Melting temperature of CoCrFeNiMn high-entropy alloys

Computational Materials Science 148 (2018) 69–75

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Melting temperature of CoCrFeNiMn high-entropy alloys M.A. Gutierrez a, G.D. Rodriguez a, G. Bozzolo d, H.O. Mosca a,b,c,⇑ a

GRUCAMM, UTN Gral. Pacheco, H. Yrigoyen 288, B1617FRP General Pacheco, Buenos Aires, Argentina Gerencia de Investigación y Aplicaciones, Comisión Nacional de Energía Atómica, Av. Gral. Paz 1499, B1650KNA San Martín, Buenos Aires, Argentina c Instituto Sabato, Universidad Nacional de San Martín (UNSAM) – Comisión Nacional de Energía Atómica (CNEA), San Martín, Buenos Aires, Argentina d Loyola University Maryland, 4501 N. Charles St, Baltimore, MD 21210, USA b

a r t i c l e

i n f o

Article history: Received 18 May 2017 Received in revised form 8 February 2018 Accepted 11 February 2018

Keywords: High entropy alloys Alloy design Melting temperature BFS method Atomistic modeling

a b s t r a c t Atomistic modeling of CoCrFeNiMn (0 < xMn < 25 at.%) high entropy alloys shows that the melting temperature as a function of Mn concentration does not follow the behavior consistent with a homogeneous solid solution, exhibiting a maximum value for 8.7 at.% Mn. Using the concepts of the BFS method for alloys, a description of the phenomenon is provided, showing that sluggish diffusion generates changes in the atomic distribution that lead to this anomalous behavior. This theoretical analysis is meant to provide a template for studying complex compositions and fine effects in high entropy alloys, which would be hard to detect experimentally but that could have an impact on potential applications of these complex materials. Ó 2018 Elsevier B.V. All rights reserved.

1. Introduction The CoCrFeNi and CoCrFeNiMn high-entropy alloys (HEA) have been the subject of numerous recent studies [1–28]. Partly due to their specific properties but also to the abundance of experimental [1–15,20–25] and theoretical [16–19] work performed on them, they have quickly become standards within the emerging field of HEAs. These two equiatomic alloys have enough elements to be considered high-entropy alloys, as they both form (for the most part [2,10,14,26–28]) homogeneous fcc solid solutions, and they both have remarkable mechanical and thermal properties [2,3,5–7]. In addition, the limited number of elements also makes them easier to investigate with standard theoretical or modeling tools [16–19]. Overall, these model systems provide excellent testing grounds for new approaches, capitalizing on the extended body of experimental work that already exists. In terms of alloy design, however, one obstacle still remains. Alloys, usually based on an original binary structure, become more complex as minority elements are added in order to induce desired changes in their properties. This process, however, increases the risk of losing the essential features that characterize their basic formulation. With the extensive

⇑ Corresponding author at: Gcia.Investigación y Aplicaciones, CNEA, Av. Gral Paz 1499, (B1650KNA), San Martín, Argentina. E-mail address: [email protected] (H.O. Mosca). https://doi.org/10.1016/j.commatsci.2018.02.032 0927-0256/Ó 2018 Elsevier B.V. All rights reserved.

knowledge on binary and, in some cases, ternary combinations, it is a sometimes a lengthy but generally successful approach to determine the final composition that meets desired criteria. However, uncertainties multiply as the number of elements increases, because of the intensive and expensive experimental work needed to validate any new choice or change in composition. At the same time, theoretical background becomes also more difficult to obtain. In this sense, HEAs present a distinctive challenge, testing the limitations of any current approach to alloy design. With HEAs, the starting point is already widely different from that of a more typical binary mix: in a multi-dimensional phase diagram, all that is known is that in the vicinity of its center (i.e., equiatomic composition), the system has a high probability of forming a (mostly) single-phase disordered alloy due to the maximum high configurational entropy, although this feature does not, by itself, guarantee such outcome [4]. As recent research shows, several other factors come into play [20], which could have a major impact on the resulting phase structure. Beyond that point, either changing the concentration of each element or adding/removing others in order to change the properties, becomes a much more complex task, as there is practically no information available, starting with the simple fact that it is not known what the range of concentrations for which the original single-phase structure exists [4], and whether substituting just a single element could lead to a different phase structure [20]. As such, the experimental work needed in order to gain more insight or fine tune properties is nearly prohibitive, while it is also true that the number of options and the

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paths to be pursued are substantially less restrictive than in the case of binary or ternary based alloys. One additional feature is prevalent in HEAs. While it is true that many of the compositions studied tend to exhibit the expected main and dominant feature (a continuous solid solution), it is also true that the formation of small intermetallic precipitates is mostly unavoidable, even if their presence does not preclude the alloy from having strong HEA characteristics [2,10,14]. It is also possible that the mix of so many elements could lead to subtle changes in the apparently uniform solid solution matrix (i.e., subtle differences in the concentration of all elements within the alloy), without altering too much its naturally homogenous distribution of the different atomic species. For HEAs, such fine effects, enhanced or diminished by the varied diffusion rates within such complex environment, could be as relevant in determining their final properties, in a way similar to how the inclusion of small amounts of a given minority addition could have in a two-element based alloy. One of the many questions that arise is then how to predict, visualize, or model such relatively minor changes, that could have rather significant consequences on the final properties of the HEA, and thus minimize the necessarily extensive experimental work. The purpose of this communication is to shed some light on that question. To answer it, a quantum approximate method which, due to its formulation, provides an efficient framework for dealing with multicomponent systems, is applied to the study of CoCrFeNi and its derivative system CoCrFeNiMn, in order to determine changes in properties due to the addition of Mn. Covering the range between 0 and 25 at.% Mn, modeling of these systems shows that there is a peculiar behavior of the melting temperature as a function of Mn concentration, which exhibits a maximum value for a relatively low Mn concentration (8.7 at.%), somewhat away from the equimolar case but still within the range of the fcc solid solution (which, as modeling shows, is conserved for all values of Mn concentration). This feature has not been determined experimentally, as studies are limited to only two cases: xMn = 0 and 20 at.% [15,26,27]. Other than the serendipitous finding while examining this 5-element system theoretically, there is no reason why any experimental work should or could be intended to find such unexpected anomaly. The fact that it is predicted by the calculations, suggests that as the concentration of Mn increases, processes are set in motion that manifest themselves via one very important macroscopic property of the HEA. The modeling shows that this can be explained with subtle changes in the atomic distribution within the solid solution. As such, and somewhat regardless of whether this feature is desired or not for any specific application, this test case constitutes an example of how simple atomistic modeling could help the alloy design process by minimizing the costly and lengthy experimental work. This test system, CoCrFeNiMn, besides the reasons listed above, is particularly suited for this study, as several previous studies provide an experimental basis for validation of the parameterization of the modeling tool. It is also interesting to see that while changes on the concentration of the variable element (in this case, Mn) may lead to rather noticeable changes in the melting temperature, these are not reflected in other properties. This could be a point of interest in the process of HEA design, as it provides a path to follow in the otherwise blind search for additions that will not alter the basic phase structure of the alloy but focus on desired changes in specific properties. Regarding the modeling tool used in this work, the BozzoloFerrante-Smith (BFS) method for alloys [29,30] (using parameters determined via first principles calculations [31]), this analysis may prove it to be a versatile tool for dealing with such complex systems, as it has been shown that it is equally accurate for simple binary systems as well as for multicomponent alloys. As a proof of concept, the BFS method has been applied to ternary, quaternary, and quinary alloys [33,37–40], and it has been also used to predict

complex phenomena such as lanthanide migration in nuclear fuels, involving as many as 15 elements [41]. 2. The BFS method for alloys The BFS method is a quantum approximate method suitable for applications to multicomponent systems [29,30]. The method is based on the notion that the energy of formation of a given atomic configuration (with unrestricted number and type of elements) can be defined as the superposition of the individual atomic contribuP tions,
H ¼ ei . Each individual contribution ei consists of a strain energy term,eSi , which accounts for the change in geometry relative to a single monoatomic crystal of the reference atom i, and a chemical energy term, eCi , where every neighbor of the reference atom i is in an equilibrium lattice site of a crystal of species i, but retaining their chemical identity. To completely separate the effect of changes in geometry (strain energy) from changes in chemical composition (chemical energy), a reference term, eCi 0 , is added in the calculation of the chemical energy, computed in the same way as eCi , but where the neighbors of the reference atom have the same identity as the reference atom. A coupling function, g i , ensures the correct volume dependence of the chemical energy contribution (i.e., the chemical energy vanishes at large interatomic distances). The net contribution ei to the total energy of formation is then

ei ¼ eSi þ g i ðeCi  eCi 0 Þ

ð1Þ

The parameters needed for the calculation of the different contributions are easily determined using the Linearized Augmented Plane Wave method [31]. The single element parameters are obtained from the zero temperature equation of state of the pure fcc solids, while the interaction parameters needed for the calculation of the chemical energy are obtained from the energy of formation as a function of volume curves for each and every one of the fcc-based binary combinations of all the elements. A full description of the steps needed to compute the different terms in Eq. (1), as well as the parameters, can be found in Ref. [29]. The bulk modulus can be computed from the energy of formation as a function of volume. Using the expression for the universal binding energy relationship (UBER) [32], the scaling length l, which describes the curvature of the curve at equilibrium, can be computed and related to the bulk modulus as

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ea l¼ 12pB0 r WSE

ð2Þ

where Ea is the cohesive energy per atom, B0 is the bulk modulus, and rWSE is the equilibrium Wigner-Seitz radius. Ea , r WSE and l can be obtained by fitting the energy of formation as a function of volume to the universal expression. Finally, temperature effects are obtained from large scale Monte Carlo simulations, described in detail in Ref. [33]. For the coefficient of thermal expansion (CTE), we implemented the approach described and illustrated in [34]. We refer the reader to Ref. [34], as a full description of the methodology and the operational equations proves to be too lengthy for this paper. The heat capacity, C v , was computed using the expression in Ref. [35].

Cv ¼

aB0 V c

ð3Þ

where a is the CTE, B0 is the bulk modulus, V is the atomic volume, and c is Grunesein’s constant. This can be written in terms of previously computed quantities [36], leading to

Cv ¼

Ea aV 4:56p r 2WSE l

ð4Þ

M.A. Gutierrez et al. / Computational Materials Science 148 (2018) 69–75

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The melting temperature, T m , is determined from the known concept that the inflexion point in the UBER of the alloy is a measure of the thermal energy necessary for melting, as described in Ref. [36], leading to the simple expression

Tm ¼

0:032Ea kB

ð5Þ

where kB is Boltzmann’s constant. Previous relevant applications of the BFS method to varied multicomponent systems can be found in Refs. [38–41]. It should be noted that all calculations of the multicomponent systems in this work are performed in rigid cells (i.e., uniform lattice parameter) but, for each system, the lattice parameter is optimized at every temperature. This restriction is hardly relevant if the system evolves into a solid solution or does not form several phases. 3. Results and discussion Simulations were performed in several Co-Cr-Fe-Ni-Mn computational cells, containing 3200 atoms and with varying Mn concentration (0 < xMn < 25 at.%), using a descending temperature cycle that ends at room temperature. The cells remained disordered for all cases (i.e., all Mn concentrations) and for all temperatures. In spite of the severe restriction of using an underlying fcc lattice in the simulations, no hints were found that such constraint elicits the formation of a second phase, whether it is an ordered multicomponent precipitate or segregation of any particular atomic species. This is apparent in Fig. 1, which shows a snapshot of the computational cell for T = 300 K. Optimized at each temperature step by letting the cell expand or contract isotropically, it is observed that the lattice parameter increases linearly with increasing temperature, with a small and slightly increasing gap between the two extreme cases (xMn = 0 and 20 at.%), as shown in Fig. 2. These values are in excellent agreement (2% difference) with experimental results at room temperature [1,26] both for assolidified and annealed equiatomic CoCrFeNi and CoCrFeNiMn alloys. The temperature dependence of the thermal expansion shows, particularly for lower temperatures, excellent agreement with experimental results (15% difference). Fig. 3.a displays the BFS-based results for the lattice expansion and experimental results for CoCrFeNiMn and also recent ab initio results, which overestimate the experimental values by up to 60% [3]. Fig. 3.b displays the theoretical results for the specific heat, C v , for which no experimental results exist. Both T m and C v were computed following the guidelines in Ref. [38]. These comparisons between experiment and modeling results are not surprising, given the known disordered structure of the experimental case, reproduced in the simulation. The computed value of the melting temperature of CoCrFeNiMn, T m = 1679 K, is in good agreement with available experimental measurements (1543 K [26], 1607 K [15,27]). This is true also for the CoCrFeNi alloy, where the experimental value, 1717 K [27], is in excellent agreement with the computed value of 1707 K. These comparisons raise the necessary confidence on the ability of the theoretical tool to determine this property. It is found, however, that while other bulk and thermal properties follow a uniform behavior as a function of Mn concentration, the melting temperature does not, exhibiting a clear maximum for xMn = 8.7 at.%, as shown in Fig. 4. This suggests that in spite of being a disordered solid solution for every concentration, there might be subtle changes that do not translate into the formation of second phases but that do have some effect on other properties. In this case, although it is not a huge change, it is then possible to obtain a Co-Cr-Fe-Ni-Mn alloy that has a melting temperature higher than the maximum pure value (i.e., equiatomic CoCrFeNi).

Fig. 1. Computational cell for T = 300 K and 20 at.% of Mn. Cr, Mn, Fe, Ni and Co are represented by spheres with decreasing shades of grey (black for Cr and white for Co).

The process can be explained in terms of the short-range order correlation matrices, rAB and sAB, that denote the probability that an A atom has a B atom as a nearest-neighbor (rAB) or as a nextnearest-neighbor (sAB), as a function of Mn concentration. Fig. 5 displays the behavior of some of these matrix elements as a function of Mn concentration for T = 300 K. The simulations performed for all CoCrFeNiMn alloys (0 < xMn < 25 at.%) corroborate the existence of a disordered solid solution,

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3,60

202

Bulk modulus (GPa)

3,56

3,54

3,52

1710

198 196

1700

194

1690

192

188 186

3,50

184

0

200

400

600

800

1000

1200

1400

1660

10

15

20

25

Mn concentration (at%)

Fig. 2. Lattice parameter (in Å) for equiatomic CoCrFeNi and CoCrFeNiMn alloys as a function of temperature.

but they also indicate that in spite of the lack of long or short range order, there are subtle changes in the atomic distribution that aggregate and manifest themselves as a noticeable change of a macroscopic property. The evolution with xMn of the short-range order parameters indicates that as the concentration of Mn increases, at equal expense of the other four elements, there is a distinction that separates the behavior of Co and Ni from that of Fe and Cr. For low xMn, Mn has a high probability of having Ni as a nearest-neighbor (NN) and nearly none of having Co as one. This is suddenly reversed at precisely xMn = 8.7 at.%, when a replacement process starts and Mn-Co NN pairs become steadily more likely to exist, as Mn-Ni pairs become correspondingly less likely. No such drastic change is observed in the probability of Mn-Fe and Mn-Cr NN pairs, making Fe and Cr inert to this process. This is supplemented with a decreasing probability for the existence of Co-Ni NN pairs. The process can be described as if the disordered solid solution was actually the superposition of two different domains that evolve as the concentration of Mn increases, by means of diffusion from one to the other of mostly Co and Ni (with minor displacement of Cr and Fe). The evolution of rMnX with xMn, shown in Fig. 5, indicates that for low xMn, both rMnCo and sMnCo are nearly zero, rMnFe (and sMnFe) and rMnCr (and sMnCr) remain relatively constant, and rMnNi is very high and stable. This means that for low xMn, Mn tends to locate itself in a CrFeNi domain, void of Co. However, in the vicinity of xMn = 9 at.%, a sudden change takes place. Beyond this critical value

Fig. 4. Modeling results for the melting temperature Tm (in K) and bulk modulus B0 (in GPa) for different temperatures, as a function of Mn concentration (in at.%) for CoCrFeNiMn alloys.

of xMn, rMnCo starts to increase steadily, at the expense of rMnNi. This substitution of Ni for Co in the nearest-neighbor layer is accompanied by a slight decrease in sMnFe and sMnCr, and a corresponding increase in sMnCo and sMnMn, indicating that Mn populates one specific sublattice while Co triggers an outflow of Ni from the domain where Mn originally resides. The sudden transition in behavior can be traced back to the preference for Mn-Co bonds over Mn-Ni bonds. No evidence can be found to show that Fe or Cr have any role in this sudden change. The lack of Mn-Co bonds for low xMn also indicates that there must be a Co-rich domain elsewhere. The correlation matrix elements rCoX and sCoX show that such domain is of the type (disordered) Co3(CrFeNi) (for low xMn), which evolves to CoCrFeNi (for high xMn). It is then clear that increasing Mn concentration is not, by itself, the reason for the increase in melting temperature. If this was the case, then it should grow steadily for all concentrations. It does so, but only up to the critical value of 8.7 at.% Mn, when the Co M Ni interdiffusion process is triggered between the different regions of the alloy. Once this process starts, the melting temperature starts decreasing to levels that are even lower than that of the quaternary CoCrFeNi alloy. Therefore, if the goal of Mn addition to CoCrFeNi was to enhance the chance of a disordered solid solution while also increasing the melting temperature of the alloy, this example shows that a fully disordered and homogenous solid solution is not necessarily fully achieved and that sluggish diffusion [15] processes triggered by the Mn addition could easily revert and even negate the original goal.

0,0104

a)

b)

experiment [3] this work ab initio theory [18]

0,0102

0,0100

Cv (eV/K)

(a-a0)/a0*103

5

1600

T(K)

20

1670

1650 0

3,48

1680

300 K 500 K 1000K 1300 K 1600 K Melting Temperature

190

Melting temperature (K)

3,58

Lattice parameter (Å)

1720

200

10

0,0098

0,0096

0 300

0,0094 600

900

T(K)

1200

0

5

10

15

20

25

Mn concentration (at%)

Fig. 3. (a) Temperature dependence of the thermal expansion from experimental (open circles), ab initio (triangles) and BFS results (solid squares) for CoCrFeNiMn. (b) Specific heat as a function of Mn concentration for T = 300 K.

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0,4 0,0

Chemical energy (eV/atom)

The simple formulation of the BFS method in terms of strain and chemical energy sheds some light on the observed process, beyond the standpoint of the correlation matrices. As mentioned in Section 2, the contribution of each atom to the total energy of the alloy is divided into two terms: the strain energy, which describes structural effects, and the chemical energy, which describes the interaction between atoms of different species. Both terms can be instrumental in deciding the final atomic distribution: atoms will locate themselves so as to minimize DH, with a balance between the always positive strain energy, and lowering the chemical energy, which plays a role in favoring specific pairs of atoms in neighboring sites. Fig. 6 shows the behavior of the chemical energy. The total strain energy is always small and positive and it increases monotonously with increasing Mn concentration. For this particular alloy, however, the behavior of the chemical energy (per atomic species and as a function of Mn concentration) is quite different, as two very distinct regimes can be observed. For low Mn concentration there is a rapid decrease in chemical energy with increasing Mn,

-0,8

Co Cr Fe Ni Mn

-1,2 -1,6 -2,0 -2,4 -2,8 -3,2 0

5

10

15

20

Fig. 6. Chemical energy for each atomic species as a function of Mn concentration.

35

Cr Co Fe Ni Mn

70 60

30 25

50

r(MnX)

20

40 30

15 10

20

Cr Co Fe Ni Mn

5

10 0 0 0

5

10

15

20

0

25

5

10

Cr Co Fe Ni Mn

20

r(NiX)

25

20 15

10

5

5

0

10

15

20

Cr Co Fe Ni Mn

15

10

5

25

30

25

0

20

35

0

25

0

5

10

Mn concentration (at%)

15

30 25

Cr Co Fe Ni Mn

20 15 10 5

0

20

Mn concentration (at%)

35

s(CrX)

r(FeX)

30

15

Mn concentration (at%)

Mn concentration (at%)

35

5

10

25

Mn concentration (at%)

80

r(CoX)

-0,4

15

20

25

Mn concentration (at%) Fig. 5. Short-range order correlations as a function of Mn concentration for T = 300 K.

25

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M.A. Gutierrez et al. / Computational Materials Science 148 (2018) 69–75

indicating that the growing presence of Mn triggers rearrangement of atoms within the lattice that, while paying a small energy price by increasing their strain energy contribution, it is handily offset by a substantial decrease in chemical energy, thus minimizing the total energy of the alloy. Such rapid decrease in chemical energy slows down significantly as the concentration of Mn approaches the aforementioned critical value (8.7 at.% Mn). Beyond this point, any further (minor) decrease in chemical energy is compensated by a corresponding increase in strain energy, leading to a rather stable alloy in terms of its atomic distribution. It is clear then that introducing Mn leads to a rapid redistribution of the different atomic species in the alloy, so as to favor the formation of specific pairs, magnified by the fact that its microstructure is not an ideal disordered solid solution. The chemical energy contribution from atoms in different A-B pairs is by no means comparable: pairs such as Cr-Mn, Cr-Fe, and Fe-Mn contribute very little to the total chemical energy of the alloy. Their contribution is not just small, but also relatively constant as a function of Mn concentration, indicating that they have no role in the diffusion process induced by the presence of Mn. The main contribution comes from Co-Ni pairs, which is larger as xMn increases, but becomes stable once xMn exceeds its critical value. This indicates that the Co M Ni interdiffusion ends at that point, as there are no more energy gains to be realized by further accommodation of Co or Ni in different regions of the alloy (i.e., closer or not to Mn-rich regions). These results are consistent with the description of the system provided by the correlation matrices, and indicate that different elements have different roles and influence in the ensuing process triggered by increasing amounts of Mn. As described, Cr and Fe play only a secondary role in the change of Tm. These elements are needed to facilitate disorder within the alloy and sustain it as Mn is added. They also prove to be somehow inert to the presence of Mn. But the original quaternary alloy is not a fully realized solid solution and imbalances exist within the system. Therefore, the addition of Mn, while contributing favorably to the configurational entropy that promotes a homogeneous solid solution, capitalizes on such imbalances by inducing a rearrangement of the different atomic species that manifests itself prominently in one specific property of the alloy (in this case, the melting temperature) without a major (if any) effect on the others. This simple analysis sheds some light on the otherwise unpredictable nature of these changes, as it is possible to identify the mechanisms needed to fine-tune changes in specific properties of a given HEA without having to go through the costly process of investigating all possible options experimentally.

4. Conclusions While the CoCrFeNi system has long been used as a model highentropy alloy, relatively little is known about the change in properties when a fifth element is added. This work shows how a theoretical/modeling study of the quinary CoCrFeNiMn system singles out an interesting behavior due to subtle changes in the atomic distribution of all five elements in the solid solution as the concentration of Mn changes, resulting into a critical Mn concentration (8.7 at.%) for which the melting temperature of the alloy is maximized, a fact that could be missed in a traditional alloy design program restricted to a limited number of concentrations. While this result awaits experimental verification, the modeling technique allows for a quick and comprehensive description of the process in terms of statistical factors such as the nearest-neighbor and next-nearest-neighbor probability distributions, as well as energy concepts. With no limitations on the number or type of elements that could be included in the analysis, it is possible that the combined experimental and modeling approach could lead to further

understanding of high entropy alloys and facilitate the analysis of more complex and interesting systems. Acknowledgements Fruitful discussions acknowledged.

with

N.

Bozzolo

are

gratefully

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