Available online at www.sciencedirect.com ScienceDirect Acta Materialia 88 (2015) 180–189 www.elsevier.com/locate/actamat Ab initio description of segregation and cohesion of grain boundaries in W–25 at.% Re alloys Daniel Scheiber,a,b Vsevolod I. Razumovskiy,a Peter Puschnig,b Reinhard Pippanc and Lorenz Romanera, ⇑ a Materials Center Leoben Forschung GmbH, Roseggerstrasse 12, 8700 Leoben, Austria University of Graz, Institute of Physics, NAWI Graz, Universitätsplatz 5, 8010 Graz, Austria c Erich Schmid Institut of Materials Science, Austrian Academy of Sciences, Jahnstrasse 12, 8700 Leoben, Austria b Received 22 October 2014; revised 23 December 2014; accepted 29 December 2014 Abstract—We investigate grain boundaries (GBs) in W–25 at.% Re alloys with special focus on the segregation of Re to the GBs and the changes in their cohesive properties that arise therefrom. Our simulations rely on density functional theory and a mean-field approximation, the virtual crystal approximation, to model the highly alloyed system. Based on a gamma-surface approach, the geometry of a wide range of GBs, including tilt, twist and mixed GBs, is obtained. Segregation energies are found to vary strongly between different segregation sites, with the strongest segregation energy amounting to 0.75 eVs. We show that bond-order parameters are able to identify strong segregation sites based on purely geometric information only. With a thermodynamic model based on a Bragg–Williams approach, the concentration of Re atoms is calculated as a function of GB character at 1913 K and compared with atom probe experiments. The segregation levels are similar, with a trend towards higher segregation levels observed in the calculations for certain misorientation angles. Finally, Re alloying is found to lead to a general increase in the work of separation of the GBs, suggesting a reduction in the tendency for intergranular fracture. Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Grain boundary structure; Grain boundary segregation; Ab initio calculations; Rhenium effect; Virtual crystal approximation 1. Introduction The cohesive properties of grain boundaries (GBs) are of great relevance for the mechanical properties of materials [1,2]. Decisive in this connection are effects resulting from solute segregation towards the GBs, since the associated local modifications in chemical composition can drastically alter the strength of the interatomic bonding. In extreme cases, the fracture behaviour of a material can be completely modified via segregation, and a prominent and well-investigated example of the radical embrittlement of an otherwise ductile material is given by bismuth alloyed into copper [3,4]. A question of great technological importance is whether the opposite effect, i.e. an increase in cohesion and the resulting ductilization and toughening, can be achieved via solute segregation. This is particularly relevant for the refractory body-centred cubic (bcc) metals W and Mo and their alloys, for which grain boundaries feature a prominently low fracture toughness [5]. The origins of the brittleness have been debated; either it results from lowering of cohesion due to impurity segregation to GBs [6,7] or it is an intrinsic property [8,9] arising from the high ⇑ Corresponding romaner@mcl.at author. Tel.: +43 3842 45922 74; e-mail: lorenz. degree of covalency in the interatomic bonding. In this context, the “Re ductilizing effect” [10,11] is a long known phenomenon and is observed for several bcc elements, such as Cr, Mo and W. Although the effect is partly governed by solute solution softening [10,11] and the modification of the properties of screw dislocations [12,13], the role of segregation of Re to the GBs is of equal importance. Atom probe field-ion microscopy (APFIM) measurements carried out by Seidman et al. [14,15] have revealed a pronounced segregation of Re atoms to the GBs, increasing the local concentration of Re atoms in W up to threefold. Such high concentrations can be expected to modify the cohesive properties of the GBs; however, the experimental picture in this connection is controversial. While, in an early work [16], it was stated that Re favours intergranular fracture (suggesting a weakening of the GB), later investigations have stated exactly the opposite [17–19]. On the computational side, the most accepted and accurate approach to studying segregation on an atomistic level is density functional theory (DFT). To date, a few works on segregation in tungsten have been published [20–23], but there are none on highly alloyed W–25 at.% Re. Setyawan et al. [22] have carried out an investigation into the dilute limit at which many transition metals, including Re, are substituted into a certain geometric variant of a R27 GB [24] in W. It was found that Re segregates to the GB and http://dx.doi.org/10.1016/j.actamat.2014.12.053 1359-6462/Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. D. Scheiber et al. / Acta Materialia 88 (2015) 180–189 that, for the specific GB considered, an increase in cohesion is expected. In this work we will take a close look at segregation in the W–25 at.% Re alloy with the use of DFT. We will establish a close connection to experimental investigations [14,15] which have revealed that segregation is anisotropic, i.e. it depends on the character of the GB. Therefore, we will treat not just one or two GBs (which is traditionally done in DFT investigations), but a set of 15. Furthermore, as already shown in Ref. [22], strong changes in segregation energy between individual sites at a GB can arise. Therefore, an extensive study of segregation energies will be carried out and we will provide an analysis revealing which geometric arrangement of atoms around a segregation site attracts a solute and which arrangements do not. Finally, segregation in highly alloyed materials differs from segregation in the dilute limit due to Re–Re interactions. A meanfield framework for treating segregation in highly alloyed material was outlined in Ref. [25] on the basis of the embedded atom method. To the best of our knowledge, this method has never been adapted for DFT calculations. All previous DFT investigations of GB segregation that we are aware of treat segregation in the dilute limit, i.e. assume that the bulk concentration of the solute elements is very small. In contrast, we will adopt the virtual crystal approximation (VCA) [26,27], which provides an ab initio implementation of the mean-field methodology described previously [25]. With this method, we will finally also investigate the impact of Re segregation on the cohesive properties of the GB by calculating changes in the work of separation, which is a fundamental thermodynamic quantity controlling the mechanical strength of an interface [28,7]. 2. Theoretical framework for segregation in highly alloyed materials The theoretical framework for an atomistic treatment of segregation in a highly alloyed material has been outlined previousely [1]. Two methods can be used: a full Monte-Carlo approach based on the Metropolis algorithm or a direct minimization of the grand canonical potential. Both methods are exact in principle. In practice, however, approximations are invoked to make the methods tractable. An assumption commonly made when minimizing the grand canonical potential is the mean-field approximation, which reduces the computational effort considerably but still provides reliable results. as investigated in detail in Ref. [25] on the basis of a semi-empirical potential. We will shortly repeat the essential elements of this approach (see Refs. [1,25] for details) to establish the connection to the present ab initio mean-field treatment and elucidate the underlying approximations. We consider the GB and the surrounding bulk material as an open system which can exchange atoms with a reservoir. The open system consists of N atomic sites and the microscopic state of the system is fully defined by the coordinates Ri and the site occupancies pi of all sites. pi is 0 if a W atom occupies the site and 1 if an Re atom occupies the site. In the mean-field approximation, the grand canonical potential X is expressed as a function of the expectation values ri = hRi i and ci = hpi i. ci is the average occupancy of Re atoms at site i and corresponds to the 181 solute concentration of the site, while ri is the mean position of atomic site i. Hence, X ¼ E  TS v  TS c  lW N W  lRe N Re ð1Þ where E is the internal energy of the system, S c is the configurational part of the entropy, S v is the vibrational part of the entropy, T is the temperature, lW and lRe are the chemical potentials of W and Re, and N W and N Re are the number of W and Re atoms in the system. The solution of the mean-field segregation problem is obtained by minimizing X with respect to the variables fci g and fri g. Since the three terms E; S c and S v depend on fci g and fri g, this is a heavy task. Hence, it is common to invoke approximations for the calculations of E; S c and S v , and our choices will be outlined in the following. In an ab initio description, the dependency of E on the fci g can be calculated in several ways: (i) using supercells with random distribution of atoms [29]; (ii) within the coherent potential approximation (CPA) [30]; or (iii) with the VCA [26,27]. The first approach is computationally rather demanding, which makes it virtually infeasible in connection to the GB modelling, where rather large supercells with up to 260 atoms even for a unary system are needed. The CPA would be the method of choice; however, its currently available implementations do not allow one to take local lattice relaxations into account in an efficient manner, and this significantly reduces the accuracy of the defect calculations. For this work, we decided to use the conceptually simpler VCA, which allows the treating of defects in binary alloys, including lattice relaxations. The idea of the VCA is to substitute each atom in the lattice with a virtual Ax B1x atom. The fictitious atom has an effective intermediate electron and a nucleus number that interpolates between the two elements. Although, in general, the VCA “is not a good approximation for transition metal alloys” [27], it has been shown to yield not only lattice parameters, bulk moduli [12] and phonons [31] of WRe alloys with reasonable accuracy, but also other phenomena, such as core structures of screw dislocations [12,13]. This good description of the effective interatomic bond can be attributed to the fact that, for the WRe system, the neighbouring W and Re atoms have strongly overlapping d-bands and that the band structure of their alloys is very insensitive to the d-band filling. We are unaware of any investigation employing the VCA for calculating segregation energies. Therefore, we provide validation with a supercell approach for a model GB in the supporting data. For the treatment of S c , we assume a regular solution, hence, X S c ¼ k B ½ci ln ci þ ð1  ci Þ ln ð1  ci Þ ð2Þ i where k B is the Boltzmann constant. This approximation is widely used and is reasonable for WRe where interaction coefficients, and consequently short-range-order effects, are comparatively small [32]. Regarding S v , we assume that it does not depend on fci g; fri g, i.e. it is zero. This is a quite common approximation in DFT investigations because a treatment of S v on the basis of, for example, the quasiharmonic approximation is computationally rather demanding. In a previous investigation carried out for many different solute elements in Cr2 O3 , segregation tendencies have been found to decrease with increasing temperature [33]. We expect 182 D. Scheiber et al. / Acta Materialia 88 (2015) 180–189 the same effect to occur in WRe alloys, but leave an explicit treatment to future investigations. With these approximations, only Eðfci g; fri gÞ depends on the fri g. Hence, the minimization with respect to the fri g is carried out with the structural optimization algorithm provided by the ab initio code. The minimization with respect to the fci g, on the other hand, is carried out by setting the first derivative of X with respect to ci equal to zero. @E ci ¼ lRe  lW þ k B T ln 1  ci @ci ð3Þ Since this relationship is valid for every site in the system, it is convenient to get rid of the chemical potentials by equating the left-hand side of this equation for an interface site i with a bulk site 0 to obtain the Bragg–Williams equation:    ci c0 1 @E @E : ð4Þ ¼ exp   k B T @ci @c0 1  ci 1  c0 The term in the exponential Eseg ¼ @E @E  @ci @c0 ð5Þ defines the segregation energy which, according to this definition, is negative if segregation is favourable. The minimization of X requires a simultaneous solution of Eq. (4) for all N sites and the minimization with respect to the fri g. Due to the dependence of Eseg on all ci s and ri s, the minimization procedure can be involved, requiring, e.g., a self-consistent treatment [34]. However, we anticipate that the dependency of Eseg on the ci s is moderate in our case; therefore, a non-self-consistent approach will be employed here. In summary, the difference between the present approach and traditional DFT investigations in the dilute limit is that Eseg is calculated in the effective medium of W–25 at.% Re VCA atoms rather than in pure W. 3. Methodology A schematic representation of the unit cells involved in the calculations is shown in Fig. 1. The unit cell for the GB calculations is shown in panel (a). It contains two grains, highlighted in red and blue, with a vacuum layer on top. The minimum amount of vacuum needed to exclude interactions between the two surfaces was found to be 6 Å and the minimum length of the grains is 30 Å. The geometry of the GBs was determined using a c-surface procedure. The two grains are shifted with respect to each other on a grid on the GB plane and at every point of the grid a complete ionic relaxation is carried out. The structure of lowest energy is selected as the GB structure. For GBs with large unit cells (R17; R19; R33; R43 and the asymmetric twist GBs), the c-surface procedure was first carried out with semi-empirical 2NN-MEAM potentials [35] and the energetically lowest structures were then validated by DFT. For all other GBs, the c-surface procedure was computed directly by DFT. The free surface structures (Fig. 1b) were obtained from a single ionic relaxation. All GB starting geometries were built using the python ASE package [36]. All computations presented in this study were performed with the DFT implementation of the VASP software and its supplied environment [37–44]. This includes projectoraugmented wave functions (PAW) and the exchange correlation functional PBEsol [45]. The VCA is implemented in the PAW approach as described previously [12,13]. PAWs are created for virtual atoms of intermediate electron and nucleus number that interpolate between the two elements. To find suitable parameters for the calculations, we performed a series of convergence tests that are presented in detail elsewhere [46]. For an accurate treatment of tungsten and rhenium, the 5p6 semicore electrons are included as valence electrons, leading to 12 (5p6 6s2 5d 4 ) and 13 (5p6 6s2 5d 5 ) valence electrons for W and Re, respectively. Thus, in the VCA treatment of W–25 at.% Re, 12.25 electrons were used in the valence. The k-point mesh was 12  12  1 for a 4:47Å  4:47Å  66Å cell and was rescaled for larger cells to have comparable sampling in the reciprocal space. The ionic relaxations were terminated 1 when all forces were less than 0:01eVÅ per atom. The cutoff energy was 226 eV for all calculations. The lattice constants for the PBEsol potentials were derived from the Birch–Murnaghan equation of state, yielding 3.156 Å for pure W and 3.135 Å for W–25 at.% Re within VCA. 4. Results and discussion 4.1. Structure and energetics of GBs without segregation Fig. 1. Schematic representation of the unit cells used in this work. Panel (a) depicts the simulation cell used for GB simulations. The minimum dimensions of grains and vacuum are indicated on the left. Panels (b) and (c) depict the unit cells used for the calculations of GB energy and surface energy, respectively. The GBs investigated in this paper were selected to establish a close connection with the experimental work of Seidman et al. [14,15]. The grains were misoriented by rotation around the h110i axis, where the misorientation angle / varied in the regime between 20 and 110 . We chose nine symmetric tilt GBs, one twist and five mixed twist-tilt GBs to explore systematic trends. All mixed GBs were of predominant twist character. Using the c-surface procedure, the groundstate structures shown in Figs. 2 and 3 were obtained. The concentrations of every site were kept fixed at 0.25 throughout the procedure. As can be recognized from the figures, the GBs reveal quite different structural motifs at the interface plane. For the smallest unit cells, which are given by the D. Scheiber et al. / Acta Materialia 88 (2015) 180–189 183 Fig. 2. Relaxed GB structures of the nine tilt grain boundaries labelled by coincidence site lattice (CSL) notation and misorientation angle /. The view is for either grains onto the f110g plane and the atom colour distinguishes between the the two different lattice planes. The sites for which segregation energies are calculated are denoted by numbers. Fig. 3. Relaxed GB structures of six CSL mixed grain boundaries. The different sizes of the circles and shades of grey refer to different lattice planes. The angle at the top right of each structure is the misorientation angle and the angle at the top left specifies the tilt component, i.e. the angle between the GB normal and the h110i axis. R3h110if112g GB (48 atoms) and the R3h110if111g GB (72 atoms), our geometries are in agreement with previous works [20,23] on W. To the best of our knowledge, the other GBs have not been investigated by DFT in any bcc metal yet; they contain up to 260 atoms in the unit cell and are computationally more demanding. We note that, for the R17; R9 and R43 tilt GBs, the structural optimization has shifted the grains with respect to each other in such a way that the GB has lost its mirror symmetry. This is a phenomenon which has been observed in other GBs in bcc metals [23,47–51]. Starting from the relaxed geometries, the GB energies were calculated using: cGB ¼ 2EGB  EFS;1  EFS;2 A ð6Þ Here, EGB denotes the total energy as obtained from the relaxation procedure with cell a in Fig. 1 and EFS;i corresponds to the total energy of cell b, which has the same dimensions and number of atoms as cell a but contains only one grain. The results are presented in Fig. 4. It can be seen that the tilt GB energies are centred around a mean value of about 2 Jm2 except for the low-energy R3h110if112g GB. This GB is of particular high symmetry, exhibiting an almost bulklike environment. Hence, this GB is energetically more stable with respect to the other GBs which do not exhibit significant energetic variations. This general energetic difference between certain R3 GBs and other GBs is confirmed by measurements of grain boundary character distributions [52,53]. Furthermore, the mixed GBs exhibit lower GB energies compared to the pure tilt GBs. 184 D. Scheiber et al. / Acta Materialia 88 (2015) 180–189 Dci =0.75. Other choices for Dci only weakly affect the segregation energy, as shown by the supporting data. For the bulk terms, EsBulk ; EBulk , an equivalent procedure is applied. Note that in the bulk case simulation cells of quadratic shape are used which are sufficiently large to exclude interactions of the solute atom with its periodic replicas. Alternatively, it is possible to place the solute in a bulk site of the GB simulation cell. The corresponding energy is denoted as EsGB;0 . Then EsBulk  EBulk ¼ EsGB;0  EGB and, by substituting this into Eq. (9) and assuming Dci ¼ Dc0 ,  1  s ¼ Eseg EGB;i  EsGB;0 ð10Þ i Dci Fig. 4. cGB ; cFS and W 0sep for different GBs as a function of the misorientation angle /. The surface energy cFS;i is obtained via cFS;i ¼ FS EFS;i  NNBulk EBulk 2A ð7Þ where EBulk is the total energy from cell c in Fig. 1 and the FS accounts for the fact that the two unit cells confactor NNBulk tain different numbers of atoms. The surface energies cFS;1 þ cFS;2 show values around 8 J m2 for all GBs. These values are in good agreement with other calculations [22,54,55]. The work of separation W sep is obtained from the expression [56]: W sep ¼ cFS;1 þ cFS;2  cGB ð8Þ Note that in this section W sep is evaluated without segregation. We denote this quantity by W 0sep in the following, while the work of separation including segregation is denoted by W sep . W 0sep displays values of around 6 J m2 for the tilt GBs, though the value for the R3h110if112g is substantially higher, indicating its high stability. Also, the mixed GBs display higher values for W 0sep due to their higher stability with respect to the tilt GBs. Calculating the work of separation for pure W (data not shown) produces values which are on average 0:12 J m2 lower compared to the results above. This reveals that an increase in cohesion with Re alloying is obtained in the unsegregated state. How segregation can further change this result will be investigated below. 4.2. Segregation profiles Segregation leads to a redistribution of Re atoms in the GB structures considered so far. According to Eq. (5), the segregation energies are calculated by evaluating the deriv@E @E atives @c . They are approximated numerically by and @c i 0 ¼ Eseg i EsGB;i  EGB EsBulk  EBulk  Dc0 Dci ð9Þ Here, EsGB;i is the total energy of the GB structure from Fig. 1a, where the concentration at site i has been increased from the bulk value by Dci , while EGB is the GB total energy where all atoms have bulk concentration. The concentration is increased by substituting a bulk VCA atom of ci =0.25 with a pure Re atom of ci =1, resulting in Note that both definitions have been used in previous DFT investigations carried out in the dilute limit where c0 ! 0 and Dci ¼ 1 (see e.g. Ref. [56,50] for Eq. (9) or Ref. [23] for Eq. (10)). In the present case, investigations on the R3h110if111g GB have shown that the two definitions agree within 0.02 eV; therefore, the second approach will be used as it is computationally more convenient. The site-dependence of the segregation energy is investigated by calculating the segregation energy individually for every segregation site with Eq. (10). The resulting segregation profiles are presented by the red circles in Fig. 5 for all GBs. For the tilt GBs, the numbers in the figures allow one to relate the segregation energies with the corresponding sites in Fig. 2. The first observation is that the segregation profile is symmetric for most of the considered tilt and twist GBs with the exception of the R9; R17 and R43 GBs, which are of lower symmetry. We see that most tilt GBs have at least one strong segregation site, with a segregation energy of about 0:75 eV. For the low-energy R3h110if112g GB, the pure twist GB and the mixed GBs, segregation is generally weaker. This is consistent with the observation that the GB energies for these GBs are also smaller than for the tilt GBs. Another important result is that the segregation profile can be strongly jagged and that the decay of the profile with the distance to the GB is not monotonic, as would be expected based on some experimental studies (e.g. [57]). Especially for the structurally more complex GBs, there exist sites, directly at the GB, with a coordination geometry similar to the ideal bulk, while sites with a reduced number of atoms in the first coordination shell can be found farther away from the GB. For future modelling, it is important to understand why a specific site is attractive to a solute or not. One of the most important factors to be taken into account is the deviation from the host crystal structure. The question remains of how this can be quantified. For that purpose, we analysed the GB structures with different structure identification methods (Voronoi volume, centrosymmetry parameter, bond order analysis, bond angle analysis and common neighbour analysis, as recently reviewed in Ref. [58]) and correlated the parameters of the different methods to our calculated segregation energies. Plotting the segregation energy vs. Voronoi volume does not reveal a clear trend (see Fig. 6a), indicating that the segregation energy is determined not just by size mismatch; rather, the local arrangement of atoms around a segregation site seems to be decisive. Since Re is a hexagonal close-packed (hcp) element, it could be expected that a coordination geometry similar to hcp or face-centred cubic (fcc) would give highly attractive segregation energies. Contrary to this expectation, fcc or hcp environments identified by the common D. Scheiber et al. / Acta Materialia 88 (2015) 180–189 185 Fig. 5. Eseg as a function of distance for all GBs. The red circles are from DFT calculations, while the blue joined circles are the predicted segregation i energies from bond order analysis. All sites within  2Å of the GB plane (see vertical lines) are considered to be GB sites in the calculation of the enhancement factor b. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) neighbour analysis were not found to correlate with strong segregation energies. Instead, we have found a good quantitative correlation for bond order parameters. The idea of bond order analysis is to define for each site a set of vectors ri pointing to all neighbouring sites contained in a sphere of radius rcut and to evaluate the spherical harmonics for these vectors. The lth order parameter for a site is then given as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u þl u 4p X jq j2 ð11Þ Ql ¼ t 2l þ 1 m¼l lm with qlm ¼ 1 X m Y ðri Þ N ri <rcut l ð12Þ Here, Y ml ðrÞ ¼ Y ml ðh; uÞ are the spherical harmonics and N is the number of neighbouring sites within the sphere. Further details are given in Ref. [58]. We chose rcut ¼ 3:7 Å since this includes first and second nearest neighbours of the bcc host lattice. For different crystal structures, Ql assumes different values. Usually the so-obtained parameters Q4 and Q6 are used to differentiate between fcc, hcp and bcc structures, yet we found an intimate correlation between the Q5 parameter and the segregation energy. This is shown in Fig. 6b, where a clear linear trend emerges. We have thus fitted the segregation energies with two parameters, Q5 and the Voronoi volume, which allows us to estimate the segregation energy of Re to all sites of our GBs based on geometric information only. The results are shown as joined blue dots in Fig. 5. The agreement is quite satisfactory. Therefore, with additional improvements, which we will leave to future investigations, fits based on bond order parameters might provide a highly convenient way to obtain segregation profiles on the basis of only a very reduced number of DFT calculations. Indeed, fitting Eseg for only the two R3 tilt GBs and then using this relation to predict the segregation energies for all other GBs already gives very good results in the present case (see the dashed line in Fig. 6c. 4.3. Concentration of Re atoms at the GBs To establish a connection with the measurements from Refs. [14,15], the concentration of Re atoms at the GBs is estimated in the following. In thermal equilibrium, the Re concentration for every site of the GBs is obtained from Eq. (4). It therefore depends on the temperature, bulk concentration and segregation energy of site i. In general, the 186 D. Scheiber et al. / Acta Materialia 88 (2015) 180–189 Fig. 6. Correlation of the Voronoi volume, and of the bond-order parameters Q4 ; Q5 ; Q6 . with the segregation energies. The red dots correspond to values from the two R3 tilt GBs only. In (c) the fit-results when using all points (solid) or just the red points (dashed) are also shown. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) solution of these equations is complicated by the fact that the segregation energy itself can depend on the concentrations, making a self-consistent solution procedure necessary, as described earlier. In the case of repulsion between solutes, the segregation energy for a specific site will become less negative if neighbouring sites are strongly enriched. Hence, depletion effects can occur, as has been discussed, for example, for GBs and for surfaces in CuNi [25,59]. To investigate the importance of repulsive interactions between Re atoms, we calculated segregation profiles for two GBs (R3h110if111g and R19h110if116g) where a strong segregation site was occupied by an Re atom, i.e. ci was changed from 0.25 to 1, while the segregation profile on the other sites was calculated as described in the previous section. As shown by the supporting data, the segregation profile changes appreciably only for the next nearest neighbours, for which the segregation energy increases up to about 0.15 eV. In general, ci will be smaller than 1 (somewhere between 0.25 and 1, depending on the temperature) and the segregation energy increase will be smaller than 0.15 eV. In the following, we do not employ a self-consistent procedure since the resulting depletion effects are not expected to greatly influence the overall enrichment of the GBs. We leave a detailed investigation of such effects to future investigations. The Re concentration ci for every site is thus calculated by Eq. (4) at the same temperature, i.e. 1913 K, as in experiments [14,15], using the segregation energies from Fig. 5. The Re concentration at the GB is obtained by averaging over all fci g at the interface C GB ¼ X 1 GBsites ci N GB i ð13Þ where N GB is the number of sites at the GB. All sites within 2 Å from the GB plane are considered to belong to the GB. The enhancement factor b ¼ C GB =c0 is plotted against Fig. 7. Enhancement factor b as a function of misorientation angle / at 1913 K. D. Scheiber et al. / Acta Materialia 88 (2015) 180–189 the misorientation angle for tilt, twist and mixed GBs in Fig. 7, where the experimental values from [14,15] are also shown. The same characteristic progression of b with misorientation angle is recognized in theory and experiment. The marked minimum around 70 results from the low segregation energies of the R3 GBs at this angle. The remaining features arise from variations of the areal density of highly attractive segregation sites. This density is small close to 0 or 70 and increases as one moves away from these points. The twist and mixed GBs reveal generally smaller segregation levels. Between 0 and 70 the calculated segregation levels appear somewhat overestimated in comparison to experiment, while between 75 and 100 the agreement is very good. The approximations inherent in the simulations are expected to affect the segregation levels in a similar way, and therefore the relative heights of the b vs. / curve are not expected to change with a higher-level theoretical treatment. The critical factor for the discrepancy between theory and experiment is represented by the assignment of atoms belonging to the GB. A clear connection to experiment cannot be made in this respect, so we have adopted a simple criterion based on the distance to the GB. Indeed, differences in this assignment are likely to dominate the observed differences between theory and experiment. 4.4. Change in the work of separation To calculate the work of separation, we discuss segregation energies to the free surface (FS) Eseg FS;i and compare Fig. 8, where the them with the already presented Eseg in GB;i same symbols are used for GB and FS sites corresponding to each other. The FS segregation profiles are smoother than the GB segregation profiles and the minima are not as deep, i.e. not below 0.5 eV. However, there also exist sites for which the surface segregation is more pronounced than the corresponding GB segregation. These values for the segregation energies on surfaces can be compared to previous results obtained for the f110g surface, for which Fig. 8. Segregation energies to the surface (left) and the GB (right) in W25 at%Re for four GBs (top 4 panels). The bottom 2 panes show the segregation energies in pure W for two selected GBs. Matching symbols denote sites that correspond to each other. 187 a segregation energy of 0.27 eVs was reported for W–Re [60]. For the f110g surface we obtained a segregation energy of 0.23 eVs, which is very close to the previous results, considering the substantially different methodologies involved. Assuming the fast separation limit where no diffusion takes place during cleavage, the change in the work of separation due to segregation to site i is given by the following expression [56]: seg DW sep;i ¼ CGB;i ðEseg FS;i  E GB;i Þ ð14Þ where CGB;i ¼ ci =A is the solute excess at the GB per area. Segregation to GB sites with stronger segregation to the GB compared to the FS leads to an enhancement in cohesion, while the opposite is the case if segregation to the surface is stronger. With a summation over all sites at the GB, the cumulative change in the work of separation can be calculated as: DW sep ¼ GBsites X DW sep;i ð15Þ i Note that Eq. (14) includes the temperature dependence arising from configurational disorder via ci ðT Þ. The work of separation so calculated therefore corresponds to the situation where the alloy is kept sufficiently long at temperature T to establish thermodynamic equilibrium between the bulk and the GB chemical composition, which is then suddenly quenched down to very low temperatures to preserve the disordered state of the elevated temperature. In Fig. 9 this temperature dependence of the work of separation W sep ¼ W 0sep þ DW sep is shown for four different GBs. First we focus on the results for T ¼ 0K. All considered segregation sites are fully occupied and the total change in work of separation is seen to be positive for all four GBs. The largest effect can be seen for the high-energy R3h110if111g, where a 13% increase in work of separation arises from segregation. Raising the temperature decreases the concentration until, at the upper limit of T, it reaches the bulk value and W sep decays to W 0sep . In the intermediate regime, however, nonmonotonic effects can arise, i.e. W sep increases with T. In fact, raising T in a regime where the concentration decreases for a weak segregation site with negative DW sep;i while the concentration remains high for a strong segregation site with positive DW sep;i results in an increase in work of separation, thereby explaining the non-monotonic behaviour. As an overall effect, Re segregation increases the cohesion in W–25 at.% Re, and the effect is most pronounced Fig. 9. Temperature dependence of W sep for four GBs. The dashed line denotes W 0sep . 188 D. Scheiber et al. / Acta Materialia 88 (2015) 180–189 the precise knowledge of the segregation profiles, a close connection to APFIM measurements was established where the Re content was measured as a function of misorientation angle. Reasonable agreement is found, although a trend towards higher segregation levels for some misorientation angles was observed in the calculations. Furthermore, we have shown that segregation of Re increases the work of separation for both W and W–25 at.% Re. This is consistent with experimental works that have reported an increased tendency for transgranular cleavage in WRe alloys. Acknowledgements Fig. 10. Comparison of the temperature dependence of W sep for the two R3 GBs, once in a pure tungsten matrix and once in the W–25 at.% Re matrix. The dashed line denotes W 0sep . for the GBs with low W 0sep . Therefore, especially the weak GBs are reinforced by the Re atoms, and the present calculations support experiments [17–19] which have reported a transition to more transgranular fracture with Re alloying. 4.5. Effects from VCA embedding As the final aspect of this investigation, we focus on the question of how large the difference in segregation and change in work of separation would be for a treatment in the dilute limit. Fig. 8 shows that segregation energies can be larger in magnitude in pure W compared to the ones in W–25 at.% Re (pure tungsten Eseg max ¼ 0:7 eV, W–25 at.% Re: Eseg max ¼ 0:58 eV). This overestimation of segregation energies has an influence on the enhancement factor, which would be about 15% larger for the highenergy R3 and about 5% for the low-energy R3 in pure W. Fig. 10 compares DW sep for the two R3 GBs. We observe that W sep for W–25 at.% Re alloy with segregation computed in pure tungsten shows some differences if computed in the matrix with VCA. For the high-energy R3 there is a constant deviation for all temperatures of about 0:2 J m2,while for the low-energy R3 the results differ from each other only slightly. Hence, the use of VCA is more important when calculating the properties of the highenergy GB. In summary, the embedding in the W–25 at.% Re VCA matrix does not give a different qualitative picture of segregation compared to an embedding in pure W. However, considering the fact that the two calculations have the same computational cost, the non-negligible quantitative differences are certainly worth taking into account. 5. Conclusion We have carried out an ab initio investigation of Re segregation in the W–25 at.% Re alloy. We have used a c-surface approach to obtain reliable geometries and have used the VCA to treat segregation in the highly alloyed material. The obtained segregation profiles reveal a strong site dependence, with the strongest segregation energies found for tilt GBs (0:75 eV). We have identified a procedure to predict high segregation energies from structural parameters based on bond order analysis, which could provide a highly convenient method to get reliable segregation energies in the future with much less computational effort. With We would like to thank G. Kresse for providing VCA PAWs. We thank the IFERC-CSC Helios supercomputer and the Vienna Supercomputing center for providing computing resources. 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