T = 0 K (dot- is shown as a function of volume at T = 0 K. The energies ted lines) and for the experimental annealing temperature of have been rescaled such that the ground state configuration 600◦C(solidlines)arecompared. Thecolorshadingindicates of Fe MnAlC is taken as a reference. 2 the phase stability at 600◦C as a function of the Mn chemi- calpotentialwithrespecttothereferencepotentialdescribed in appendix A.The chemical potentialscorrespondingto the orderedstructure(AFMD,yieldsavanishingnetmagne- compositionoftheexperimentalalloy(Fe-29.8Mn-7.7Al-1.3C tization) and a completely non-magnetic (NM, unrealis- in wt.%) are for both temperatures shown as a red dash- ticscenarioofvanishinglocalatomicmagneticmoments) dotted lines. For Fe MnAlC the ground state energy of an 2 configuration for comparison. ordered Mn arrangement (thick green line) and of an SQS The results for T = 0 K (Fig. 4) show that the FM disordered structure (thin green line) are compared. phase in chemically ordered κ carbide is energetically most favourable and therefore indeed the correct choice for ground state ab initio calculations. However, some concentration of x=1, one obtains a T of approx. 75 OD of the disordered structures are energetically very close K. Therefore, any chemical ordering will be lost at room to the ground state. In particular, the difference of the temperature, which is in agreement with observations in PM to the FM state is approx. 75 meV/unitcell, which experiment,buthasnotbeenconsideredinpreviousthe- is smaller than that of the AFMD and NM states. This oretical studies24,25. indicates,ontheonehand,alowCurietemperature,T . C To complete the considerations on chemical order and Using Eq. (4), the Curie temperature T for a transi- C to emphasize the crucial role played by the chemical or- tion from chemically disordered FM to the PM phase is dering in the Al sub-lattice for the formation of κ car- approx. 60 K. Even a combined magnetic and chemical bides, we discuss the free energies with chemical dis- disorderingofanoriginallyFMorderedstatewouldonly order in both Fe-Mn and Al sub-lattices (Fig. 4). We require 90 K. This number is only an estimate, because find that the additional chemical disorder in the Al sub- Eq. (3) is based on a mean-field approximation and does lattice makes the formation of κ carbides substantially notdistinguishbetweenFeandMnatoms. Nevertheless, less favourable. Using Eq. (7), the corresponding order- our study supports those experiments44 that do not ob- disorder transition temperature is ≈ 1400 K. Further serve any macroscopic magnetic order in κ carbides at calculations showed that these findings are qualitatively room temperature. On the other hand, we observe very similar for other compositions of κ carbide. little difference between structural properties (e.g. equi- In the following we extend the concept of disorder librium lattice constant) of a FM and a PM material in also to the magnetic degrees of freedom. Experimen- contrast to, e.g., a NM calculation (see also Sec. IIIC). tally, the relevance of magnetic disorder for this carbide This justifies the application of the FM approach, if a isinconclusive. Afewexperimentalworksindicateκcar- PM calculation is not feasible. bides to be ferromagnetic43, in agreement with theoreti- We can now investigate the stability of κ carbides as calcounterparts24. Ontheotherhand,someexperiments a function of its composition (as given by Eq. (1)) to ex- suggestκcarbidenottobemagnetic44. Inourtheoretical plain the experimentally observed C off-stoichiometry in approach,wecomparetheab initiofreeenergies,accord- κ carbides. Due to the computational effort, no thermo- ingtoEq.(1), correspondingtoordered(FM)anddisor- dynamic excitations such as lattice vibrations and mag- dered (PM) spin configurations. In order to evaluate the netic entropy are taken into account. Their impact on, energy difference, we further add another magnetically e.g.,vacancyformationenergiesistypicallysmallatroom 7 FIG. 5: (Color online) Area modulus48,49 of (a) a cubic-symmetry approximant of Fe MnAlC with disorder Fe-Mn sublattice 2 inFMstate,(b)Fe MnAlC ,i.e. withreducedCcontentinFMstate,(c)Fe AlC,i.e. withoutMninFMstate,(d)Fe AlC 2 0.625 3 3 in PM state. The calculation (values in GPa) is based on the determined elastic constants C , C and C summarized in 11 12 44 Table I (visualization by the SC-EMA software package50–52). temperature45. Due to the low order-disorder transition (001)/(001), this misfit is obtained from our DFT calcu- temperature,nochemicalsuperstructure/orderingonthe lations to be in the stoichiometric case as high as 9%. Fe-Mn sublattice can be expected. In principle, also the This value is too large to allow coherent interfaces with- magnetic disorder should be taken into account. Due to out misfit dislocations. the fluctuating moments in this phase, however, the nec- Synchrotron diffraction experiments (not shown here) essary relaxations, e.g., for a vacancy calculation would have indicated a reduction in the lattice misfit to 1.4% require sophisticated approaches such as the spin-space between off-stoichiometric κ carbide and γ matrix in the averaging (SSA) method46. This goes beyond what is grain interior. In order to enforce a completely coher- currently feasible for a complex alloy like the κ carbides. entinterfacewithoutmisfitdislocations,asitisobserved Having in addition the limited impact of magnetism on for GI carbides (at least for the small channels of γ structural properties in mind (Fig. 4), we restrict most material6), a compromise of the lattice constant of both of our calculations to the magnetic ground state (FM). phases is required. It will depend on the volume fraction of the phases and the elastic energy associated with a compression or elongation. C. Elastic properties Togetadeeperunderstandingoftheelasticproperties of κ carbides, we determined its elastic tensor. Accord- The κ carbides have so far been considered as an in- ing to the investigations of the previous section, we first dividual bulk phase. However, the experimental find- use the composition Fe MnAlC with chemical disorder 2 ings provided above clearly indicate that the C off- and ferromagnetic order for this purpose. The results stoichiometry is strongly related to the microstructure. aresummarizedinTableI.Acomparisonoftheseelastic The main difference between GI and GB precipitates is constants with those of a cubic elastic approximant38,47 the coherency to the matrix material. We argue that based on the values reported in Ref. 24 for an ordered, next to configurational entropy also the strain caused by ferromagnetic unitcell shows that the chemical disor- the degree of coherency drives the C out of the carbide. der has only limited impact on elastic properties of the The coherency is related to the lattice parameter mis- studied κ carbide. The directional dependence of the match between κ carbides and the γ matrix material. corresponding single-crystalline Youngs modulus yields For the experimentally observed orientation relationship a significant anisotropy of the ferromagnetic κ carbide. 8 The hard (cid:104)001(cid:105) direction has an almost twice as large TABLE I: Single-crystalline elastic constants (C , C , C , Youngs modulus (394 GPa) as the soft (cid:104)111(cid:105) direction 11 12 44 B) calculated for different chemical compositions and mag- (215 GPa). For our considerations, however, the area netic states of κ carbide. The selections are identical with modulus of elasticity48,49, which provides the amount of those shown in Fig. 5. In the cases (a) and (b), a disor- energyneededforcoherentplanarloadingwithinaplane dered configuration of Fe and Mn is considered. For com- normaltothevectorn,ismorerelevant. Thedirectional parison, elastic constants of a cubic elastic approximant38,47 dependence of these normal vectors, n, is visualized in basedonresultsobtainedfororderedFe MnAlCfromRef.24 2 Fig. 5a, which still shows an anisotropy. The Youngs are shown. All values are in GPa. modulus is highest for the {001} planes, i.e. the cor- responding energy required for epitaxial loadings within Composition Magn. C11 C12 C44 B theplanesthatarerelevantfortheκ/γcoherencyishigh- (a) (Fe2,Mn)AlC FM 418 77 82 191 est. Thisobservationtogetherwiththelargemisfitof9% (b) (Fe2,Mn)AlC5/8 FM 282 167 94 205 (c) Fe AlC FM 446 109 72 221 makes the stabilization of an (001)/(001) very unlikely, 3 (d) Fe AlC PM 439 90 96 206 in puzzling disagreement with experiment. 3 Ref. 24: Fe MnAlC FM 436 80 92 199 2 A reduction of the C content is expected to yield a smaller misfit. The question is, however, how it influ- encestheelasticproperties. Thechallengeofcorrespond- ing calculations of the elastic tensor is to ensure a cubic crystal structure of the 2×2×2 supercell. A tetragonal distortion would not only increase the numerical effort significantly,itisalsoinconflictwiththephysicalexpec- tationforaninfinitelylargesystem. Theonlyreasonable choice that fulfills this constraint is the presence of three C vacancies. The resulting area modulus of elasticity is shown in Fig. 5b. It reveals that some of the elastic constants are softer, as expected from the high vacancy concentration, whilethebulkmodulusishardlychanged (Tab. I). More important is the observation that (cid:104)001(cid:105) has now turned into the elastically soft direction, there- with resolving the before mentioned puzzle. Due to the central importance of the elastic proper- tiesfortheupcominginvestigations,wealsoinvestigated the impact of the assumptions formulated at the end of Sec. IIIB. Figure 5c allows a comparison of the area FIG.6: (Coloronline)Volumedependenceoftheenergycon- modulus for Fe MnAlC with the Mn-free version, while 2 tributions to the formation energies of C, Fe and Al vacan- Fig.5dshowstheresultsofafullyparamagneticcalcula- cies in FM κ carbide according to Eq. (8). The energies are tion. Inbothcasesaclosesimilaritytotheresultsforthe rescaled such that the perfect Fe AlC (filled symbols) at its 3 FM Mn-containing version shown in Fig. 5a is obtained. equilibrium lattice constant is taken as a reference. The ver- For the area modulus as well as the bulk modulus the ticaldashedlinemarkscompromisingvolumebetweenκcar- maximum changes are of the order of 10 %. This jus- bide and γ matrix, if both have a volume fraction of 50% tifies our choice for the chemical and magnetic degrees (compare with Fig. 8). of freedom. In addition, we note that the area modulus does not show a strong anisotropy, if the C content is re- for species X, according to the expression duced (Fig. 5b). We therefore consider in the upcoming calculations the bulk modulus instead of the area modu- Ef (V)=ESC (V)−ESC (V)+µ (T) (8) lus. X-Vac X-Vac Fe3AlC X analogous to Eq. (5) where µ (T) is obtained at 600◦C X asdefinedinappendixA.Thevolumedependenceofthe supercell energies entering Eq. (8) are shown in Fig. 6, D. Vacancy formation energy where the energies of the perfect carbide (filled symbols) shouldbecomparedwiththeenergiesofthedefectstruc- Due to the coherency strain, we expect a driving force tures (open symbols). Since these defect energies are for forCtoleavetheκcarbideanddissolveinthematrix. A FeandAlvacanciesformostvolumessubstantiallyhigher Cdepletion(asdescribeinSec.IIIE)isexpectediftheC thanthoseofFe AlC,theirvacancyformationisunlikely. 3 vacancy formation is exothermic, or if the energy loss is ThesituationisdifferentforthecaseofC.Attheequilib- small enough to be compensated by a gain in configura- rium volume of κ carbide, for example, the red symbols tional entropy at finite temperature. We have therefore are below the green symbols, i.e. the C vacancy forma- investigatedthecorrespondingvacancyformationenergy tionenergyisnegative(-290meV).Hence,theremovalof 9 asingleCatomfromanotherwiseperfectκcarbideisan exothermicprocess. Theoriginofthenegativeformation energy lies largely in the large configurational entropy in the γ matrix, where the C concentration is low. The consequences of this driving force will be discussed in the next subsection. Second, there is a remarkable volume dependence of the C vacancy formation energy, yielding a substantial reduction to even more negative values under volumet- ric compression. This reduction is a consequence of the largenegativevacancyformationvolumeofapprox. 7A˚3, which allows the system to efficiently release strain en- ergy by creating C vacancies. As a consequence, the for- mation of C vacancies is more feasible in κ carbides that areformedascoherentprecipitatesintheFematrixthan in incoherent particles as formed near grain boundaries. FIG. 7: (Color online) Schematic picture of C partitioning As discussed at the end of Sec. IIIB, the calculations betweenκcarbideandγ matrix: AssuminganequalCdistri- bution in the as-cast state (dashed line), there is a chemical are performed for Mn-free κ carbide. This is mainly drivingforceforCtoaccumulateinAl-orderedregions. Since due to the fact that we would otherwise need to treat thisimposesanelasticenergypenalty,thedecompositionwill the Fe-Mn sublattice as a disordered alloy, which results remain incomplete. in a huge increase in the number of configurations to be considered for the calculation of (multiple) vacancies. WhileweshowedinSec.IIICthattheeffectontheelas- formκcarbide(whichisanexothermicprocess). TheAl tic energy is small, we have also tested the impact for ordering,whichisusedinthisworktodefinetheregionof the chemical part of the vacancy formation. We realize κcarbide, isvolumeconserving, whiletheCpartitioning that the difference in formation energies of a C vacancy isnot. Sincethecoherencyconditionpreventsanyrelease in Mn free (Fe AlC) and Mn containing (Fe MnAlC) κ 3 2 of elastic energy by plastic relaxation via misfit disloca- carbides can be up to 0.24 eV. We have further consid- tions at the interface, partitioning unavoidably increases ered the impact of magnetism on the vacancy formation the misfit and thus the elastic energy. This mechanism energies, by performing a fully paramagnetic calculation prevents a complete filling of the Al-ordered region (i.e. for a single chemical configuration. These calculations the κ carbide) with C. are extremely challenging and prone to errors, but the These considerations show that the required energy obtained deviations from the FM calculation are in the minimization also needs to take the chemical and elas- same order of magnitude as the chemical difference. It tic energy of the γ matrix into account. In principle, we is therefore clear that the upcoming calculations cannot shoulddeterminethevolumedependentCsolubilityina aim at a quantitative reproduction of the experimental disorderedFe-Al-Mnmatrix. AsmentionedinSec.IIIB, results,sincethenumericalefforttoachievethisaccuracy Mn has been removed from the considerations, but even would be enormous. However, the general mechanisms the treatment of Al disorder in the γ matrix would re- for the C partitioning discussed in the following are not sult into a large configuration space. Two limiting cases affected by these approximations. canbeconsideredinstead: (i)theγ matrixconsistsofFe and C only, or(ii) the γ matrix is itself an ordered Fe-Al phase. For reasons that will be discussed below, we only E. Partitioning between κ and γ use the first scenario. The optimization of the composite consisting of κ pre- Given that the combination of configurational entropy cipitate and γ matrix does not only affect the C concen- and the coherency constraint results in negative vacancy trations, but also the coherent lattice parameters. We formationenergies,itisclearthatthecommonlyapplied expressthelatterbyVκ+γ,thecoherentvolumeperunit- concepts of dilute point defects cannot be used for the cell, which is an intensive thermodynamic variable. It present study. Rather, since the concentration changes capturesboththehydrostaticchangeoflatticeconstants are well above a few percent, thermodynamic concepts of cubic nano-precipitates and the volume change in a developed for alloy decomposition become appropriate. tetragonal distortion, if the coherency is only assumed In this sense we discuss the problem as an incomplete C for the in-plane lattice constant (biaxial strain) and the partitioning between κ carbide and γ matrix as shown normal component is relaxed. Therefore, the Helmholtz in Fig. 7, i.e. we have in upcoming considerations the freeenergyofthecompositeofκprecipitateandγmatrix following physical picture in mind: After casting, C and is given by Al are homogeneously distributed in the sample. During annealing the onset of Al ordering occurs along with a Ftot(T,V,cκ,cγ) = vκFκ(T,V,cκ) chemical driving force for C to enter these regions and + v Fγ(T,V,c ), (9) γ γ 10 where v and v are volume fractions of κ and γ respec- determines a concentration c and a corresponding con- κ γ κ tively and c and c are their corresponding C concen- centration c as given by Eq. (10). Since c cannot be κ γ γ γ trations. It can be split into an elastic, a chemical and a represented by a 2 × 2 × 2 supercell and since Veg- a configurational part. In the case of the κ carbide (ex- ardslawisfulfilled,alinearinterpolationisemployedfor pressions for γ matrix are similar) the definition of the eachV inordertodetermineFγ(T,V,c ). Subsequently, γ first two terms is given by theequilibriumcoherentvolumeV isobtainedbythe κ+γ minimizationofthetotalfreeenergy(Eq.(9))oftheκ-γ Eelas(V,cκ) = Eκ(V,cκ)−Eκ(Vκ(cκ),cκ) composite with respect to the volume V which is com- Echem(Vκ(cκ),cκ) = Eκ(Vκ(cκ),cκ) mon for both the phases. The results for Vκ+γ are again linearly interpolated. − Eκ(V (c ),c ). κ exp exp Theprocedureisrepeatedforvariousvolumefractions The chemical part covers the change of the concentra- of the phases which enter Eq. (10). Fig. 8 shows the tion at equilibrium cubic volume as obtained from the resulting Vκ+γ together with the unstrained equilibrium Murnaghan equation of state53,54. The elastic part cov- volumes of the individual phases. The results of Vγ indi- ersthevolumedeformation(hydrostaticorbiaxial). The catesonceagainthattheCconcentrationintheγ matrix reference is the homogeneous C distribution (see Fig. 7) dependsonthevolumefractionvκ foragivenconcentra- with a concentration determined by experiment cexp at tion cκ due to Eq. (10). The lower the C concentration its respective equilibrium volume V (c ). inκcarbide,themoresimilarthelatticeconstantsofthe κ exp The C concentrations are not independent, but are unstrained phases get. coupledduetothefactthatthetotalnumberofCatoms Assuming coherency of the carbide in all three dimen- during the partitioning must be conserved: sions, the common volume per unitcell of the composite V will be closer to that of the κ phase than of the κ+γ c v +c v =c . (10) γ phase, because the former is stiffer and has the larger κ κ γ γ exp bulkmodulus. Nevertheless,theκcarbideshowsasignif- The C concentrations c and c are both defined with icant adaptation of its lattice constant, too. The impact κ γ respect to the octahedral sublattice that corresponds to ofthevolumefractiononV issmallandcanbesafely κ+γ the body-centered positions in Fig. 7, i.e. one per four neglected. A larger volume fraction v (larger impact γ metal atoms. If the C concentration in this sublattice is on V ) is compensated by a higher C concentration κ+γ 100 at.% (complete filling of this sublattice), then the C (i.e. increasingV )oftheγ matrix. Further,V shows γ κ+γ concentration per unitcell would be 20 at.% (the other hardly any concentration dependence, since the effects 80 at.% are metal atoms). Since the experimentally de- of V and V cancel each other. Therefore, the value of κ γ termined C concentration per unitcell (averaged over κ V ≈ 49˚A3/unitcell can be safely used as a universal κ+γ and γ) is only 9 at.%, the sublattice C concentration is parameter of the system. cexp =9/20=45 at.%. After the volume optimization, we now perform the Thepossibilitytooccupyonlyonesublatticelimitsthe energy minimization with respect to C concentration. In numberofconfigurationsandhasthusastrongimpacton principle,therearetwoprocedurespossibleandbothare the configurational entropy. This is taken into account compared for a volume fraction v = 0.5 and the an- κ by including the number of available sublattices (sγ=4 nealing temperature of T = 600 ◦C. First, one can in- and sκ=1). Therefore, the overall expression for the free troduce temperature dependent chemical potentials for energyoftheindividualphasesσ (=κorγ)inEq.(9)is C (see Eq. (A.11) of appendix B), which we now treat as formally independent in both phases and which have Fσ(T,V,c )=E (V,c )+E (V (c ),c ) σ elas σ chem σ σ σ both been plotted in Fig. 9d with a common x axis (c κ (cid:20)c c (cid:18) c (cid:19) (cid:18) c (cid:19)(cid:21) and c are coupled by Eq. (10)). The thermodynamic +k Ts σ ln σ + 1− σ ln 1− σ . (11) γ B σ s s s s equilibrium is then determined by the intersection point σ σ σ σ of these two lines. We note that exactly the same result The particle conservation (10) enables us to express is obtained, if the particle conservation (10) is used and c in terms of c . Under these circumstances, Eq. (9) the Helmholtz free energy is directly minimized. γ κ simplifies, i.e., Ftot(T,V,c ,c ) = Ftot(T,V,c ) imply- The dependence of the free energies on c is shown in κ γ κ κ ing that we have to perform the minimization only over Figs. 9b and 9c. Similarly to Fig. 8, DFT data points a single concentration c . Before doing so, we consider can only be provided for the individual phases, while an κ the minimization with respect to the volume V in order interpolation (polynomial fit) is used for the composite. to obtain V for different values of concentration c . Thefreeenergy Ftot(T,V,c )inFig.9c(solidline)illus- κ+γ κ κ Forthispurpose, theMurnaghanequationofstateisap- trates that starting from the homogeneous distribution, plied to the energy-volume curve for an integer number the partitioning of C atoms yields first a gain in energy ofCatomsin2×2×2supercells. Thisprocedureisper- before it increases again when too many C atoms are formedforbothphasesseparately. IfseveralCconfigura- transferred into the carbide. The minimum energy is tions are possible, an averaging of the energies has been achieved at an equilibrium sublattice C concentration in performed. EachCatomremovedfromthesupercellofκ the κ carbide of approx. 55 at.%.

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