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Site occupancies in sigma-phase Fe–Cr–X (X = Co, Ni) alloys: Calculations versus experiment
Computational Materials Science 122 (2016) 229–239
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Site occupancies in sigma-phase Fe–Cr–X (X = Co, Ni) alloys: Calculations versus experiment J. Cies´lak ⇑, J. Tobola, S.M. Dubiel AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, PL-30-059 Krakow, Poland
a r t i c l e
i n f o
Article history: Received 4 February 2016 Received in revised form 9 May 2016 Accepted 11 May 2016
Keywords: Sigma phase FeCrNi alloys FeCrCo alloys Electronic structure calculations Neutron diffraction measurements
a b s t r a c t Ternary sigma-phase Fe–Cr–X (X = Co, Ni) alloys were studied theoretically (electronic structure calculations, Gibbs free energy analysis) and experimentally (X-ray diffraction, neutron diffraction, Mössbauer spectroscopy) in order to determine sublattice site occupancies by alloying elements. In general, good agreement between the predictions and experimental data was achieved. The obtained results agree reasonably well with expectations i.e. both Co and Ni atoms substitute for Fe atoms which predominantly occupy the sites A and D. Ó 2016 Elsevier B.V. All rights reserved.
1. Introduction The sigma (r) phase is one of many Frank–Kasper (FK) phases known also as topologically close-packed (TCP) phases that occur in alloys [1]. Originally it was discovered in a ternary Fe–Cr–Ni alloy [2] and its crystal structure was definitely identified in the Fe–Cr system [3]. Most of the FK-phases are detrimental to the high-temperature performance of technologically important structural materials, hence its presence in these materials is highly undesired (see e.g. Ref. [4]). The knowledge and understanding of mechanism(s) responsible for the formation of these phases is crucial in designing FK-free structural materials. In the course of time, several methods toward this end have been developed. They can be divided into three groups: (1) phenomenological, (2) computeradded and (3) theoretical (ab initio calculations). Concerning (1), the first attempt was based on the concept of the valence electron concentration (VEC) [4–6]. However, the VEC model could not explain satisfactorily all cases in which the FK phases were formed. The VEC scheme, although it can neither explain all the cases in which the FK-phases occur nor those for which the criterion was fulfilled but no FK-phases were formed, is still in use. Recently, it was successfully applied to study stability of the r-phase in high entropy alloys [7]. Laves and Wallbaum [8] and Duwez and Bean
⇑ Corresponding author. E-mail address: [email protected] (J. Cies´lak). http://dx.doi.org/10.1016/j.commatsci.2016.05.008 0927-0256/Ó 2016 Elsevier B.V. All rights reserved.
[9] introduced the difference in atomic size of elements constituting an alloy as a dominant factor responsible for the formation and stability of the FK-phases. Two-dimensional model, in which both the concentration of the holes as well as the atomic size are considered as important factors was proposed by Watson and Bennett for two-component alloys [10], and it was recently extended for multicomponent systems [11]. The computer-added methods include (a) PHACOMP, based on a concept of the average number of holes, hNhi, was originally introduced to predict formation of r in commercial Ni–Co-based super alloys [12], (b) newPHACOMP, in which instead of hNhi the average energy of the d-orbital was used as the dominant factor responsible for the r-phase formation [13], and (c) CALPHAD, based on thermodynamic modeling of a phase combined with equilibrium calculations [14]. There are numerous examples of its applications related to the FK-phases e.g. [15]. Theoretical ab initio calculations were carried out to give information on various issues related to the FK-phases like site occupancies e.g. [16,17], phase diagrams and structural trends e.g. [18,19], electronic and magnetic properties e.g. [20,21]. Whatever the method, its relevance can be ultimately verified by comparison of its predictions with experimental data. Studies of disordered FK phases have been carried out for many binary systems, both theoretically and experimentally. Ternary systems are a major challenge due to a much larger number of degrees of freedom available in filling sublattices with three elements. This variety must be taken into account in the calculation, as well as in the analysis of experimental results. Some ab initio
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2. Experimental 2.1. Samples and their characterization
Fig. 1. r-Fe–Cr–Ni samples investigated in the present study. Full symbols denote samples for which full transformation to the r-phase occurred. Dotted line stands for the r-Fe–Cr border.
calculations on stabilities of various configurations of sigma phase in subsystems of Fe–Ni–Cr i.e. in Fe–Ni and Ni–Cr were already published [22]. Other research (experimental and theoretical) was carried out for the systems r-CrMoRe [23] and r-MoNiRe [24]. The choice of elements forming these systems allow them to be easily differentiated by the X-ray (XRD) and neutron diffraction (ND) methods which, in combination with studying the formation energy, enabled determination of the sublattices occupancy yielding a good agreement between the theory and the experiment. In this paper such comparison is made as far as lattice site occupancies by Fe, Cr, Co (Ni) atoms in ternary sigma-phase systems of Fe–Cr–Co and Fe–Cr–Ni are concerned. In this case, the elements are poorly distinguishable using the XRD technique (due to similar values of the atomic number, Z). Consequently, the basic research technique used in this study was ND. On one hand, the sublattice occupancies were calculated using the ab initio calculations, and on the other hand, they were determined experimentally using neutron diffraction measurements.
Ternary sigma-phase alloy systems Fe–Cr–X (X = Co, Ni) were the subject of the present study. Concerning samples containing Co, they had the formula Fe52.51.4xCr47.5+0.4xCox, with x = 5, 10, 15, 20, 25, 30, 35. It should be added that Co atoms substitute for Fe ones without any limit [25 and references therein]. In the case of Fe–Cr–Ni alloy system, the r-phase can be formed with Ni concentration limited to 10 at.% [26 and references therein]. In the latter case 14 different compositions – see Fig. 1 – were synthesized for the present study. The master alloys of both series were obtained by melting appropriate amounts of constituting elements in an arc furnace. The melting process was carried out in a protective atmosphere of argon and it was repeated few times to ensure a better homogeneity. The master alloys of the Fe–Cr–Co series were transformed into the sigma-phase by isothermal annealing in a quartz tube, kept under a dynamical vacuum, in two steps: first at 1273 K for 1 day followed by the second one at 973 K for 7 days. The latter was terminated by quenching the samples onto a block of brass kept in a cold (300 K) end of the tube. The transformation of the Fe–Cr–Ni alloys into the sigma-phase turned out to be more difficult. Since exact concentration ranges in which the r-phase exists in this system are not precisely known, the prepared alloys covered a wider composition range. After homogenization (one day at 1273 K) the samples were annealed at 973 K for 7 days. For all not fully transformed samples the procedure was repeated, but this time the annealing temperature was 1073 K. Finally, 6 samples were found to be totally transformed to the r-phase, see Fig. 1 and Table 1. For both series the verification of the alpha-to-sigma phase transformation was done by recording room temperature XRD patterns as well as Mössbauer spectra. The XRD patterns, examples of which are shown in Fig. 2, also permitted determination of lattice parameters, a and c. Chosen samples of both series were investigated also with the ND techniques. Examples of the ND patterns recorded at room temperature at ILL Grenoble (D2B) can be seen in Fig. 3. Their
Table 1 Alloy compositions and lattice parameters for the Fe–Cr–Ni and Fe–Cr–Co r-phases. Fe
Cr
Ni
a (Å)
c (Å)
Experimental 0.460 0.458 0.507 0.407 0.525 0.475
0.501 0.490 0.467 0.489 0.455 0.505
0.039 0.052 0.026 0.104 0.020 0.020
8.797(2) 8.795(2) 8.791(2) 8.794(2) 8.790(2) 8.798(2)
4.559(1) 4.559(1) 4.560(1) 4.559(1) 4.560(1) 4.558(1)
Used for calculations 11–16 atoms (36.7–53.3 at.%)
13–16 atoms (43.3–53.3 at.%)
1–3 atoms (3.3–10 at.%)
488 different atomic arrangements were calculated
Fe
Cr
Co
a (Å)
c (Å)
Experimental 0.455 0.385 0.315 0.245 0.175 0.105 0.035
0.495 0.515 0.535 0.555 0.575 0.595 0.615
0.050 0.100 0.150 0.200 0.250 0.300 0.350
8.792(2) 8.792(2) 8.791(2) 8.788(2) 8.787(2) 8.784(2) 8.783(2)
4.556(1) 4.553(1) 4.550(1) 4.547(1) 4.548(1) 4.541(1) 4.539(1)
Used for calculations 1–13 atoms (3.3–43.3 at.%)
14–19 atoms (46.7–63.3 at.%)
2–10 atoms (6.7–33.3 at.%)
672 different atomic arrangements were calculated
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Fig. 2. XRD patterns recorded at room temperature on the r-Fe38.5Cr51.5Co10 (top) and the r-Fe47.5Cr50.5Ni2 (bottom) samples.
Fig. 3. ND patterns recorded at room temperature on the r-Fe38.5Cr51.5Co10 (top) and the r-Fe47.5Cr50.5Ni2 (bottom) samples.
analysis yielded probabilities of finding Fe, Cr and Co(Ni) atoms at particular lattice sites which was the main aim of the present study. In practice, the diffractograms were analyzed by means of the Rietveld method (FULLPROF program) [27]. In the calculations, the alloying element concentrations were held fixed to the nominal values, while the concentrations related to the particular sublattices were treated as free parameters. The occupancy of the elements on the fifth lattice site was constrained by the occupancies on other four sites. In total, 4 parameters were used to refine the occupancy, 9 parameters for atomic positions and lattice
parameters, the background intensity was refined with a polynomial (6 parameters) and the peak shape was approximated by the pseudo-Voigt function. 3. Theoretical As it was already mentioned, binary r-phases are chemically disordered systems which means that the translational symmetry of the lattice is maintained, but the sublattices are more or less randomly occupied by two elements, usually with different
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Fig. 4. Lattice parameters a and c as determined from the XRD patterns for the r-Fe–Cr–Co alloys. The lines represent the best linear fits to the data. Error bars are indicated, too.
probabilities. Similar type of disorder should be expected for more complicated ternary r-systems (e.g. mixing three elements on one site). Their theoretical analysis, and especially determination of the preferences of sublattice occupation, should be based on the analysis of the Gibbs energy variation in different stoichiometries of sigma-phase, i.e. dG = dE TdS + pdV. Since dV can be expected to be negligible, the aforementioned analysis can be based on the analysis of the free energy variation dF = dE TdS. It contains (besides the total energy, E) also the entropy term, S, scaled by the temperature of the system. In general, the total entropy consists of four contributions, namely configuration, Sconf, magnetic, Smag, phonon, Sphon, and electronic Sel, which should be considered to investigate relative crystal stability. The total energy of the system, Etot, can be understood as the work to be done to build the system at 0 K from single atoms of constituting elements, atom by atom. In practice, the value of Etot can be obtained e.g. from the DFT-based ground state selfconsistent electronic structure calculations. On that basis one can define the formation energy, Eform, as the difference between the total energy, Etot, of the investigated phase and the sum of total energies of its constituents (as bulk):
Eform ðAm Bn Þ ¼ Etot ðAm Bn Þ m Etot ðAÞ n Etot ðBÞ
ð1Þ
Fig. 5. Lattice parameters a (top) and c (bottom) in Å as determined from the XRD patterns for the r-Fe–Cr–Ni alloys. Those lying on the straight line correspond to binary r-FeCr alloys.
Smag ¼ kB
3 X 5 X xij lnðlij þ 1Þ
ð3Þ
i¼1 j¼1
In principle, all Etot values in Eq. (1) should be determined for elements A and B, assuming the same crystal structure as that of the investigated AmBn phase. Since in the case of the r-phase its one-component analogs does not crystallize in reality, an alternative way is to determine the total energy of single elements for real structures occurring at 0 K. The latter definition of Eform can be understood as the work to be done to change the pure elemental crystalline structures (but not single atoms) to build the system under consideration (also at 0 K). The configuration entropy, Sconf, is related to the degree of disorder of a system and for the three component r-phase can be expressed by the known formula (2), where index i denotes the constituent element (i = Fe, Cr or Co/Ni), index j denotes sublattice (A, B, C, D and E in the case of r-phase), and x stands for the site occupancy:
Sconf ¼ kB
3 X 5 X xij lnðxij Þ
ð2Þ
i¼1 j¼1
As can be seen from Eq. (2), Sconf can be calculated for any configuration of the sublattice occupancies. The magnetic entropy, Smag, can be described using Eq. (3), where lij denotes a magnetic moment:
In order to calculate this contribution, magnetic moments of atoms in various atomic surroundings should be known. However, for the purpose of this study it was assumed that the magnetic entropy in the unit cell is determined by the occupancy of the sites, xij, so it should be a continuous function of the number of atoms of different types on all five sites. Although the analytical form of such dependence is not known, one can approximate this function using a polynomial of n-th degree, W(xij), the coefficients of which can be determined by its fitting to the Smag values, calculated for selected unit cells with known atomic arrangements. Unfortunately, since one should take into account the occupancy of each site separately, the number of the coefficients to be determined rapidly increases with the degree of the polynomial, n, and, in practice, such analysis seems to be reliable only for n < 3. However, applying this model to the studied ternary r systems one can find the differences between the values determined using electronic structure calculations. Noteworthy, the values of the such determined polynomials are 0.05kB/atom, while the values found for the different arrangements of the atoms differ from each other by 0.3kB/atom, on average. This means that determination of the magnetic entropy in this case may be vitiated by a significant error.
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Fig. 6. Histograms of the difference Ecalc–Efit for the studied systems: (a) r-Fe–Cr–Ni and (b) r-Fe–Cr–Co. The solid lines stand for the Gaussian approximations.
Detailed analysis of Sphon-term contribution would require more experimental and theoretical data on the lattice dynamics in the r phase to figure out its role. However, Sphon-values determined experimentally and from the ab initio modeling [28] for the r-FeCr (and the corresponding bcc-alpha) phases are quite similar, so their difference presumably neither affects the alpha-sigma transformation process nor physical properties of the r-FeCr phase. Moreover, it is expected to be less influential than other forms of entropy, since Sphon determined for the r-FeCr is 0.1kB/atom whereas the value of Sconf 0.5kB/atom. The electronic entropy, Sel, at low temperature is a linear function of T (with the coefficient gamma related to density of states near the Fermi level) due to the linear dependence of specific heat [29]. In general, Sel values can be only slightly different for different atomic arrangements, which was also confirmed by our computations. Therefore, comparing different r-phases having the same alloy stoichiometry, one can neglect possible Sel differences in the analysis of the site occupancy, also when extrapolating to high temperature range i.e. where r-phases precipitate. Alike the magnetic entropy, also the total energy is a function of the site occupancies. In a simplified approach, Etot can be determined for the unit cells characterized by the absence of disorder on sublattices. Such assumption significantly simplifies the calculations that should be carried out as only the configurations of sites that are fully occupied by single type of atoms need to be taken into account [30,31]. For a ternary r-phase there are 35 = 243 different configurations to analyze. On the basis of such calculated values of the total energy one can construct diagrams of the formation energy versus concentrations of each element in the unit cell. The envelope determining the bottom of the Eform points on these diagrams (so-called convex-hulls) indicates energetically the most favorable configurations in the range of the considered concentrations [24]. Real crystalline systems may be, however, more precisely described when the disorder on the sublattices is taken into account. This can be done, for example, by calculating the total energy for one selected atomic arrangement and setting it as the reference system. Next, similar calculations should be performed for other atomic arrangements, differing from the reference one in a pre-specified way (e.g. by variation of the occupancy on one site only). Finally, such computations have to be performed for a number of modified arrangements that are sufficient to determine the shape of Etot function versus the occupancy’s variation on all sublattices. In the next step one can approximate this shape using an analytical function. The second-order polynomial in the five dimensional space (due to A–E sites) proved to be very effective. The values of the total energy per one atom in the r-phases are of the order of tens keV/atom, while the differences in energies for different arrangements of atoms in the unit cell are about
1 eV/atom. However, an adjustment of this relationship by the mentioned polynomial allowed to minimize the differences between the polynomial and the calculated values to few meV/atom. Actually, such calculations carried out for a number of selected arrangements are time-consuming but they result in the analytical approximation of the formation energy which can be easily used for further calculations. The knowledge of the analytical forms of Sconf, Smag as well as Eform as a function of the site occupancies allows to express the thermodynamic Gibbs function G of the system also analytically both versus site occupancies and versus temperature. For a ternary r-phase this kind of relationship is an 11-dimensional function. It can be further analyzed analytically and its lowest values (at various T) are especially interesting, since they correspond to the real sublattice occupancies. For the present analysis of the structural stability of the rphase we have employed the Korringa–Kohn–Rostoker (KKR) technique, as outlined in [32,33], to calculate the electronic structure and the total energy in the framework of the non-relativistic LDA theory and using the muffin-tin approximation for a crystal potential. It is worth noticing that depending on employed model of chemical disorder, Etot of the sigma phase was computed either using KKR (in case of ordered approximants of alloys) or KKR with the coherent potential approximation, CPA method (in case of chemically disordered alloys). All five crystallographic sites were assumed to be occupied by all three kinds of atoms i.e. a chemical disorder on these sublattices has been taken into account. Computations were carried out for a number of ordered unit cells with atomic arrangements selected randomly for all five sublattices and concentration ranges presented in Table 1. The electronic structure calculations were performed independently for each unit cell with a symmetry reduced to a simple tetragonal one (P1 instead of P42/mnm) i.e. each position was occupied by one type of atoms. In the case of the r-Fe–Cr–Ni system altogether 488 different atomic configurations of atoms in unit cell were analyzed, while 672 were taken into account for the r-Fe–Cr–Co system. Resulting magnetic moments (on particular sites) and formation energies (for particular unit cells) were used to construct analytical formulas for Smag and Etot, as described above. Actually, as will be presented below, the comparison among Etot of different atomic configurations within the same alloy stoichiometry allowed to achieve accuracy of the order of mRy (or even better). Next, collecting the large number of results obtained for different stoichiometries and atomic arrangements, also allowed to maintain such accuracy when fitting with polynomial functions. The comparison between computed and fitted values are given in the Supplementary Materials. As far as the KKR-CPA method is concerned, which actually accounts for random distribution of atoms on considered
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Fig. 7. Probabilities, Ak, of finding constituting elements on particular lattice sites (k = A, B, C, D, E) in the r-FeCrCo alloys with 3 different compositions, as indicated, vs. temperature, T, as calculated based on the analytical form of the total energy, Efit, (solid lines) and experimental data obtained from the ND measurements (symbols).
sublattices, the xi concentrations in Eq. (2) are exactly the same as those used for atoms occupancies on five inequivalent sites (A, B, . . . , E) in KKR-CPA calculations. It should be emphasized, however, that the KKR-CPA technique results in ground state (T = 0 K) Etot values for such disordered alloy, but not entropy values itself. So, in spite of the coincidence of xi defining chemical disorder in unit cell and those defining configuration entropy, the procedures to get Sconf (Eq. (2)) and Etot (KKR-CPA) are fully independent. For a given alloy stoichiometry and assuming atomic occupancies of sublattices, actually only one self-consistent result is possible to be obtained from the KKR-CPA calculation (let us remind that chemical disorder is random). On the other hand, the choice of ordered approximants for KKR computations is more subtle. Assuming that each atomic position is occupied by single atom (i.e. Fe or Cr or X), but with assumed sublattices occupancies, many atomic arrangements are possible. In practice for the given alloy stoichiometries and sublattices occupancies, ca. 20 different
atomic arrangements were computed. In the next step the obtained G functions for particular systems were minimized for various temperatures, yielding the sublattice occupancy. Actually, a graphical presentation of such multidimensional function fails in the most cases. On the other hand the calculated lowest values of G enabled determination of the sublattice occupancies. The latter agrees quite well with the values determined experimentally (see below), giving us confidence that the adopted model works correctly. 4. Results and discussion The lattice parameters a and c obtained by the analysis of the XRD patterns with the Rietveld method [27] are shown in Fig. 4 for the r-Fe–Cr–Co system, and in Fig. 5, for the r-Fe–Cr–Ni one. Concerning the former, the data lie on straight lines having a negative slope what is in line with the fact that the atomic radius of Co
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Fig. 8. Probabilities, Ak, of finding constituting elements on particular lattice sites (k = A, B, C, D, E) in the r-Fe–Cr–Ni alloys, as specified, vs. temperature, T, as calculated based on the analytical form of the total energy, Efit, (solid lines) and experimental data obtained from the ND measurements (symbols).
is smaller than the one of Fe for which it substitutes. In the case of the r-Fe–Cr–Ni system there is no possibility to present them as a function of one parameter only, so they are presented as 2dimensional maps (Fig. 5). As can be seen, lattice parameters change in the opposite way i.e. increase of a corresponds to decrease of c. Consequently, volume of the unit cell has a tendency to remain constant. Regarding the total energy calculations, the analysis of the computed data in terms of the second order polynomial was successful. Histograms illustrating the differences between the calculated, and the fitted values of the total energies for the two studied systems are shown in Fig. 6. In both cases they have the Gaussian shape
with the half width of 0.003 eV/atom for Fe–Cr–Ni and 0.007 eV/atom for Fe–Cr–Co, being much smaller than the differences between particular configurations. In a similar way was treated the relationship between the magnetic entropy and the atomic configurations. Here, the differences between the fitted and calculated Smag-values were of the order of 0.05kB/atom for the Fe–Cr–Co system and 0.02kB/atom for the Fe– Cr–Ni one i.e. relatively higher than in the case of the total energies (the average values of the magnetic entropy were 0.3kB/atom for the former and 0.5kB/atom for the latter). On the basis of the determined Gibbs function the site occupancy versus temperature was calculated. A comparison between
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Fig. 9. Probabilities, Ak, of finding constituting elements on particular lattice sites (k = A, B, C, D, E) in the r-Fe52.51.4xCr47.5+0.4xCox alloys vs. Co-concentration, x, as calculated based on the analytical form of the total energy for the temperature of 1000 K (solid lines), and the experimental data obtained from the ND measurements (symbols).
the calculated and the measured data for different sites in three different r-Fe–Cr–Co samples can be seen in Figs. 7 and 9. Generally, the measured values are in line with the predictions. For all sublattices, except C, the agreement is remarkably good. For C, a discrepancy is more visible and its reason is not clear. The corresponding comparison for the r-Fe–Cr–Ni system is presented in Fig. 8. Here, the relationship between the calculated and experimentally found values is similar as in the previous case i.e. the strongest deviation exists again for the site C. However, in this case the trend is reversed viz. the experimental data are close to each other while the theoretical ones are more distant.
The calculated and experimentally determined site occupancies confirmed that larger atoms show preference to occupy sites with larger available volumes (and higher coordination numbers). Atomic radii of substituent atoms (RNi = 149 pm, RCo = 152 pm) are meaningfully smaller than those of iron (RFe = 156 pm) and chromium (RCr = 166 pm), and it is obvious that the substituents follow smaller Fe atoms in the unit cell. This is well evidenced in Figs. 7–9, where the differences of Cr occupancy between samples with different stoichiometry are small. Consequently, changes of the substituent concentration is mainly visible on sublattice A and D (in r-FeCr predominantly populated by Fe-atoms) where it increases at the expense of iron. For additional verification of the obtained theoretical results the binary r-FeCr system was characterized on the basis of the results obtained for the ternary r-Fe–Cr–X ones. In fact the analytical representation of the Eform function allows to determine its value for any configuration of site occupancy, and, in particular, for the cases without third element. Such obtained values, can be, in turn, compared with those determined experimentally for the r-FeCr system [34]. The comparison is shown in Fig. 10. The theoretical predictions of the site occupancies in the r-FeCr obtained from the Eform approximation applied to the Fe–Cr–Ni and Fe–Cr–Co system differ from each other. The lack of consistency may be not so surprising since the results were obtained as the projections of complicated 11-dimensional functions approximated at different ranges of their arguments. It should also be borne in mind that the projected Eform function was obtained by calculations performed on the systems being often very different than the two-component ones. Consequently, the values of the Eform function may not properly approximate the values of the function in the absence of the substituent. In spite of this drawback, a rather good qualitative agreement between the theoretically and experimentally determined occupancies were found i.e. we were able to reproduce the experimental results with an accuracy of 20% (see Fig. 10). The predictions on the site preference of substituted Co/Ni atoms in the FeCrX-sigma phases, based on the analysis of the total energy calculations of ordered approximants, to some extend can be verified in parallel by the KKR computations combined with the coherent potential approximation (CPA) [35,36]. The CPA model allows to treat chemical disorder (as random) in selfconsistent way by taking into account all possible scatterings among atoms on the same sublattice as well as among inequivalent sublattices (theoretical details on KKR-CPA application can be found e.g. in Refs. [21,37,38]). But the fact that in the investigated FeCrX sigma-phases chemical disorder appears on all crystallographic sites, makes the computations very complex and timeconsuming. In order to facilitate this issue we have performed KKR-CPA calculations for two selected Co-concentrations of Fe16xCr14Cox with x = 1 and 2, and Fe15Cr14Ni (well corresponding to real alloy compositions), where Fe was substituted with Co/Ni selectively on five inequivalent sites. The fact that the same stoichiometry was maintained in all considered cases (isoelectronic systems) allowed for highly accurate comparison of the total energy (below mRy). One should bear in mind that the ground state total energy, calculated here with the KKR and KKR-CPA methods, includes various terms (e.g. kinetic, potential, exchange–correlation, Madelung, band) which may also differ from each other in individual cases, but, in general, features of the density of states observed on inequivalent sites quite well reflect electronic reasons for the site preference. Fig. 11a presents total energy differences (compared to the highest value obtained for B site) computed for Fe15Cr14Co (Etot(B) = 69790.034 Ry) and Fe14Cr14Co2 (Etot(B) = 70027.916 Ry) assuming that Co replaces Fe on A, B, C, D and E sites, respectively. We clearly see that whatever the Co content, substitution of Fe
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Fig. 10. Probabilities, Pk, of finding Fe atoms on particular lattice sites (k = A, B, C, D, E) in the r-Fe100xCrx (45 6 x 6 50) alloys vs. temperature, T, as determined based on the projection of the total energy, Efit, calculated (solid lines) for the r-Fe–Cr–Co (left panel) and for the r-Fe–Cr–Ni (right panel). The curves from top to bottom correspond to alloys with increasing x. Experimental data obtained from the ND measurements are shown as vertical bars.
Fig. 11. Total energy differences, DE (relative to the B site) computed for (a) r-Fe–Cr–Co and (b) r-Fe–Cr–Ni sigma phases, when Fe atoms are substituted by Co/Ni atoms. Calculations were done using the KKR-CPA method (squares/triangles stand for one/two substituted atoms), the results from the KKR method for one substituted atom (green circles) as well as their average values (red circles) are presented for r-Fe–Cr–Ni system for comparison. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
with Co is highly unfavorable on B and C sites, for which computed total energies are the highest. In view of the KKR-CPA results the most beneficial for Fe substitution appears to be the A site and next the D site, while the E site is an intermediate case. With increasing Co content, the overall trends in site preference for Co atoms in Fe–Cr–Co-sigma phases remain similar, but replacement of Fe on D site seems to be energetically slightly more favorable than on A site. This result can be tentatively attributed to the fact that A site has the multiplicity 2 (against multiplicity 8 for D site) and for the Fe14Cr14Co2 stoichiometry the A site must be fully occupied, which decreases scatterings due to chemical disorder. The reason for the site preference in the above-mentioned Fe16xCr14Cox alloy can also be examined by the comparison of
the site-decomposed DOS for Fe and Co on all five sites. The most favorable sites (A and D) show relatively lower DOS (on both atoms) in the vicinity of the Fermi level (EF) with respect to B and C sites, where high peaks of d-states just below EF are seen both for Fe and Co atoms (see Fig. 12). Such DOS features observed on A and D sites tend to decrease the band contribution to the total energy. Conversely, in the case of the B site the contribution of Co to total DOS near EF is even higher than that of Fe one, which is uncommon and energetically unfavorable situation, since d-states of Co are expected to be more filled than those of Fe, which generally produces lower DOS at EF. In view of DOS criteria, the E site appears to be an ambiguous case, since quite low DOS both on Fe and Co atoms are seen. Noteworthy, two sharp d-DOS peaks of
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Fig. 12. Non-spin polarized KKR-CPA DOS in FeCo r-phases. Fe- and Co-decomposed DOS are plotted. DOS contributions on five sublattices where single Fe atom was replaced by Co one are labelled by A, B, C, D and E. The Fermi energy (EF) marked by vertical line is shifted to zero.
Co are larger than those of substituted Fe, observed just below EF, may also enhance the band energy in the case of the E site. Similar KKR-CPA calculations performed for Fe–Cr–Ni-sigma phases also indicate a strong preference of the substituted Ni for A and D sites (Fig. 11b), which also remains in line with the results obtained for 400 ordered approximants achieved from the KKR computations. One sees that average total energy values obtained for inequivalent sites well match the corresponding values from
the KKR-CPA method, but of course due to different atomic configurations, the resulting total energy values are quite scattered.
5. Conclusions The results obtained in this study and reported in this paper permit the following conclusions to be drawn:
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1. It is possible to calculate site occupancies for complex and disordered ternary r-phases by analysis of the Gibbs function of the system. Theoretically calculated and experimentally determined occupancies stay in fairly good agreement with each other. 2. Calculated and experimentally determined site occupancies confirmed that for the r-Fe–Cr–Co and r-Fe–Cr–Ni systems larger atoms preferentially occupy the sites with the higher coordination numbers and larger available volumes. 3. Substituent atoms (Co, Ni) are located mainly on A and D sites i.e. they replace Fe atoms. Chromium occupancy changes slightly on all sites.
Acknowledgements This work was supported by the Ministry of Science and Higher Education of the Polish Government and by the National Science Centre, Poland (Grant DEC-2012/05/B/ST3/03241). E. Suard and C. Ritter are thanked for help in ND measurements. References [1] F.C. Frank, J.S. Kasper, Complex alloy structures regarded as sphere packings. I. Definitions and basic principles, Acta Cryst. 11 (1958) 184; F.C. Frank, J.S. Kasper, Complex alloy structures regarded as sphere packings. II. Analysis and classification of representative structures12 (1959) 483. [2] E.C. Bain, The nature of solid solution, Chem. Met. Eng. 28 (1923) 23. [3] G. Bergman, D.P. Shoemaker, The determination of the crystal structure of the r phase in the iron-chromium and iron-molybdenum systems, Acta Cryst. 7 (1954) 857. [4] A.H. Sully, T.J. Heal, An electron compound in alloys of the transition metals, Research 1 (1948) 288. [5] S.P. Rideout, W.D. Manly, E.L. Kamen, B.S. Lement, P.A. Beck, Intermediate phases in ternary alloy systems of transition elements, Trans. AIME 191 (1951) 872. [6] D.K. Das, S.P. Rideout, P.A. Beck, Intermediate phases in the molybdenum– iron–cobalt, molybdenum–iron–nickel and molybdenum–nickel–cobalt ternary systems, Trams. AIME 194 (1952) 1071. [7] Ming-Hung Tsai, Kun-Yo Tsai, Che-Wei Tsai, Chi Lee, Chien-Chang Juan, JienWei Yeh, Criterion for sigma phase formation in Cr- and V-containing highentropy alloys mater, Sci. Lett. 1 (2013) 207. [8] F. Laves, H.J. Wallbaum, Ueber den Einfluss geometrischer Factoren auf die stoechiometrische Formel metallischer Verbindungen gezeigt an der Kristallstruktur des KNa2 (in German), Z. Anorg. Allg. Chem. 250 (1942) 110. [9] P. Duwez, S.R. Bean, Symposium on the Nature, Occurrence, and Effects of the Sigma Phase, in: Proc. ASTM, vol. 51, 1950, p. 48. [10] R.E. Watson, L.H. Bennett, The occurence of topologically close-packed phases, Acta Metall. 32 (1984) 477. [11] B. Seiser, R. Drautz, D.G. Pettifor, TCP phase predictions in Ni-based superalloys: structure maps revisited, Acta Mater. 59 (2011) 749. [12] L.R. Woodyatt, C.T. Sims, H.J. Bettie, Prediction of sigma-type phase occurence from compositions in austenitic superalloys, TMS-AIME 236 (1966) 519. [13] H. Ezaki, M. Moringa, N. Yukawa, Prediction of the occurrence of the r phase in Fe–Cr–Ni, Alloys Phil. Mag. A 53 (1986) 709. [14] N. Saunders, P. Miodownik, CALPHAD – A Comprehensive Guide, Elsevier, London, 1998.
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Site occupancies in sigma-phase Fe–Cr–X (X = Co, Ni) alloys: Calculations versus experiment J. Cies´lak ⇑, J. Tobola, S.M. Dubiel AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, PL-30-059 Krakow, Poland
a r t i c l e
i n f o
Article history: Received 4 February 2016 Received in revised form 9 May 2016 Accepted 11 May 2016
Keywords: Sigma phase FeCrNi alloys FeCrCo alloys Electronic structure calculations Neutron diffraction measurements
a b s t r a c t Ternary sigma-phase Fe–Cr–X (X = Co, Ni) alloys were studied theoretically (electronic structure calculations, Gibbs free energy analysis) and experimentally (X-ray diffraction, neutron diffraction, Mössbauer spectroscopy) in order to determine sublattice site occupancies by alloying elements. In general, good agreement between the predictions and experimental data was achieved. The obtained results agree reasonably well with expectations i.e. both Co and Ni atoms substitute for Fe atoms which predominantly occupy the sites A and D. Ó 2016 Elsevier B.V. All rights reserved.
1. Introduction The sigma (r) phase is one of many Frank–Kasper (FK) phases known also as topologically close-packed (TCP) phases that occur in alloys [1]. Originally it was discovered in a ternary Fe–Cr–Ni alloy [2] and its crystal structure was definitely identified in the Fe–Cr system [3]. Most of the FK-phases are detrimental to the high-temperature performance of technologically important structural materials, hence its presence in these materials is highly undesired (see e.g. Ref. [4]). The knowledge and understanding of mechanism(s) responsible for the formation of these phases is crucial in designing FK-free structural materials. In the course of time, several methods toward this end have been developed. They can be divided into three groups: (1) phenomenological, (2) computeradded and (3) theoretical (ab initio calculations). Concerning (1), the first attempt was based on the concept of the valence electron concentration (VEC) [4–6]. However, the VEC model could not explain satisfactorily all cases in which the FK phases were formed. The VEC scheme, although it can neither explain all the cases in which the FK-phases occur nor those for which the criterion was fulfilled but no FK-phases were formed, is still in use. Recently, it was successfully applied to study stability of the r-phase in high entropy alloys [7]. Laves and Wallbaum [8] and Duwez and Bean
⇑ Corresponding author. E-mail address: [email protected] (J. Cies´lak). http://dx.doi.org/10.1016/j.commatsci.2016.05.008 0927-0256/Ó 2016 Elsevier B.V. All rights reserved.
[9] introduced the difference in atomic size of elements constituting an alloy as a dominant factor responsible for the formation and stability of the FK-phases. Two-dimensional model, in which both the concentration of the holes as well as the atomic size are considered as important factors was proposed by Watson and Bennett for two-component alloys [10], and it was recently extended for multicomponent systems [11]. The computer-added methods include (a) PHACOMP, based on a concept of the average number of holes, hNhi, was originally introduced to predict formation of r in commercial Ni–Co-based super alloys [12], (b) newPHACOMP, in which instead of hNhi the average energy of the d-orbital was used as the dominant factor responsible for the r-phase formation [13], and (c) CALPHAD, based on thermodynamic modeling of a phase combined with equilibrium calculations [14]. There are numerous examples of its applications related to the FK-phases e.g. [15]. Theoretical ab initio calculations were carried out to give information on various issues related to the FK-phases like site occupancies e.g. [16,17], phase diagrams and structural trends e.g. [18,19], electronic and magnetic properties e.g. [20,21]. Whatever the method, its relevance can be ultimately verified by comparison of its predictions with experimental data. Studies of disordered FK phases have been carried out for many binary systems, both theoretically and experimentally. Ternary systems are a major challenge due to a much larger number of degrees of freedom available in filling sublattices with three elements. This variety must be taken into account in the calculation, as well as in the analysis of experimental results. Some ab initio
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2. Experimental 2.1. Samples and their characterization
Fig. 1. r-Fe–Cr–Ni samples investigated in the present study. Full symbols denote samples for which full transformation to the r-phase occurred. Dotted line stands for the r-Fe–Cr border.
calculations on stabilities of various configurations of sigma phase in subsystems of Fe–Ni–Cr i.e. in Fe–Ni and Ni–Cr were already published [22]. Other research (experimental and theoretical) was carried out for the systems r-CrMoRe [23] and r-MoNiRe [24]. The choice of elements forming these systems allow them to be easily differentiated by the X-ray (XRD) and neutron diffraction (ND) methods which, in combination with studying the formation energy, enabled determination of the sublattices occupancy yielding a good agreement between the theory and the experiment. In this paper such comparison is made as far as lattice site occupancies by Fe, Cr, Co (Ni) atoms in ternary sigma-phase systems of Fe–Cr–Co and Fe–Cr–Ni are concerned. In this case, the elements are poorly distinguishable using the XRD technique (due to similar values of the atomic number, Z). Consequently, the basic research technique used in this study was ND. On one hand, the sublattice occupancies were calculated using the ab initio calculations, and on the other hand, they were determined experimentally using neutron diffraction measurements.
Ternary sigma-phase alloy systems Fe–Cr–X (X = Co, Ni) were the subject of the present study. Concerning samples containing Co, they had the formula Fe52.51.4xCr47.5+0.4xCox, with x = 5, 10, 15, 20, 25, 30, 35. It should be added that Co atoms substitute for Fe ones without any limit [25 and references therein]. In the case of Fe–Cr–Ni alloy system, the r-phase can be formed with Ni concentration limited to 10 at.% [26 and references therein]. In the latter case 14 different compositions – see Fig. 1 – were synthesized for the present study. The master alloys of both series were obtained by melting appropriate amounts of constituting elements in an arc furnace. The melting process was carried out in a protective atmosphere of argon and it was repeated few times to ensure a better homogeneity. The master alloys of the Fe–Cr–Co series were transformed into the sigma-phase by isothermal annealing in a quartz tube, kept under a dynamical vacuum, in two steps: first at 1273 K for 1 day followed by the second one at 973 K for 7 days. The latter was terminated by quenching the samples onto a block of brass kept in a cold (300 K) end of the tube. The transformation of the Fe–Cr–Ni alloys into the sigma-phase turned out to be more difficult. Since exact concentration ranges in which the r-phase exists in this system are not precisely known, the prepared alloys covered a wider composition range. After homogenization (one day at 1273 K) the samples were annealed at 973 K for 7 days. For all not fully transformed samples the procedure was repeated, but this time the annealing temperature was 1073 K. Finally, 6 samples were found to be totally transformed to the r-phase, see Fig. 1 and Table 1. For both series the verification of the alpha-to-sigma phase transformation was done by recording room temperature XRD patterns as well as Mössbauer spectra. The XRD patterns, examples of which are shown in Fig. 2, also permitted determination of lattice parameters, a and c. Chosen samples of both series were investigated also with the ND techniques. Examples of the ND patterns recorded at room temperature at ILL Grenoble (D2B) can be seen in Fig. 3. Their
Table 1 Alloy compositions and lattice parameters for the Fe–Cr–Ni and Fe–Cr–Co r-phases. Fe
Cr
Ni
a (Å)
c (Å)
Experimental 0.460 0.458 0.507 0.407 0.525 0.475
0.501 0.490 0.467 0.489 0.455 0.505
0.039 0.052 0.026 0.104 0.020 0.020
8.797(2) 8.795(2) 8.791(2) 8.794(2) 8.790(2) 8.798(2)
4.559(1) 4.559(1) 4.560(1) 4.559(1) 4.560(1) 4.558(1)
Used for calculations 11–16 atoms (36.7–53.3 at.%)
13–16 atoms (43.3–53.3 at.%)
1–3 atoms (3.3–10 at.%)
488 different atomic arrangements were calculated
Fe
Cr
Co
a (Å)
c (Å)
Experimental 0.455 0.385 0.315 0.245 0.175 0.105 0.035
0.495 0.515 0.535 0.555 0.575 0.595 0.615
0.050 0.100 0.150 0.200 0.250 0.300 0.350
8.792(2) 8.792(2) 8.791(2) 8.788(2) 8.787(2) 8.784(2) 8.783(2)
4.556(1) 4.553(1) 4.550(1) 4.547(1) 4.548(1) 4.541(1) 4.539(1)
Used for calculations 1–13 atoms (3.3–43.3 at.%)
14–19 atoms (46.7–63.3 at.%)
2–10 atoms (6.7–33.3 at.%)
672 different atomic arrangements were calculated
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Fig. 2. XRD patterns recorded at room temperature on the r-Fe38.5Cr51.5Co10 (top) and the r-Fe47.5Cr50.5Ni2 (bottom) samples.
Fig. 3. ND patterns recorded at room temperature on the r-Fe38.5Cr51.5Co10 (top) and the r-Fe47.5Cr50.5Ni2 (bottom) samples.
analysis yielded probabilities of finding Fe, Cr and Co(Ni) atoms at particular lattice sites which was the main aim of the present study. In practice, the diffractograms were analyzed by means of the Rietveld method (FULLPROF program) [27]. In the calculations, the alloying element concentrations were held fixed to the nominal values, while the concentrations related to the particular sublattices were treated as free parameters. The occupancy of the elements on the fifth lattice site was constrained by the occupancies on other four sites. In total, 4 parameters were used to refine the occupancy, 9 parameters for atomic positions and lattice
parameters, the background intensity was refined with a polynomial (6 parameters) and the peak shape was approximated by the pseudo-Voigt function. 3. Theoretical As it was already mentioned, binary r-phases are chemically disordered systems which means that the translational symmetry of the lattice is maintained, but the sublattices are more or less randomly occupied by two elements, usually with different
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Fig. 4. Lattice parameters a and c as determined from the XRD patterns for the r-Fe–Cr–Co alloys. The lines represent the best linear fits to the data. Error bars are indicated, too.
probabilities. Similar type of disorder should be expected for more complicated ternary r-systems (e.g. mixing three elements on one site). Their theoretical analysis, and especially determination of the preferences of sublattice occupation, should be based on the analysis of the Gibbs energy variation in different stoichiometries of sigma-phase, i.e. dG = dE TdS + pdV. Since dV can be expected to be negligible, the aforementioned analysis can be based on the analysis of the free energy variation dF = dE TdS. It contains (besides the total energy, E) also the entropy term, S, scaled by the temperature of the system. In general, the total entropy consists of four contributions, namely configuration, Sconf, magnetic, Smag, phonon, Sphon, and electronic Sel, which should be considered to investigate relative crystal stability. The total energy of the system, Etot, can be understood as the work to be done to build the system at 0 K from single atoms of constituting elements, atom by atom. In practice, the value of Etot can be obtained e.g. from the DFT-based ground state selfconsistent electronic structure calculations. On that basis one can define the formation energy, Eform, as the difference between the total energy, Etot, of the investigated phase and the sum of total energies of its constituents (as bulk):
Eform ðAm Bn Þ ¼ Etot ðAm Bn Þ m Etot ðAÞ n Etot ðBÞ
ð1Þ
Fig. 5. Lattice parameters a (top) and c (bottom) in Å as determined from the XRD patterns for the r-Fe–Cr–Ni alloys. Those lying on the straight line correspond to binary r-FeCr alloys.
Smag ¼ kB
3 X 5 X xij lnðlij þ 1Þ
ð3Þ
i¼1 j¼1
In principle, all Etot values in Eq. (1) should be determined for elements A and B, assuming the same crystal structure as that of the investigated AmBn phase. Since in the case of the r-phase its one-component analogs does not crystallize in reality, an alternative way is to determine the total energy of single elements for real structures occurring at 0 K. The latter definition of Eform can be understood as the work to be done to change the pure elemental crystalline structures (but not single atoms) to build the system under consideration (also at 0 K). The configuration entropy, Sconf, is related to the degree of disorder of a system and for the three component r-phase can be expressed by the known formula (2), where index i denotes the constituent element (i = Fe, Cr or Co/Ni), index j denotes sublattice (A, B, C, D and E in the case of r-phase), and x stands for the site occupancy:
Sconf ¼ kB
3 X 5 X xij lnðxij Þ
ð2Þ
i¼1 j¼1
As can be seen from Eq. (2), Sconf can be calculated for any configuration of the sublattice occupancies. The magnetic entropy, Smag, can be described using Eq. (3), where lij denotes a magnetic moment:
In order to calculate this contribution, magnetic moments of atoms in various atomic surroundings should be known. However, for the purpose of this study it was assumed that the magnetic entropy in the unit cell is determined by the occupancy of the sites, xij, so it should be a continuous function of the number of atoms of different types on all five sites. Although the analytical form of such dependence is not known, one can approximate this function using a polynomial of n-th degree, W(xij), the coefficients of which can be determined by its fitting to the Smag values, calculated for selected unit cells with known atomic arrangements. Unfortunately, since one should take into account the occupancy of each site separately, the number of the coefficients to be determined rapidly increases with the degree of the polynomial, n, and, in practice, such analysis seems to be reliable only for n < 3. However, applying this model to the studied ternary r systems one can find the differences between the values determined using electronic structure calculations. Noteworthy, the values of the such determined polynomials are 0.05kB/atom, while the values found for the different arrangements of the atoms differ from each other by 0.3kB/atom, on average. This means that determination of the magnetic entropy in this case may be vitiated by a significant error.
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Fig. 6. Histograms of the difference Ecalc–Efit for the studied systems: (a) r-Fe–Cr–Ni and (b) r-Fe–Cr–Co. The solid lines stand for the Gaussian approximations.
Detailed analysis of Sphon-term contribution would require more experimental and theoretical data on the lattice dynamics in the r phase to figure out its role. However, Sphon-values determined experimentally and from the ab initio modeling [28] for the r-FeCr (and the corresponding bcc-alpha) phases are quite similar, so their difference presumably neither affects the alpha-sigma transformation process nor physical properties of the r-FeCr phase. Moreover, it is expected to be less influential than other forms of entropy, since Sphon determined for the r-FeCr is 0.1kB/atom whereas the value of Sconf 0.5kB/atom. The electronic entropy, Sel, at low temperature is a linear function of T (with the coefficient gamma related to density of states near the Fermi level) due to the linear dependence of specific heat [29]. In general, Sel values can be only slightly different for different atomic arrangements, which was also confirmed by our computations. Therefore, comparing different r-phases having the same alloy stoichiometry, one can neglect possible Sel differences in the analysis of the site occupancy, also when extrapolating to high temperature range i.e. where r-phases precipitate. Alike the magnetic entropy, also the total energy is a function of the site occupancies. In a simplified approach, Etot can be determined for the unit cells characterized by the absence of disorder on sublattices. Such assumption significantly simplifies the calculations that should be carried out as only the configurations of sites that are fully occupied by single type of atoms need to be taken into account [30,31]. For a ternary r-phase there are 35 = 243 different configurations to analyze. On the basis of such calculated values of the total energy one can construct diagrams of the formation energy versus concentrations of each element in the unit cell. The envelope determining the bottom of the Eform points on these diagrams (so-called convex-hulls) indicates energetically the most favorable configurations in the range of the considered concentrations [24]. Real crystalline systems may be, however, more precisely described when the disorder on the sublattices is taken into account. This can be done, for example, by calculating the total energy for one selected atomic arrangement and setting it as the reference system. Next, similar calculations should be performed for other atomic arrangements, differing from the reference one in a pre-specified way (e.g. by variation of the occupancy on one site only). Finally, such computations have to be performed for a number of modified arrangements that are sufficient to determine the shape of Etot function versus the occupancy’s variation on all sublattices. In the next step one can approximate this shape using an analytical function. The second-order polynomial in the five dimensional space (due to A–E sites) proved to be very effective. The values of the total energy per one atom in the r-phases are of the order of tens keV/atom, while the differences in energies for different arrangements of atoms in the unit cell are about
1 eV/atom. However, an adjustment of this relationship by the mentioned polynomial allowed to minimize the differences between the polynomial and the calculated values to few meV/atom. Actually, such calculations carried out for a number of selected arrangements are time-consuming but they result in the analytical approximation of the formation energy which can be easily used for further calculations. The knowledge of the analytical forms of Sconf, Smag as well as Eform as a function of the site occupancies allows to express the thermodynamic Gibbs function G of the system also analytically both versus site occupancies and versus temperature. For a ternary r-phase this kind of relationship is an 11-dimensional function. It can be further analyzed analytically and its lowest values (at various T) are especially interesting, since they correspond to the real sublattice occupancies. For the present analysis of the structural stability of the rphase we have employed the Korringa–Kohn–Rostoker (KKR) technique, as outlined in [32,33], to calculate the electronic structure and the total energy in the framework of the non-relativistic LDA theory and using the muffin-tin approximation for a crystal potential. It is worth noticing that depending on employed model of chemical disorder, Etot of the sigma phase was computed either using KKR (in case of ordered approximants of alloys) or KKR with the coherent potential approximation, CPA method (in case of chemically disordered alloys). All five crystallographic sites were assumed to be occupied by all three kinds of atoms i.e. a chemical disorder on these sublattices has been taken into account. Computations were carried out for a number of ordered unit cells with atomic arrangements selected randomly for all five sublattices and concentration ranges presented in Table 1. The electronic structure calculations were performed independently for each unit cell with a symmetry reduced to a simple tetragonal one (P1 instead of P42/mnm) i.e. each position was occupied by one type of atoms. In the case of the r-Fe–Cr–Ni system altogether 488 different atomic configurations of atoms in unit cell were analyzed, while 672 were taken into account for the r-Fe–Cr–Co system. Resulting magnetic moments (on particular sites) and formation energies (for particular unit cells) were used to construct analytical formulas for Smag and Etot, as described above. Actually, as will be presented below, the comparison among Etot of different atomic configurations within the same alloy stoichiometry allowed to achieve accuracy of the order of mRy (or even better). Next, collecting the large number of results obtained for different stoichiometries and atomic arrangements, also allowed to maintain such accuracy when fitting with polynomial functions. The comparison between computed and fitted values are given in the Supplementary Materials. As far as the KKR-CPA method is concerned, which actually accounts for random distribution of atoms on considered
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Fig. 7. Probabilities, Ak, of finding constituting elements on particular lattice sites (k = A, B, C, D, E) in the r-FeCrCo alloys with 3 different compositions, as indicated, vs. temperature, T, as calculated based on the analytical form of the total energy, Efit, (solid lines) and experimental data obtained from the ND measurements (symbols).
sublattices, the xi concentrations in Eq. (2) are exactly the same as those used for atoms occupancies on five inequivalent sites (A, B, . . . , E) in KKR-CPA calculations. It should be emphasized, however, that the KKR-CPA technique results in ground state (T = 0 K) Etot values for such disordered alloy, but not entropy values itself. So, in spite of the coincidence of xi defining chemical disorder in unit cell and those defining configuration entropy, the procedures to get Sconf (Eq. (2)) and Etot (KKR-CPA) are fully independent. For a given alloy stoichiometry and assuming atomic occupancies of sublattices, actually only one self-consistent result is possible to be obtained from the KKR-CPA calculation (let us remind that chemical disorder is random). On the other hand, the choice of ordered approximants for KKR computations is more subtle. Assuming that each atomic position is occupied by single atom (i.e. Fe or Cr or X), but with assumed sublattices occupancies, many atomic arrangements are possible. In practice for the given alloy stoichiometries and sublattices occupancies, ca. 20 different
atomic arrangements were computed. In the next step the obtained G functions for particular systems were minimized for various temperatures, yielding the sublattice occupancy. Actually, a graphical presentation of such multidimensional function fails in the most cases. On the other hand the calculated lowest values of G enabled determination of the sublattice occupancies. The latter agrees quite well with the values determined experimentally (see below), giving us confidence that the adopted model works correctly. 4. Results and discussion The lattice parameters a and c obtained by the analysis of the XRD patterns with the Rietveld method [27] are shown in Fig. 4 for the r-Fe–Cr–Co system, and in Fig. 5, for the r-Fe–Cr–Ni one. Concerning the former, the data lie on straight lines having a negative slope what is in line with the fact that the atomic radius of Co
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Fig. 8. Probabilities, Ak, of finding constituting elements on particular lattice sites (k = A, B, C, D, E) in the r-Fe–Cr–Ni alloys, as specified, vs. temperature, T, as calculated based on the analytical form of the total energy, Efit, (solid lines) and experimental data obtained from the ND measurements (symbols).
is smaller than the one of Fe for which it substitutes. In the case of the r-Fe–Cr–Ni system there is no possibility to present them as a function of one parameter only, so they are presented as 2dimensional maps (Fig. 5). As can be seen, lattice parameters change in the opposite way i.e. increase of a corresponds to decrease of c. Consequently, volume of the unit cell has a tendency to remain constant. Regarding the total energy calculations, the analysis of the computed data in terms of the second order polynomial was successful. Histograms illustrating the differences between the calculated, and the fitted values of the total energies for the two studied systems are shown in Fig. 6. In both cases they have the Gaussian shape
with the half width of 0.003 eV/atom for Fe–Cr–Ni and 0.007 eV/atom for Fe–Cr–Co, being much smaller than the differences between particular configurations. In a similar way was treated the relationship between the magnetic entropy and the atomic configurations. Here, the differences between the fitted and calculated Smag-values were of the order of 0.05kB/atom for the Fe–Cr–Co system and 0.02kB/atom for the Fe– Cr–Ni one i.e. relatively higher than in the case of the total energies (the average values of the magnetic entropy were 0.3kB/atom for the former and 0.5kB/atom for the latter). On the basis of the determined Gibbs function the site occupancy versus temperature was calculated. A comparison between
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Fig. 9. Probabilities, Ak, of finding constituting elements on particular lattice sites (k = A, B, C, D, E) in the r-Fe52.51.4xCr47.5+0.4xCox alloys vs. Co-concentration, x, as calculated based on the analytical form of the total energy for the temperature of 1000 K (solid lines), and the experimental data obtained from the ND measurements (symbols).
the calculated and the measured data for different sites in three different r-Fe–Cr–Co samples can be seen in Figs. 7 and 9. Generally, the measured values are in line with the predictions. For all sublattices, except C, the agreement is remarkably good. For C, a discrepancy is more visible and its reason is not clear. The corresponding comparison for the r-Fe–Cr–Ni system is presented in Fig. 8. Here, the relationship between the calculated and experimentally found values is similar as in the previous case i.e. the strongest deviation exists again for the site C. However, in this case the trend is reversed viz. the experimental data are close to each other while the theoretical ones are more distant.
The calculated and experimentally determined site occupancies confirmed that larger atoms show preference to occupy sites with larger available volumes (and higher coordination numbers). Atomic radii of substituent atoms (RNi = 149 pm, RCo = 152 pm) are meaningfully smaller than those of iron (RFe = 156 pm) and chromium (RCr = 166 pm), and it is obvious that the substituents follow smaller Fe atoms in the unit cell. This is well evidenced in Figs. 7–9, where the differences of Cr occupancy between samples with different stoichiometry are small. Consequently, changes of the substituent concentration is mainly visible on sublattice A and D (in r-FeCr predominantly populated by Fe-atoms) where it increases at the expense of iron. For additional verification of the obtained theoretical results the binary r-FeCr system was characterized on the basis of the results obtained for the ternary r-Fe–Cr–X ones. In fact the analytical representation of the Eform function allows to determine its value for any configuration of site occupancy, and, in particular, for the cases without third element. Such obtained values, can be, in turn, compared with those determined experimentally for the r-FeCr system [34]. The comparison is shown in Fig. 10. The theoretical predictions of the site occupancies in the r-FeCr obtained from the Eform approximation applied to the Fe–Cr–Ni and Fe–Cr–Co system differ from each other. The lack of consistency may be not so surprising since the results were obtained as the projections of complicated 11-dimensional functions approximated at different ranges of their arguments. It should also be borne in mind that the projected Eform function was obtained by calculations performed on the systems being often very different than the two-component ones. Consequently, the values of the Eform function may not properly approximate the values of the function in the absence of the substituent. In spite of this drawback, a rather good qualitative agreement between the theoretically and experimentally determined occupancies were found i.e. we were able to reproduce the experimental results with an accuracy of 20% (see Fig. 10). The predictions on the site preference of substituted Co/Ni atoms in the FeCrX-sigma phases, based on the analysis of the total energy calculations of ordered approximants, to some extend can be verified in parallel by the KKR computations combined with the coherent potential approximation (CPA) [35,36]. The CPA model allows to treat chemical disorder (as random) in selfconsistent way by taking into account all possible scatterings among atoms on the same sublattice as well as among inequivalent sublattices (theoretical details on KKR-CPA application can be found e.g. in Refs. [21,37,38]). But the fact that in the investigated FeCrX sigma-phases chemical disorder appears on all crystallographic sites, makes the computations very complex and timeconsuming. In order to facilitate this issue we have performed KKR-CPA calculations for two selected Co-concentrations of Fe16xCr14Cox with x = 1 and 2, and Fe15Cr14Ni (well corresponding to real alloy compositions), where Fe was substituted with Co/Ni selectively on five inequivalent sites. The fact that the same stoichiometry was maintained in all considered cases (isoelectronic systems) allowed for highly accurate comparison of the total energy (below mRy). One should bear in mind that the ground state total energy, calculated here with the KKR and KKR-CPA methods, includes various terms (e.g. kinetic, potential, exchange–correlation, Madelung, band) which may also differ from each other in individual cases, but, in general, features of the density of states observed on inequivalent sites quite well reflect electronic reasons for the site preference. Fig. 11a presents total energy differences (compared to the highest value obtained for B site) computed for Fe15Cr14Co (Etot(B) = 69790.034 Ry) and Fe14Cr14Co2 (Etot(B) = 70027.916 Ry) assuming that Co replaces Fe on A, B, C, D and E sites, respectively. We clearly see that whatever the Co content, substitution of Fe
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Fig. 10. Probabilities, Pk, of finding Fe atoms on particular lattice sites (k = A, B, C, D, E) in the r-Fe100xCrx (45 6 x 6 50) alloys vs. temperature, T, as determined based on the projection of the total energy, Efit, calculated (solid lines) for the r-Fe–Cr–Co (left panel) and for the r-Fe–Cr–Ni (right panel). The curves from top to bottom correspond to alloys with increasing x. Experimental data obtained from the ND measurements are shown as vertical bars.
Fig. 11. Total energy differences, DE (relative to the B site) computed for (a) r-Fe–Cr–Co and (b) r-Fe–Cr–Ni sigma phases, when Fe atoms are substituted by Co/Ni atoms. Calculations were done using the KKR-CPA method (squares/triangles stand for one/two substituted atoms), the results from the KKR method for one substituted atom (green circles) as well as their average values (red circles) are presented for r-Fe–Cr–Ni system for comparison. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
with Co is highly unfavorable on B and C sites, for which computed total energies are the highest. In view of the KKR-CPA results the most beneficial for Fe substitution appears to be the A site and next the D site, while the E site is an intermediate case. With increasing Co content, the overall trends in site preference for Co atoms in Fe–Cr–Co-sigma phases remain similar, but replacement of Fe on D site seems to be energetically slightly more favorable than on A site. This result can be tentatively attributed to the fact that A site has the multiplicity 2 (against multiplicity 8 for D site) and for the Fe14Cr14Co2 stoichiometry the A site must be fully occupied, which decreases scatterings due to chemical disorder. The reason for the site preference in the above-mentioned Fe16xCr14Cox alloy can also be examined by the comparison of
the site-decomposed DOS for Fe and Co on all five sites. The most favorable sites (A and D) show relatively lower DOS (on both atoms) in the vicinity of the Fermi level (EF) with respect to B and C sites, where high peaks of d-states just below EF are seen both for Fe and Co atoms (see Fig. 12). Such DOS features observed on A and D sites tend to decrease the band contribution to the total energy. Conversely, in the case of the B site the contribution of Co to total DOS near EF is even higher than that of Fe one, which is uncommon and energetically unfavorable situation, since d-states of Co are expected to be more filled than those of Fe, which generally produces lower DOS at EF. In view of DOS criteria, the E site appears to be an ambiguous case, since quite low DOS both on Fe and Co atoms are seen. Noteworthy, two sharp d-DOS peaks of
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Fig. 12. Non-spin polarized KKR-CPA DOS in FeCo r-phases. Fe- and Co-decomposed DOS are plotted. DOS contributions on five sublattices where single Fe atom was replaced by Co one are labelled by A, B, C, D and E. The Fermi energy (EF) marked by vertical line is shifted to zero.
Co are larger than those of substituted Fe, observed just below EF, may also enhance the band energy in the case of the E site. Similar KKR-CPA calculations performed for Fe–Cr–Ni-sigma phases also indicate a strong preference of the substituted Ni for A and D sites (Fig. 11b), which also remains in line with the results obtained for 400 ordered approximants achieved from the KKR computations. One sees that average total energy values obtained for inequivalent sites well match the corresponding values from
the KKR-CPA method, but of course due to different atomic configurations, the resulting total energy values are quite scattered.
5. Conclusions The results obtained in this study and reported in this paper permit the following conclusions to be drawn:
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1. It is possible to calculate site occupancies for complex and disordered ternary r-phases by analysis of the Gibbs function of the system. Theoretically calculated and experimentally determined occupancies stay in fairly good agreement with each other. 2. Calculated and experimentally determined site occupancies confirmed that for the r-Fe–Cr–Co and r-Fe–Cr–Ni systems larger atoms preferentially occupy the sites with the higher coordination numbers and larger available volumes. 3. Substituent atoms (Co, Ni) are located mainly on A and D sites i.e. they replace Fe atoms. Chromium occupancy changes slightly on all sites.
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