A first principles study of the stacking fault energies for fcc Co-based binary alloys

Acta Materialia 136 (2017) 215e223

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A first principles study of the stacking fault energies for fcc Co-based binary alloys
rraga a, *, Henrik Larsson a, Erik Holmstro € m b, Levente Vitos a, c, d Li-Yun Tian a, Raquel Liza a

Department of Materials Science and Engineering, Royal Institute of Technology (KTH), Stockholm, SE-100 44, Sweden Sandvik Coromant R&D, Stockholm, SE-12680, Sweden c Department of Physics and Astronomy, Division of Materials Theory, Uppsala University, Box 516, SE-75121, Uppsala, Sweden d Research Institute for Solid State Physics and Optics, Wigner Research Center for Physics, Budapest, H-1525, P.O. Box 49, Hungary b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 4 May 2017 Received in revised form 29 June 2017 Accepted 5 July 2017 Available online 6 July 2017

The stacking fault energy is closely related to structural phase transformations and can help to understand plastic deformation mechanisms in materials. Here we perform first principles calculations of the stacking fault energy in the face centered cubic (fcc) Cobalt-based binary alloys Co1
x Mx, where M ¼ Cr, Fe, Ni, Mo, Ru, Rh, Pd and W. We investigate the concentration range between 0 and 30 at.% of the alloying element. The results are discussed in connection to the phase transition between the lowtemperature hexagonal close packed (hcp) and the fcc structures observed in Co and its alloys. By analyzing the stacking fault energies, we show that alloying Co with Cr, Ru, and Rh promotes the hcp phase formation while Fe, Ni and Pd favor the fcc phase instead. The effect of Mo and W on the phase transition differs from the other elements, that is, for concentrations below 10% the intrinsic stacking fault energy is lower than that for pure fcc Co and the energy barrier is higher, whereas above 10% the situation reverses. We carry out also thermodynamic calculations using the ThermoCalc software. The trends of the ab initio stacking fault energy are found to agree well with those of the molar Gibbs energy differences and the phase transition temperature in the binary phase diagrams and give a solid support for the phase stability of these alloys. © 2017 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: First principles calculations Stacking fault energies Cobalt-based alloys Thermodynamic calculations

1. Introduction A stacking fault (SF) is a disruption in the order of the perfect stacking of crystallographic planes; a planar defect in a material [1]. The energy associated with the formation of a SF, the stacking fault energy (SFE) is a fundamental parameter in the design of alloys. It has a large impact on the creep properties of materials in temperature and stress ranges where the dominant deformation mechanism is by dislocation glide [2]. Low SFE leads to a large separation between Shockley partial dislocations which hinders cross-slip and thereby effectively reduces the creep rate. In an fcc material, a SF can be regarded as an embedded hcp slice in the fcc matrix and high SFE indicates that martensitic (hcp) transformation is not feasible [3,4]. Consequently, by tuning the SFE the plastic deformation properties of materials can be varied in a wide range. Cobalt and Co-based alloys are known for their excellent

* Corresponding author. E-mail address: [email protected] (R. Liz
arraga). http://dx.doi.org/10.1016/j.actamat.2017.07.010 1359-6454/© 2017 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

performance in many industrial applications because of their highstrength, corrosion resistance and the ability to retain hardness at high temperatures [5]. Co is the most commonly used binding material in the cemented-carbide industry, superior in many respects to Fe and Ni [6]. The Co-based alloys are also employed in sliding components in nuclear power plants [7]. However, serious health-concerns associated to Co powder [8] have prompted a lot of research aiming to find Co-free alternatives. This has proven to be a difficult task and it is still an unresolved issue [3,9,10]. In order to take an assertive decision regarding where research efforts should be directed to in order to find a successful substitution, one needs first to identify the properties that make Co well suited for a specific application. Several observations indicate that the SFE and the closely related structural phase transition hcp%fcc observed in Co at ~700 K [11] may be in the list of such properties. For example, the high resistance to galling in the so-called Stellite, a family of Cobased alloys, is known to be related to the low SFE [10] and the fcc-hcp transition in these materials [12]. Furthermore, the exceptional erosion resistance of the Stellite alloys can be ascribed to the

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formation of mechanical twins and by the presence of platelets of the hcp phase, which in turn are controlled by the SFE [5,13]. The phase transition temperature can be tuned by alloying Co with other elements as seen in the Co-Fe and Co-Ni alloys, where a small concentration of Fe or Ni lowers the transition temperature below room temperature [14,15] and therefore the SFE increases in these alloys. Since the SFE and its relation to the structural phase transition have a strong impact in the excellent performance of these materials, here we investigate the alloying effect on the generalized SFE curve of Co and the deformation mechanisms behind the fcc-hcp phase transition. We perform first principles calculations of the SFE and thermodynamic calculations of the Gibbs free energy. These approaches complement each other to bring about a realistic picture of the phase stability in fcc Co-based alloys. Despite the large influence the SFE has on the remarkable properties observed in these alloys there are, to our knowledge, only a few first principles calculations of the SFE of pure Co and Cobased alloys [16e18]. The paper is organized as follows: in Sec. 2 we introduce the model used to calculate SFE, in Sec. 3 we describe the computational methods, in Sec. 4 we present the results of the ab initio and thermodynamic calculations and finally in Sec. 5 we summarize our findings. 2. Model Plastic deformation in crystals is caused by shear stresses that can slide blocks of atomic planes relative to one another by rearranging the bonds between them. In fcc materials, the most common planes on which such displacements or slips take place are the {111} planes and the glide occurs along the 〈110〉 crystallographic directions [1]. The shortest atom to atom translation vector, b1 ¼ h110i=2, connects the atoms in the corner with the atoms in the center of the cubic faces and it corresponds to the Burgers vector of a perfect or full dislocation. Under certain circumstances it can be energetically favorable to dissociate the full dislocation into two partial dislocations with Burgers vectors b2 and b3 , respectively. In particular, the reaction displayed in Fig. 1a) can be realized in fcc materials. Between these two Shockley partial dislocations a SF ribbon is created.

i 1h i 1h i 1h 101 / 112 þ 211 ; 2 6 6

(1)

A SF is produced in a crystal when a local deviation occurs in the stacking sequence. The hcp and the fcc structures consist of stacked (111) planes of atoms sitting at the vertexes of equilateral triangles in a net. These structures share a common base of two layers of atoms, A and B which are not stacked one on top of the other but shifted relative to each other. If the third layer comes directly above the first layer A (…ABABABAB…), the hcp structure is formed. On the other hand, if the atoms of the third layer occupy the other available position C, the resulting stacking …ABCABCABC … is the fcc. Fig. 1b) shows the fcc unit cell and the colored balls represent atoms belonging to the A, B and C atomic planes. Here we model a single-layer ideal SF caused by shear on the (111) plane in fcc alloys by generating first a supercell of atoms arranged in the perfect fcc stacking (Fig. 1c) and then we use a triclinic supercell that follows the shear along a displacement vector b2

b2 ¼

i afcc h 112 : 6

(2)

The vector b2 is the Burgers vector of the Shockley partial dislocation in Eq. (1) and afcc is the lattice parameter of the fcc

unitcell. In this way, the supercell is tilted along the vector b2 and keep the atoms fixed in their positions (perfect fcc lattice). The slanted supercell is obtained by adding the vector b2 to the lattice pffiffiffi vector T ¼ 3 3 af cc ½111,

T0 ¼ T þ b2 :

(3)

Fig. 1d) shows two repetitions of the tilted supercell with 9 (111)-planes and one atom per layer. A SF is created at the cell boundary as a consequence of the shear, where an A atom reaches a B position. The color of the A atom in the tenth layer has been changed from cyan to red in the color scheme in Fig. 1d) to illustrate the SF. The layers above the SF follow and also change color. Effectively, there is an A layer missing at the SF. This approach has proven to be efficient and reliable because it allows for smaller supercells and the introduction of only one SF per cell [19,20]. The generalized stacking fault energy (GSFE) or g-surface, denoted by gðbÞ, corresponding to different sheared configurations along the direction b ¼ bafcc =6½112, can be computed as a function of the dimensionless displacement b,

gðbÞ ¼

EðbÞ
Efcc ; A

(4)

where Efcc is the total energy of the ideal fcc structure b ¼ 0 (the reference), EðbÞ is the energy of the sheared lattice and pffiffiffi A ¼ 3a2fcc =4 is the area of the supercell basal plane. This area is highlighted in Fig. 1a) and b). When b ¼ 1, b is the Burgers vector b2 and gðbÞ becomes the energy of the intrinsic SF (ISF). A micro-twin fault can be realized by successively deforming the layer just above the SF, between the tenth and the eleventh layers (Fig. 1e) and is called extrinsic SF (ESF). This corresponds to an extra B layer in the perfect fcc stacking. Here the color of the eleventh layer (C) has changed to cyan in the color scheme of Fig. 1e) to highlight the second SF. The layers above also change color accordingly. 3. Computational method 3.1. Total energy calculations We performed total energy calculations in the framework of density functional theory (DFT) [21,22] as implemented in the exact muffin-tin orbital (EMTO) method [23e26]. EMTO is an improved screened Korringa-Kohn-Rostoker (KKR) method [27], where the exact one electron potential is represented by large overlapping muffin-tin potential spheres. By using overlapping spheres the exact crystal potential can be described more accurately when compared to the conventional muffin-tin or not overlapping approaches [25,28]. The Perdew-Burke-Ernzerhof generalized gradient approximation (PBE) was used for the exchange correlation functional [29]. The chemical disorder was treated within the coherent-potential approximation (CPA) [30e32]. Nowadays, CPA is one of the most powerful techniques to treat random alloys due to the possibility to calculate properties of materials accurately using the unit cell instead of employing large and time consuming supercells [26]. The EMTO method together with CPA have been successfully applied to calculate the ground state properties of ordered [33,34] and disordered alloys [35]. The full charge density (FCD) technique is implemented for accurate total energies [36,37]. The EMTO basis includes s, p, d and f orbitals. The one-electron equations were solved within the scalar relativistic approximation and the soft-core scheme. The k-mesh was carefully tested and the 12  24  3 mesh was adopted for all calculations. During the GSFE calculations we used the triclinic supercell

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Fig. 1. Cyan, red and blue balls represent atoms corresponding to layers A, B and C in the fcc stacking sequence …ABCABC …(a) Basal view of the supercell: the (111) stacking plane. The vector b1 ¼ ½101=2 corresponds to a full dislocation and b2 ¼ ½112=6 and b3 ¼ ½211=6 are two partial Shockley dislocations. b) The fcc unit cell. The colored diamond-shaped area corresponds to the basal plane of the ideal fcc and the tilted supercells (see text). c) Two repetitions along the [111] direction of the ideal fcc supercell with 9 (111)-layers and one atom per layer. d) Two repetitions along the [111] direction of the tilted supercell with 9 (111)-layers. A SF is located between the ninth (counting from the bottom layer) and the tenth layer at the cell boundary. Layers above the SF have changed color to emphasize the SF so that the original layer A becomes B, and the B layer above becomes C, etc. e) The extrinsic stacking fault configuration, which is realized by a second SF between the tenth and eleventh layers (counting from the bottom layer). Layers above the second SF (from the eleventh layer) change color to show that an original layer C becomes A and that a layer A becomes B, etc. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

described in the model section, with 9 (111)-layers and one atom per layer. The interlayer distances were kept fixed along the stacking direction to the ideal fcc cell, except the layer that contains the SF which is relaxed until the SFE is minimized. The in-plane lattice parameters are fixed to the calculated equilibrium value of the ideal fcc lattice. Previously, this approach has been successfully used to calculate the g-surface of fcc metals [20] and fcc Fe [19]. 3.2. Thermodynamic calculations Calphad [38] based thermodynamic calculations were made using the Thermo-Calc software [39] and the TCNI8 database [40]. This database contains assessments of all investigated binary systems except for the Co-Rh alloy. To the authors knowledge there is no Calphad assessment of the Co-Rh system. The TCNI8 database is particularly suitable for the type of calculations performed in the present work since the assessments are made over the whole composition range.

constants of the binary alloys become larger as the concentration increases. The greatest increment is observed in Co-W alloys since W is a large atom, whereas Ni and Cr change the volume very little because their atomic radii (1.28 Å and 1.25 Å, respectively) are very similar to the atomic radius of Co (1.25 Å) [43]. Fe, on the other hand, enlarges the volume noticeably compared to Ni and Cr. This effect may be related to magnetism because the atomic magnetic moments of Co and Fe are enhanced in Co-Fe alloys compared to the pure element materials [44]. The structural properties of fcc Co-Cr alloys were investigated by Gudzenko and Polesya [45]. They found that the lattice parameters of these alloys increased parabolically up to 70 at.% Cr, however, up to 20 at.% Cr the increment can be approximated by a line with a slope of 2  10
4 Å/at.%. The presently calculated lattice parameters

4. Results 4.1. Equilibrium lattice parameters The theoretical lattice constant for pure fcc Co is 3.53 Å, which is in excellent agreement with previous ab-initio results, 3.52 Å ([41,42]), and also with the experimental value of 3.544 Å at room temperature [11]. We calculated the equilibrium lattice parameters for fcc Cobased binary alloys, Co1
x Mx, where M ¼ Cr, Fe, Ni, Mo, Ru, Rh, Pd and W, within the concentration range from 0 to 30 at.%. These values were then used in the calculations of the generalized stacking fault energies. In Fig. 2 we display the variation of the lattice constants with respect to pure fcc Co as a function of concentration. For most of the alloying elements, we can see that the lattice

Fig. 2. The variation of the lattice constants with respect to the alloying concentration for several fcc Co-based binary alloys. The values are given with respect to the calculated value for pure fcc Co, afcc ¼ 3.53 Å.

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increased a small amount up to 9 at.% Cr and above that the volume was reduced minimally. Ohnuma et al. investigated the fcc Co-Fe alloys and found that the lattice parameters increased linearly as the Fe content increased up to 20 at.% with a slope of 1.0233  10
3 Å/at.% [46]. Our results also show a linear increment of the lattice parameter with a slope very close to the observed one, 0.9  10
3 Å/at.%. The case of the fcc Co-Ni alloys is known to exhibit very small change in the lattice constants as the Ni concentration increases [47]. Jen and Huang measured lattice constants for CoxNi1
x alloys and found a slight linear decrease in the lattice constants as the Ni content increased [48]. We observed the same trend and our calculated values are in very good agreement with the measured ones. The lattice parameters of the Co-Mo alloys in the fcc phase have been experimentally determined to increase linearly up to 20% with a slope 4  10
3 Å/at.% [49]. Our value is 7  10
3 Å/at.%. The lattice parameter for the fcc Co-Ru alloys have € ster and Horn [50] up to been experimentally determined by Ko 20 at.% Ru. The lattice parameters increase linearly with a slope of 1.5  10
3 Å/at.% and our calculated slope is 4.6  10
3 Å/at.%. In the case of fcc Co-Rh alloys, the lattice parameters increase linearly with slope 2.72  10
3 Å/at.% [50], which compares well with our calculated slope 4.6  10
3 Å/at.%. Finally, the lattice parameter versus composition curve for fcc Co-Pd alloys deviates from Vegards' law. However up to 30 at.% Pd the curve can be approximated by a line with slope 3.6  10
3 Å/at.% [51] which is in reasonable good agreement with our slope 5.8  10
3 Å/at.%. In summary, for all binary systems considered here, we obtained very good agreement between the calculated and observed lattice parameters. The deviations between theoretical and experimental values are of the order of 1e3% in the lattice constants for all investigated alloys. 4.2. Generalized stacking fault energy The g
surfaces for the present Co1
x Mx alloys are shown in

Fig. 3. The results are presented in the following order (from left to right) M ¼ Cr, Fe, Ni, Mo, Ru, Rh, Pd and W. The first maximum of the g-surface corresponds to the unstable SFE (USF), which represents an energy barrier for the SF. When b ¼ 1, gðbÞ reaches a minimum, the stable or ISF, where a SF is created (between the ninth and the tenth layer in our model). The atomic configuration of the ISF can be seen in Fig. 1d). The figure also shows that a hcp nucleus of two hcp layers BCBC is formed at the SF, which is often regarded as the onset of the fcc to hcp transformation [4]. In the b ¼ 1 to b ¼ 2 regions in Fig. 3, the displacement takes place on the layer above the first SF (between the tenth and the eleventh layer in our model). The color of the background in these graphs have been changed accordingly. The second maximum in the g-surface is the unstable twinning fault (UTF), that represents the energy barrier to create a micro-twin defect. The second minimum corresponds to ESF and the atomic configuration is shown in Fig. 1e). For pure fcc Co, the USF is 290 mJ/m2, the ISF is
106.2 mJ/m2 and the UTF is 244.3 mJ/m2. The calculated ISF is in good agreement with the result estimated using the axial interaction model (AIM), which is a parametric model that maps the stacking sequence to a one-dimensional axial next-nearest neighbors Ising model [52]. In a first order approximation within AIM, ISF ¼ 2DEhcp
fcc =A ¼
105:63 mJ/m2, where A is the area of the supercell and DEhcp
fcc is presently calculated to be
1.3 mRy. This value agrees well with previous DFTcalculations [44,53]. The negative value of the ISF comes from the fact that hcp is the ground state of pure Co. Recently, Jo et al. developed a theory of plasticity that uses a parameter involving the ISF, USF and UTF to determine the leading deformation mode in fcc metals [54]. The effective energy barrier for a SF formation is given by USF/cos q, where q is the angle between the resolved shear stress and the SF slip direction. The effective barrier for twinning is (UTF
ISF)/cos q. By comparing these two energy barriers one can determine which plastic deformation mechanism is likely to be activated. In pure Co USF ¼ 290 mJ/m2< UTF
ISF ¼ 350.5 mJ/m2.

Fig. 3. g-surface for CoxM100
x alloys, where M ¼ Cr, Fe, Ni, Mo, Ru, Rh, Pd and W. Calculations were performed along the vector b ¼ b afcc =6 ½112. The colored background regions, from b ¼ 0 to from b ¼ 1 correspond to displacements on the tenth layer and the non-colored ones to displacements on the eleventh layer. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

L.-Y. Tian et al. / Acta Materialia 136 (2017) 215e223

This model predicts then SF formation upon mechanical load in pure fcc Co, which is consistent with the observed ground state hcp structure of Co. Ericsson obtained the ISF for pure Co in the hcp and fcc phases by using the node method based on the size of dislocation nodes as seen in electron micrographs [55]. The values obtained for fcc Co at 773 K and 983 K are 13.5 ± 3 mJ/m2 and 18.5 ± 5 mJ/m2, respectively. The experimental trend shows that the ISF decreases as the temperature drops and approaches the transition temperature, which is consistent with the fact that the fcc phase in Co becomes unstable below 700 K. By linear extrapolation of these data and considering the large errors involved in the experiments we estimate that at T ¼ 0 K the ISF in fcc Co would be of the order of
50 mJ/m2. This is in reasonable good agreement with our results which were obtained without strain or temperature effects. In these experiments the ISF of hcp Co were also obtained; 31 ± 5 mJ/ m2 at 293.15 K, 24.5 ± 5 mJ/m2 at 423.15 K and 20.5 ± 5 mJ/m2 at 643.15 K. Later, Hitzenberger et al. [66] showed that SFE in hcp Co decreases with increasing temperature and that ISF ¼ 3 1.5 mJ/m at 698 K. Recently, Achmad et al. investigated the g-surface of Co-based binary alloys, Co at 9 at.% with Cr, W, Mo, Ni, Mn, Al and Fe by means of a DFT-based method, CASTEP [17]. They obtained values of the ISF for pure fcc Co close to zero by using supercells with 11 layers containing 22, 44 and 66 atoms. By close inspection of their supercells one can see that when choosing to describe the ideal fcc structure with 11 layers or any number that is not a multiple of three, an unintentional SF is created at the cell boundary because of the periodicity of the cell. We believe that the problem with the reference energy is what caused their small ISF energy. 4.2.1. Co1
x Crx alloys The g-surfaces of Co1
x Crx alloys for different concentrations are shown in Fig. 3. In comparison with the values for pure fcc Co, the effect of alloying Co with Cr is that the USF becomes slightly larger whereas the ISF becomes more negative. This shows that CoCr alloys favor the formation of SFs and therefore the stabilization of the hcp structure, which is consistent with the experimental fact that the hcp-fcc transition temperature increases sharply as a result of the addition of Cr [6]. The Co-Cr phase diagram shows that the fcc phase is observed from 0 to 40 wt% Cr whereas at low temperatures, the hcp phase is present in the concentration range from 0 to 36 wt% Cr [6]. The UTF in Fig. 3 does not vary much with concentration but it is lower than the USF for all investigated concentrations. 4.2.2. Co1
x Fex alloys The GSFEs of Co1
x Fex alloys for different concentrations are displayed in Fig. 3. The trend shows that as the concentration of Fe becomes larger, USF decreases whereas the ISF becomes larger and positive. This means that alloying Co with Fe promotes the fcc phase and already at 15 at.% of Fe concentration the ISF becomes positive. One can also observe in the figure that the UTF becomes larger as the Fe concentration increases and for x > 30 at.% it becomes higher than the USF. Following the criteria in Ref. [54], the energy barriers become equal at 15 at% Fe (USF¼(UTF
ISF) ¼ 273 mJ/m2), and hence above this concentration twinning is energetically favored. Onozuka et al. investigated the phase transformation of Co alloys containing less than 8 at% Fe by means of Xray diffraction, electron microscopy and heat capacity measurements [56]. They found that Co-Fe alloys with a concentration range between 7 and 9 at.% Fe transform from hcp through dhcp to fcc. The transformation temperature was found to decrease with increasing Fe content from about 673 K to room temperature at about 6 at.% Fe [6,57]. If this trend is extrapolated down to 0 K, the

219

Fe concentration at which the fcc phase starts to be stable is 15 at.% which is consistent with our results.

4.2.3. Co1
x Nix alloys The phase transformation fcc-hcp in Co-Ni alloys has been extensively studied experimentally [58e62]. The transformation temperature is known to be lowered as the concentration of Ni increases, from ~ 700 K for pure Co to room temperature at about 30 at.% Ni [15]. The transformation occurs within the ferromagnetic regime as in pure Co [60]. Fig. 3 displays the g-surface of Co-Ni alloys for different concentrations. The effect of alloying Co with Ni is to increase the values for USF, ISF and UTF. Previously, Chandran et al. obtained a value for the ISF of
52 mJ/m2 for about 22 at.% Ni by using the Vienna Ab initio Simulation Package (VASP) [16]. This is in excellent agreement with our result
56.8 mJ/m2 at 30% Ni and the fact that below 30 at.% Ni, the hcp phase is stabilized at low temperatures [15] and hence the negative value of the ISF. By interpolating their results, they estimated that below 37 at.% Ni the ISF becomes energetically favorable. They used AIM also for comparison and the parameters in this model were calculated by means of VASP. Within this approach the ISF becomes positive between 50 and 60 at.% Ni depending on the structures included in the expansion. In our calculations, the ISF energy is already positive just below 50 at.% Ni. Other VASP calculations for Co67 Ni33 reported a positive value for the ISF 20 mJ/m2 and 205 mJ/m2 for the USF [63]. In this calculations though, the concentration of Ni may have been higher than 33% at the SF due to their supercell, which would explain the positive value of the ISF. Ericsson obtained the experimental values for the ISF for Co67 Ni33 and Co85 Ni15. The data for the alloy with 85 at.% Co shows a similar behavior as in the hcp Co case, that is, the ISF decreases as the temperature approaches the transition temperature ~ 630 K for that alloy [55]. The situation is different for Co67 Ni33, since the transition temperature is very low, 173 K and the ISF extrapolated to 0 K from Ericsson's data gives ~ 3 mJ/m2.

4.2.4. Co1
x Mox alloys In the g-surfaces of the Co-Mo alloys in Fig. 3, the USF and the UTF increase up until 10 at.% Mo, while the ISF decreases. Above 10 at.% Mo the situation reverses. The interpretation of the phase diagram of the Co-Mo alloys is very difficult because of the pronounced hysteresis in the hcp % fcc phase transformation. However, it appears that the effect of Mo on the transition temperature is to raise it [6]. This is supported by the theory of plasticity in Ref. [54] that predicts that the SF mechanism is preferred in the case of Co-Mo alloys because USF < (UTF
ISF) for all concentrations studied here. Furthermore, Mo and W are known to provide additional strength to the matrix in the Stellite family of Co-based alloys due to their large atomic size which hinders dislocation flow [64]. The typical nominal Mo-composition in the Stellite family is between 1 and 10 wt% which is consistent with our low value of the ISF below 10 at.% Mo.

4.2.5. Co1
x Rux alloys The Co-Ru alloys follow a similar pattern as Co-Cr alloys, that is, the USF becomes larger and the first minimum at ISF deepens which suggests that the hcp becomes even more stable. In fact above ~35 at.% Ru the hcp-fcc transition vanishes in these alloys and the structure is hcp until melting temperature as seen in Fig. 7. € ster, Ru raises the transformation According to the work of Ko temperature [50]. The UTF varies very little and it is ~ 100 mJ/m2 lower than the USF.

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4.2.6. Co1
x Rhx alloys When Co is alloyed with Rh the result is very similar to the case of the Co-Ru alloys. The USF becomes larger upon increments of the concentration and the ISF becomes more negative although not as much as in the case of Co-Ru alloys which suggests also a tendency to stabilize the hcp phase. The twinning barrier changes more than in the cases of Co-Cr and Co-Ru alloys. The effect of Rh on the transformation temperature is to lower it [50]. Unfortunately, we were not able to obtain a database that permitted a thermodynamic calculation of the Co-Rh alloys and hence a direct comparison was not possible. 4.2.7. Co1
x Pdx alloys The fact that Co-Pd alloys behave as the Co-Ni alloys, is likely because Ni and Pd belong to the same column in the periodic table. Both barriers, USF and UTF grow as a function of alloying concentration and the ISF comes close to zero which indicates a tendency to favor fcc with concentration. Compared with Ni though, Pd stabilizes the fcc phase in the Co-Pd alloys at much lower concentration than Ni. The phase transformation in Co-Pd alloys is said to be sluggish and the hcp / fcc occurs at much higher temperature than in pure Co, while the fcc / hcp takes place at much lower temperature than in the pure metal case [50]. 4.2.8. Co1
x Wx alloys By alloying Co with W one observes in Fig. 3 that USF initially increases and above 10 at.% W it decreases similarly to the Co-Mo case. However, the ISF increases above 5 at.% W indicating a tendency to stabilize the fcc phase. The UTF becomes higher initially and then it decreases. By using the criteria in the theory of plasticity of Ref. [54] we find that for W-concentrations below 50 at.% the SF mechanism is favored and above that it becomes more favorable to deform by twinning. 4.3. Summary of ab-initio trends Fig. 4 and Fig. 5 summarize our results for USF and ISF, respectively, for all Co-based binary alloys. The USF for Mo and W behave similarly in Fig. 4, that is, both curves increase their values with respect to the value for pure Co and above ~ 10% these values decrease. The USF for Fe decreases monotonically and for all other elements Pd, Cr, Ru, Ni and Rh, the USF becomes larger. In Fig. 5 one can see that Cr, Ru and Rh decrease the ISF which suggests that these elements when alloyed with Co favor the hcp phase, however the Co-Cr curve has a minimum at 15% of Cr. The ISF for Co-Mo

Fig. 4. Variation of the unstable stacking fault energy of Co-based alloys as a function of composition. Values are given with respect of the value in pure fcc Co. Inset shows the concentration interval between 2 and 12 at.%.

Fig. 5. Variation of the intrinsic stacking fault energy of Co-based alloys as a function of composition. Values are given with respect of the value in pure fcc Co. Inset shows the concentration interval between 2 and 12 at.%.

alloys also has a very shallow minimum between 5 and 10%, and above that it increases a little with respect to pure Co. Alloying Co with W and Ni has the same effect: to increase the ISF almost the same amount. Pd increases also the ISF but faster. but much faster. The ISF for Co-Fe alloys increases monotonically and above 15% it becomes positive. 4.4. Thermodynamic calculations hcp

The molar Gibbs energy differences DGm ¼ Gm
Gfcc m are shown in Fig. 6 as a function of composition at 298.15 K. This temperature is the lowest temperature at which the database may be used and it was chosen in order to compare the trends of the ab initio and the thermodynamic calculations. We also performed the same calculations at 500 K and no significant difference was observed. It must be emphasized, though, that performing thermodynamic calculations at 298.15 K is a quite severe extrapolation away from the experimental data on which the thermodynamic assessments are based. Stacking fault formation generally occur under isothermal conditions, but it can be discussed whether constant pressure or constant volume is the most appropriate assumption, i.e. whether a Gibbs or a Helmholtz energy difference should be considered as the most suitable approximation of the stacking fault energy. The difference is most likely negligible. We choose to calculate the Gibbs energy differences since the pressure dependence of Gibbs energy is not well assessed. The thermodynamic calculations give no information of the barriers, but should be able to give indications of the ISF. The molar Gibbs energy difference in Fig. 6 becomes more negative for Co-Cr and Co-Ru alloys which indicates that these elements tend to favor the hcp phase, whereas the curves for Fe, Ni, Pd and W have positive slopes. The Gibbs energy difference for the Co-Fe alloys becomes positive at around 8% Fe, which coincides with the destabilization of the hcp phase. In the case of Ni, DGm ¼ 0 around 33%, which is consistent with the fact that the phase transition is not observed above this concentration. For Co-Pd alloys, DGm becomes zero at much lower concentrations of Pd than in the case of Ni which was also observed in the ab-initio results. The Gibbs energy difference for Co-Mo alloys differs from the other elements, that is, it has a minimum at around 25 at.% Mo, which means that it behaves like Cr below this concentration. This behavior can also been observed in the ab-initio calculations. In general, the trends of the ab initio and the thermodynamic calculations are quite similar, that is, for all alloying elements the prediction of whether the

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fcc Fig. 6. Molar Gibbs energy difference, DGm ¼ Ghcp m
Gm , for fcc Co-based alloys in the concentration range between 0 and 30%. From left to right the alloying elements are Cr, Fe, Ni, Mo, Ru, Pd and W.

Fig. 7. Phase diagrams.

element will increase or decrease the SFE is in good agreement. There is also a general, though not perfect, agreement regarding the relative effect of the alloying elements. At 30 at-% the SFE as obtained from the ab initio calculations are in order from the highest to the lowest Fe, Pd, W, Ni, Mo, Cr, Ru. The corresponding order obtained from the thermodynamic calculations is W, Pd, Fe, Ni, Cr, Mo, Ru. The fact that the DGm curve for Co-Mo differs significantly

from the corresponding curve for Co-W is somewhat surprising considering the chemical similarity of Mo and W. In the Co-Mo phase diagram in Fig. 7d) there is a two phase fcc-hcp region in the range 15e18 at% at 1400 K, which has been fitted in the thermodynamic assessment. For Co-W there is no such a fcc-hcp two phase region. In the original assessment of the Co-W system by Guillermet it is indeed mentioned that the information used to

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assess the interaction parameter LCo;W was not certain [65]. We notice that in the low concentration range Cr and Mo show similar behavior and the same applies to Ni and Pd. Ru and Fe do not follow the same trend although they sit in the same column in the periodic table, however this is expected because Fe is ferromagnetic at room temperature, stabilizing the bcc structure instead of the hcp. In Fig. 7 we display the calculated binary phase diagrams for all investigated alloys. By inspecting the Co rich side of the phase diagrams it can be seen that the phase stability reflects the same conclusions as in Fig. 6. Here, a positive slope of the hcp-fcc phase boundary line indicates that the hcp phase is stabilized by the alloying element, which means that the SFE is lowered. A negative slope indicates that the alloying element decreases the stability of the hcp phase, resulting in a higher SFE. 5. Conclusions We have calculated stacking fault energies from first principles for fcc Co-based binary alloys. The alloying elements were Cr, Fe, Ni, Mo, Ru, Rh, Pd and W and the concentration ranged between 0 and 30 at.%. The ISF for fcc Co was calculated to be
106.2 mJ/m2. Negative values for the SFE for fcc Co and its alloys can be unintuitive, however this occurs because the fcc phase is metastable at T ¼ 0 K [53]. Our calculated ISF compares well with the extrapolated value of the experimental measurements at 773 K and 983 K of the SFE of fcc Co. Further analysis of the ISF of the Co-based alloys reveals that alloying Co with Cr, Ru and Rh lowers the SFE, whereas Fe, Ni and Pd enlarges the SFE. The cases for Mo and W are less straight forward, however they both tend to favor hcp under 10%. These findings are consistent with the relative phase stability of the phases. Thermodynamic calculations were performed by means of the CALPHAD method at 298.15 and 500 K. The results of the molar Gibbs energy difference calculated in the concentration range between 0 and 30 at.% are consistent with the ab-initio trends found for the SFE. The phase diagrams were also obtained. The trends obtained by the ab-initio and the thermodynamic calculations are also reflected in the slope of the hcp-fcc phase boundary of the phase diagram. Here a positive slope implies the stabilization of the hcp phase and therefore the ISF decreases. A negative slope indicates the opposite, such is the case for Fe, Ni and Pd. The Co-W is unclear due to the lack of experimental information on the hcp solid solution phase. We have demonstrated that the combined results of the abinitio and thermodynamic calculations give a picture of the SFE that is consistent with the phase stability of the fcc Co binary alloys. Our calculated stacking fault energies are expected to become reliable design parameters in future material development. Acknowledgments Work was supported by the Swedish Research Council, the Swedish Foundation for Strategic Research, Sweden's Innovation Agency (VINNOVA Grant No. 2014-03374), the Swedish Foundation for International Cooperation in Research and Higher Education, the Carl Tryggers Foundation, and the Hungarian Scientific Research Fund (OTKA 109570). The Swedish National Infrastructure € ping for Computing at the National Supercomputer Centers in Linko and Stockholm are acknowledged. References [1] M.N. Shetty, Dislocations and Mechanical Behaviour of Materials, PHI Learning Private Limited, Delhi, 2013.

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