**Entropy Determination of Single-Phase High Entropy Alloys with Different Crystal Structures over a Wide Temperature Range**

#### **Sebastian Haas 1, Mike Mosbacher 1, Oleg N. Senkov 2, Michael Feuerbacher 3, Jens Freudenberger 4,5, Senol Gezgin 6, Rainer Völkl <sup>1</sup> and Uwe Glatzel 1,\***


Received: 31 July 2018; Accepted: 20 August 2018; Published: 30 August 2018

**Abstract:** We determined the entropy of high entropy alloys by investigating single-crystalline nickel and five high entropy alloys: two fcc-alloys, two bcc-alloys and one hcp-alloy. Since the configurational entropy of these single-phase alloys differs from alloys using a base element, it is important to quantify the entropy. Using differential scanning calorimetry, cp-measurements are carried out from −170 ◦C to the materials' solidus temperatures TS. From these experiments, we determined the thermal entropy and compared it to the configurational entropy for each of the studied alloys. We applied the rule of mixture to predict molar heat capacities of the alloys at room temperature, which were in good agreement with the Dulong-Petit law. The molar heat capacity of the studied alloys was about three times the universal gas constant, hence the thermal entropy was the major contribution to total entropy. The configurational entropy, due to the chemical composition and number of components, contributes less on the absolute scale. Thermal entropy has approximately equal values for all alloys tested by DSC, while the crystal structure shows a small effect in their order. Finally, the contributions of entropy and enthalpy to the Gibbs free energy was calculated and examined and it was found that the stabilization of the solid solution phase in high entropy alloys was mostly caused by increased configurational entropy.

**Keywords:** HEA; entropy; multicomponent; differential scanning calorimetry (DSC); specific heat

#### **1. Introduction**

The conventional strategy of alloy design is based on the selection of one base element for the primary properties of a material, e.g., iron in steels or nickel in nickel-based superalloys. This base element dominates the chemical composition, usually representing more than 80 at. %, and relatively small amounts of other elements are added to modify the alloys' properties [1–3]. Thus the regions next to a single element of multicomponent phase diagrams have been well investigated in the past.

A novel way of alloy design concentrates on the unexplored centers of phase diagrams, where the alloys consist of elements in near equiatomic ratios. Such an alloy was pointed out by Cantor et al. [1] in 2004, when the equiatomic system of Cr, Mn, Fe, Co and Ni was found to result in a single-phase material. This simple face-centered cubic microstructure is free of any precipitates and stable over a wide temperature range [4]. Merely after long time annealing for 500 h at intermediate temperatures

precipitations may segregate, e.g., a Cr-rich phase at 700 ◦C and three different phases (NiMn-rich, FeCo-rich, Cr-rich) at 500 ◦C [5]. However, a single-phase microstructure is the key factor for a high configurational entropy. In the same year Yeh et al. [3] pointed out the concept of high entropy alloys, independent of Cantor's work. His definition of high entropy alloys is to consist of at least five or more elements with concentrations of each element between 5 and 35 at. % [6].

Many elements, and therefore a high value of configurational entropy, can lead to a more stable solid-solution phase with randomly distributed atoms [3]. A solid-solution phase with statistically distributed atoms in the crystal lattice is claimed to lead to interesting and outstanding properties, e.g., high hardness, wear resistance, high temperature strength and stability, sluggish diffusion, oxidation and corrosion resistance [6,7].

Yeh et al. [8] named four core effects, which are characteristic for microstructures and properties of high entropy alloys: the formation of one random solid-solution phase to reach a high entropy effect [8]; severe lattice distortion in the random solid-solution [9]; sluggish diffusion kinetics [6,10]; the so called "cocktail effect" [8].

Nevertheless the major part of investigated alloys with compositional requirements of high entropy alloys do not form single solid-solutions, but consist of several, mostly intermetallic phases, which can be brittle, difficult to process. This observation particularly disagrees with the crucial issue of a single-phase microstructure.

Numerous examinations deal with the prediction of the conditions when a solid-solution phase is stable or additional intermetallic compounds are forming [7,10–12], but no reliable approaches have yet been proposed. Thus we follow the idea to calculate the total Gibbs free energy of an alloy system in single-phase state to compare it with formation enthalpies of several intermetallic compounds. Therefore the determination of thermal enthalpy and especially of entropy over a wide temperature range is necessary. A short insight in thermodynamics and in the way of calculating different parts of the entropy is given in the following part of the introduction:

The terms and definitions of the entropy theory, as well as the basic approach of determinations refer to Gaskell [13]. Changes in Gibbs free energy ΔGtotal of a system at any state and temperature depends on the entropy Stotal and on the enthalpy Hthermal (see Equation (1)). The consideration of both thermochemical parameters leads to a description of the equilibrium state of an alloy system by minimization of the Gibbs free energy at a fixed temperature:

$$
\Delta \mathbf{G}\_{\text{total}} = \Delta \mathbf{H}\_{\text{thermal}} - \mathbf{T} \Delta \mathbf{S}\_{\text{total}} \tag{1}
$$

The entropy of an alloy consists of the configurational entropy Sconf [7] and the thermal entropy Sthermal [13]:

$$\mathbf{S}\_{\text{total}} = \mathbf{S}\_{\text{conf}} + \mathbf{S}\_{\text{thermal}} \tag{2}$$

$$
\Delta \mathbf{S}\_{\rm conf} = -\mathbf{R} \cdot \sum\_{i=1}^{n} \mathbf{x}\_{i} \cdot \ln \mathbf{x}\_{i\nu} \tag{3}
$$

The calculation of configurational entropy, also called mixing entropy, is given in Equation (3) with n as the number of elements, xi the concentration of each element i and R as the universal gas constant. The equation is derived from the mixing of noble gases and is adopted to fully disordered solid solution, which are assumed in our work. If elements are distributed non-equally between possible sub-lattices, then a more general equation should be used [14]. In case of an equiatomic alloy, Equation (3) is reduced to ΔSconf = R ln n. Thus the configurational entropy of 5-component equiatomic alloy is ~1.6·R. At 0 K the resulting total entropy may not be zero. This does not violate the third law of thermodynamics ( dS dT <sup>T</sup>=<sup>0</sup> <sup>=</sup> 0) which is often misinterpreted as S(T = 0) = 0. Crystalline solids may exhibit a non-zero entropy at the absolute zero point due to a randomly crystallographic orientation. A change of entropy in this point is not possible, because there is no ability of motion or diffusion.

The thermal entropy Sthermal can be directly determined by measuring the temperature dependent heat capacity at constant pressure cp(T) by differential scanning calorimetry (DSC):

$$
\Delta \mathbf{S}\_{\text{thermal}}(\mathbf{T}) = \int\_0^\mathbf{T} \mathbf{c}\_\mathbf{P}(\mathbf{T}) \frac{1}{\mathbf{T}} \mathbf{d} \mathbf{T},\tag{4}
$$

Next to the entropy, there is an enthalpy-contribution to the Gibbs free energy, mentioned in Equation (1). This thermal enthalpy can be calculated using Equation (5):

$$
\Delta \mathbf{H}\_{\text{thermal}}(\mathbf{T}) = \int\_0^\mathbf{T} \mathbf{c}\_\mathbf{P}(\mathbf{T}) \, \mathbf{d} \mathbf{T}, \tag{5}
$$

Therefore, the total change in Gibbs free energy at a certain temperature is presented in the following Equation (6). Setting the value of configurational entropy to zero, just the thermal entropy is considered and the result is the change in thermal Gibbs free energy:

$$
\Delta \mathbf{G}\_{\text{total}} = \Delta \mathbf{H}\_{\text{thermal}} - \mathbf{T} \Delta \mathbf{S}\_{\text{total}} = \int\_0^\mathbf{T} \mathbf{c}\_\mathbf{p}(\mathbf{T}) \mathbf{d} \mathbf{T} - \mathbf{T} \left( \Delta \mathbf{S}\_{\text{conf}} + \int\_0^\mathbf{T} \mathbf{c}\_\mathbf{p}(\mathbf{T}) \frac{1}{\mathbf{T}} \mathbf{d} \mathbf{T} \right), \tag{6}
$$

A schematic drawing of cp from 0 K to temperatures in the range of incipient melting after reaching the solidus temperature TS is given in Figure 1a. For lower temperatures cp(T) can be extrapolated down to 0 K following the Debye T3-law [13]. For temperatures close to room temperature (RT) the heat capacity is close to 3R if there is no change in magnetic behavior and no phase transformation. This is known as the Dulong-Petit law, which states that every solid that consists of N atoms has 3N modes of vibration (corresponding to the freedom of motion in three dimensions). Energetic considerations using the equipartition theorem lead to cp = 3·R for sufficiently high temperatures (RT). DSC measurements are carried out until the solidus temperature is reached and the cp-value rises dramatically. In Figure 1b, the calculated thermal entropy and total entropy are schematically illustrated. The thermal entropy is shifted vertically by the configurational entropy over the whole temperature range, in case there are no contributions by phase changes. This is due to the low influence of various alloys on the values of thermal entropy, while the configurational entropy has a high impact. The vertical offset between thermal and total entropy, caused by the alloys' configuration, is still holding on after melting in the liquid state.

**Figure 1.** Schematic drawing (**a**) of the heat capacity of an ideal metal from 0 Kelvin to the liquid state with no change in magnetic behavior and no phase transition over the whole temperature range and (**b**) of the calculated thermal entropy and total entropy from 0 K to the liquid state.

#### **2. Materials and Methods**

The temperature dependent molar heat capacity cp(T) of the specimen was experimentally determined in alumina crucibles using differential scanning calorimetry (DSC 204, Netzsch, Selb, Germany) in the temperature range from −170 ◦C to 600 ◦C under a flushing flow of nitrogen. In this temperature range oxidation of the samples is not critical. For T = 20 ◦C − TS (if TS < 1600 ◦C) samples were measured using a Netzsch DSC 404 F1 Pegasus under an Ar 5.0 gas flush with a rate of 70 ml/min in crucibles composed of 80% Pt and 20% Rh. The crucibles are lined with ceramic inlays to prevent interaction of the metallic specimens with the Pt-Rh crucible. The calibration of temperature and enthalpy of the two Netzsch devices was performed using the calibration set 6.239.2–91.3 under the conditions of 10 K/min and a nitrogen flow of 40 mL/min. The two measurements were evaluated for each material and the curves were connected in the common temperature range from room temperature to 600 ◦C.

Single-crystalline (SX) nickel and SX-Cantor alloy (favored equiatomic composition of Cr, Mn, Fe, Co, Ni) were cast using a proprietary Bridgman investment casting furnace. Cylindrical specimens (Ø 5 mm, height 1 mm) were then cut out of the rods by electrical discharge machining (EDM). All other alloys were in poly-crystalline state. The bcc-alloys were provided by Senkov et al. [15], the noble metal alloy by Freudenberger et al. [16] and the hcp-alloy by Feuerbacher et al. [17]. The recast layer of the EDM samples was removed by etching and the base of all samples was finely ground with SiC paper up to 2000 grit to ensure good thermal contact to the DSC-sensor. All materials tested several times showed good reproducibility in their DSC-signal, even at different heating rates of 10 K/min and 20 K/min respectively.

Except for pure nickel all materials belong to the high entropy alloys group. They form a single-phase microstructure and contain at least four elements in desired, near equiatomic composition. The single-phase solid-solution has been confirmed by the authors using X-ray diffraction experiments. Homogenization of the samples was reached by very slow cooling rates after melting, except of fcc-noble metal that was annealed for 24 h at 1000 ◦C and bcc 5-component was annealed for 24 h at 1200 ◦C.

Although the ROM may only represent a very rough estimate of the melting temperature of the alloys, it has been applied for the bcc-5 component and hcp-alloys. The reasons are twofold: (i) the melting temperature of the bcc-5 component alloy cannot be experimentally determined with our set-up and (ii) we are facing significant reactions of the metals with the crucible during investigation and, therefore, the determined values would not reflect the samples under investigation.

The Mn content of 11 at. % in the Cantor alloy is due to the single-crystal investment casting process. Mn evaporates from the melt in the vacuum of 10−<sup>2</sup> Pa of the Bridgman furnace during the slow withdrawal of the single-crystal.

#### **3. Results and Discussion**

The specific heat capacity in units of J/(g·K) has been converted into the molar heat capacity in units of J/(mol·K) by multiplication with the molar mass of the alloys corresponding to their chemical composition. Figure 2 shows the molar heat capacity of all materials as a function of temperature. For temperatures below −<sup>170</sup> ◦C, the curves were extrapolated using the fit-function a·T3 (parameter a is listed in Table 1) and cp(0 K) = 0.

**Table 1.** Values of the parameter a in J/mol·K4 in the fit-function a·T3.


**Figure 2.** Molar heat capacities of pure nickel and five high entropy alloys. Green crosses indicate TS, red lines show the beginning of oxidation or reactions with the crucible that make the data not evaluable.

**Table 2.** Sample labelling, crystal specification, chemical composition (measured by μ-XRF) and solidus temperature. In case of pure Ni, TS is equal to the melting temperature Tm. The rule of mixture (ROM) is used for calculation, if no literature data is available. All alloys are in a single-phase state over the entire temperature region.


The two single-crystal fcc-materials, Nickel and Cantor alloy, show a steady increase in cp until shortly before their melting points. Both curves are very close to each other with the exception of the Curie-peak in Ni at around 356 ◦C, which lies in excellent agreement with literature data (e.g., 354 ◦C [20]). The Cantor alloy shows a broad plateau-like peak between 600 ◦C and 800 ◦C. This peak is most likely caused by a change in magnetic behavior. Jin et al. [21] have shown that Co and Fe shift the Curie temperature of Ni to higher temperatures and their investigations on Cantor alloy show the exact same plateau-like peak starting at around 600 ◦C. Incipient melting at the solidus temperatures is indicated by excessive jumps in the curves. The solidus temperatures of fcc-nickel, fcc-Cantor and fcc-noble do not correspond to the values listed in Table 2, but differ by about 30 K in case of fcc-Cantor and fcc-nickel and by about 70 K in case of fcc-noble. The other three alloys show decreasing specific heat capacities at elevated temperatures, which indicates reactions with the atmosphere and/or the crucible. High chemical interaction and oxidation with the ceramic liners of the crucibles occur with the alloys Hf-Mo-Nb-Ti-Zr and Ho-Dy-Y-Gd-Tb, so their curves are finally cut-off at 700 ◦C and 900 ◦C respectively. Melting intervals, starting from the solidus temperatures, are not evaluable for the bcc-5 component and hcp alloy. In case of the bcc-4 component alloy the curve mistakenly suggests the solidus temperature to be at about 1520 ◦C. However, Senkov et al. [15] expect a melting temperature of about 2904 ◦C, using the rule of mixture. This issue is likely to be due to chemical reactions with the Pt-Rh crucible and therefore we have to regard the values of the heat capacity of bcc-4 component for

higher temperatures with caution. For further calculations concerning entropy, enthalpy and Gibbs free energy, the data of bcc- and hcp-alloys respectively are just used in the temperature range until the red lines in Figure 2.

Table 2 gives an overview of all tested samples, their crystal structures, chemical compositions, configurational entropies and solidus temperatures. Note that in case of bcc-5 component and the hcp alloy, there was no literature data concerning solidus temperature available. Therefore, the possible solidus temperatures were calculated using the rule of mixture (ROM).

Obvious shifts in the trend of cp indicate phase transitions or changes in the magnetic behavior, e.g., the Curie point of nickel at 356 ◦C as mentioned above. The hcp-alloy Ho-Dy-Y-Gd-Tb shows a very pronounced peak at −120 ◦C, probably due to a not investigated magnetic phase transformation, because oxidation seems to be very unlikely in such a low temperature range.

Of particular interest are the heat capacities at room temperature, where all curves seem to approach a cp value close to 25 J/(mol·K) = 3R (see Figure 3a–d), according to the Dulong-Petit-law [13]. The bcc-5 component equiatomic alloy, Hf-Mo-Nb-Ti-Zr, shows the lowest molar heat capacity at room temperature with about 23.5 J/(mol·K) (Figure 3b), the hcp-alloy Ho-Dy-Y-Gd-Tb shows the highest value at about 30.6 J/(mol·K) (Figure 3d). The high value of the molar heat capacity of the hcp-alloy might originate from magnetic ordering of the 4f electrons. This is likely the case as pure Gd shows ferromagnetic ordering below 19.9 ◦C. However, this suggestion needs to be verified in future studies. The other four alloys show RT cp-values pretty close to 25 J/(mol·K). We applied the rule of mixture (ROM) to predict molar heat capacities of the alloys and their average molar mass (see Equation (7)). The calculated data is in very good agreement with the experimental results and shows a maximum deviation of 6%:

$$\mathbf{c}\_{\text{p,alloy}} = \frac{1}{\mathbf{n}} \cdot \sum\_{i=1}^{\mathbf{n}} \mathbf{c}\_{\text{p}i} \quad \mathbf{u}\_{\text{alloy}} = \frac{1}{\mathbf{n}} \cdot \sum\_{i=1}^{\mathbf{n}} \mathbf{u}\_{\text{i}\prime} \tag{7}$$

**Figure 3.** Molar heat capacities at room temperature of the investigated alloys, both experimental and rule of mixture (ROM) data for (**a**) fcc-Cantor, (**b**) both bcc-alloys, (**c**) fcc-noble and (**d**) the hcp-alloy. Values for pure elements were taken from [20]. The red line indicates the Dulong-Petit law with cp(RT) ≈ 3·R.

Figure 4 shows the thermal entropy Sthermal determined with Equation (4). Sthermal is equial zero at −273 ◦C andincreases continuously with increasing temperature having different slopes for different alloys. The curves are drawn till shortly before the solidus temperature of the alloys, except for Ho-Dy-Y-Gd-Tb, Hf-Mo-Nb-Ti-Zr and Mo-Nb-Ta-W where chemical reactions with the crucible/environment make the high temperature regions inaccessible. The highest thermal entropy appears for the hcp-alloy, caused by the

highest heat capacity at low temperatures and additionally the early peak in the cp-curve at about −120 ◦C. The fcc-nickel, fcc-Cantor alloy and fcc noble metal alloy have almost the same temperature dependence of Sthermal, which is noticeably weaker than that of the hcp-alloy. These fcc materials have similar Sthermal values at any given temperature. The two bcc-alloys exhibit the lowest values of thermal entropy.

**Figure 4.** Thermal entropy Sthermal of nickel and the five high entropy alloys.

The total entropy Stotal = Sconf + Sthermal over the whole temperature range starting from absolute zero is plotted in Figure 5. While the curve of nickel starts at 0 J/(mol·K) at absolute zero, the bcc-4 component alloy Mo-Nb-Ta-W exhibits a configurational entropy of R ln(4) = 1.38·R = 11.5 J/(mol·K), because of its four components. All other high entropy alloys, with five different elements show an offset of R ln(5) = 1.6·R = 13.4 J/(mol·K). In theory the configurational entropy is equal for all alloys with the same number and concentrations of elements. No considerations are done with respect tosimilarities between participating atoms, like differences in the atomic radius or crystal structures of the pure metals. In our case, all elements in the hcp-alloy promote a hexagonal close packed crystal structure and exhibit quite equal atomic radii. The bcc-5 component alloy however consists of elements of different crystal structures with larger atomic size differences. Nevertheless, the same value of Sconf is assumed, as stated in the Gibbs paradox [22]. For dislocation movement, however, different atom sizes in solid-solution crystal structures indeed play an important role. Consequently, this mixing paradox will be investigated thoroughly in a future work. It can be seen from Figure 5 that at any given temperature Stotal is the smallest for fcc-Ni, followed by the bcc alloys, then the fcc alloys and being the highest for the hcp alloy.

**Figure 5.** Total entropy as a sum of temperature dependent thermal entropy Sthermal and constant composition dependent configurational entropy Sconf.

The contribution to the Gibbs free energy is the product of entropy and temperature. Using thermal and configurational entropy and their sum, we can examine their contributions to the Gibbs free energy separately. Figure 6a shows the energy contribution of thermal entropy. We can see a clear order with the alloy crystal structure: the highest contribution by thermal entropy is given by the hcp-alloy, mainly because of the high value of molar heat capacity at low temperatures (−120 ◦C) and also at room temperature. Pure nickel, noble and Cantor, all fcc-structures, overlie each other over a wide temperature range. The lowest heat capacities and therefore thermal entropy contributions are given by the bcc (4 and 5 component) alloys. Using the sum of thermal and configurational entropy, the total entropy times the temperature is shown in Figure 6b. Alloys with the same crystal structure and configurational entropy are close to each other. For example, the total entropy input to Gibbs free energy for fcc-Ni is smaller than for fcc Cantor and fcc nobel metal alloys. To explore the difference, two materials with fcc-structure, pure nickel and Cantor alloy, are drawn in Figure 7a in a temperature range from 600 ◦C to 1600 ◦C. Dashed lines show the configurational entropy contribution to Gibbs free energy of both materials, calculated by Equation (3). While the contribution of configurational entropy rises with higher temperatures for fcc-Cantor alloy, fcc-nickel does not exhibit any configurational entropy and, therefore, has no contribution to Gibbs free energy. Continuous lines show the product of the temperature and the total entropy. Thus the influence of configuration and thermal input can be quantified at certain temperatures. At 1000 ◦C the gap between continuous and dashed lines is almost similar, meaning that at this temperature there is a negligible difference in the influence of thermal entropy, but just the chemical composition of the solid-solutions affects the varying energy contributions. The energy level of total entropy contribution is about 90 kJ/mol for fcc-nickel and 108 kJ/mol for fcc-Cantor. The difference of 18.0 kJ/mol is very close to fcc-Cantor configurational entropy contribution of 1273 K·R·ln(5) = 17 kJ/mol (T·Sconf) at this energy level, resulting that thermal entropy has an equal impact for materials with the same crystal structures (in this case fcc) and thus cannot play role in stabilizing a solid solution phase in multicomponent alloys. Figure 7b shows a bigger section of the area near the melting interval of both materials. The two values of melting enthalpy 13.3 kJ/mol and 15.0 kJ/mol have been detected by DSC and it is obvious that differences in entropy contribution to Gibbs free energy still remains dependent from the configurational entropy until the end of the solid state. Investigations on the fcc-noble alloy instead of fcc-Cantor yield to similar results and are, therefore, not shown in Figure 7 in detail.

**Figure 6.** Contribution of thermal entropy (**a**) and total entropy (**b**) of nickel and all high entropy alloys to Gibbs free energy. The horizontal arrows refer to Figure 7 with magnified details of a special temperature range.

**Figure 7.** Contribution of total and configurational entropy of fcc-nickel and fcc-Cantor to Gibbs free energy in the range from (**a**) 600 ◦C to 1600 ◦C and from (**b**) 1100 ◦C to 1600 ◦C.

Gibbs energy has another input besides entropy, namely thermal enthalpy, Hthermal. The temperature dependence of the enthalpy, calculated with Equation (5), is displayed in Figure 8a, while Figure 8b,c show thermal Gibbs free energy and total Gibbs free energy, respectively. Similar to Sthermal, Hthermal slightly depends on the type of the crystal structure having smallest values for the bcc alloys and highest values for the hcp alloy and shows almost no dependence on the number of components and alloy composition (Figure 8a). As a result of such behavior, the thermal part of the Gibbs free energy, ΔGthermal, also does not depend on the number and concentration of the alloying elements, but its temperature dependence is slightly stronger for the hcp alloy and weaker for the bcc alloys relative to the fcc materials (Figure 8b). On the other hand, the total Gibbs free energy, ΔGtotal, which additionally includes the configurational term, has a stronger temperature dependence and thus becomes noticeably smaller at higher temperatures for the alloys with larger configurational entropy (Figure 8c). These observations indicate that, although the contribution of Sthermal to the Gibbs free energy is much higher than that of than Sconfig, at any given temperature ΔGthermal is nearly the same for the simple and complex alloys of the same type of crystal structure, i.e., Sthermal does not play any role in stabilizing a solid solution phase in complex, multicomponent alloys: On the other hand, Sconfig increases with the number of constituents, which noticeably decreases ΔGtotal of a multicomponent solid-solution relative to that of pure metals, especially at high temperatures. Moreover, in some specific cases ΔGtotal of a multicomponent solid-solution with high Sconfig can become smaller than ΔGtotal of competing intermetallic phases, resulting in a single-phase solid solution alloy.

**Figure 8.** Temperature dependence of (**a**) thermal enthalpy, (**b**) thermal part of Gibbs free energy and (**c**) total Gibbs free energy for the studied materials.

#### **4. Conclusions**

In this work the heat capacity, entropy, enthalpy and Gibbs free energy of six different single-phase solid solution alloys (three fcc-alloys, two bcc-alloys and one hcp-alloy) were experimentally determined and investigated over a wide temperature range.


**Author Contributions:** Conceptualization, U.G., R.V., S.H., M.M.; Methodology, S.H., M.M., S.G.; Software, S.G.; Validation, S.H., M.M., O.N.S.; Formal Analysis, S.H., O.N.S.; Investigation, S.H., M.M., O.N.S.; Resources, O.N.S., J.F., M.F., M.M.; Data Curation, S.H., O.N.S.; Writing-Original Draft Preparation, S.H.; Writing-Review & Editing, S.H., M.M., O.N.S., J.F., M.F.; Visualization, S.H., M.M.; Supervision, U.G.

**Funding:** This research was funded by German Research foundation (DFG), i.e., during the projects GL 181/50, GL 181/56, FR 1714/7 and FE 571/4, the three last within the priority programme SPP2006 "Compositionally Complex Alloys—High Entropy Alloys (CCA-HEA)". O.N. Senkov acknowledges financial support through the Air Force on-site Contract FA8650-15-D-5230 managed by UES, Inc., Dayton, OH, USA.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **About the Reliability of CALPHAD Predictions in Multicomponent Systems**

**Stéphane Gorsse 1,2,\* and Oleg N. Senkov <sup>3</sup>**


Received: 19 October 2018; Accepted: 22 November 2018; Published: 24 November 2018

**Abstract:** This study examines one of the limitations of CALPHAD databases when applied to high entropy alloys and complex concentrated alloys. We estimate the level of the thermodynamic description, which is still sufficient to correctly predict thermodynamic properties of quaternary alloy systems, by comparing the results of CALPHAD calculations where quaternary phase space is extrapolated from binary descriptions to those resulting from complete binary and ternary interaction descriptions. Our analysis has shown that the thermodynamic properties of a quaternary alloy can be correctly predicted by direct extrapolation from the respective fully assessed binary systems (i.e., without ternary descriptions) only when (i) the binary miscibility gaps are not present, (ii) binary intermetallic phases are not present or present in a few quantities (i.e., when the system has low density of phase boundaries), and (iii) ternary intermetallic phases are not present. Because the locations of the phase boundaries and possibility of formation of ternary phases are not known when evaluating novel composition space, a higher credibility database is still preferable, while the calculations using lower credibility databases may be questionable and require additional experimental verification. We estimate the level of the thermodynamic description which would be still sufficient to correctly predict thermodynamic properties of quaternary alloy systems. The main factors affecting the accuracy of the thermodynamic predictions in quaternary alloys are identified by comparing the results of CALPHAD calculations where quaternary phase space is extrapolated from binary descriptions to those resulting from ternary system descriptions.

**Keywords:** alloy design; structural metals; multi-principal element alloys; CALPHAD

#### **1. Introduction**

Computational approaches are now a vital aspect of materials science, which enable the prediction of unknown data, such as materials properties and phase stability, and the more rapid exploration of design space. CALPHAD (Calculation of Phase Diagrams) is an efficient computational thermodynamic technique which permits the prediction of the phase equilibria and thermodynamic properties of multicomponent systems from those of the respective binary and ternary subsystems. CALPHAD practices also include the simulation of solidification, mapping of onset driving forces and nucleation barriers for precipitation [1], modeling of phase transformations [2], and estimation of stacking fault energy [3], for example. The development of multicomponent thermodynamic databases caused a growing interest in the calculations of phase diagrams for high entropy alloys (HEAs) and complex concentrated alloys (CCAs) [3–7] allowing the exploration of the composition space to be accelerated [8–10]. In this context, it is of paramount importance to analyze the reliability of the predictions made using commercial dedicated HEA databases.

The reliability of the predictions depends upon the quality of the assessment of the Gibbs energy parameters of the phases stored in the databases. The CALPHAD approach faces new challenges when applied to the development of multi-principal element alloys (HEAs and CCAs) due to large extrapolations from the thermodynamic models which may lead to inaccurate evaluations of the Gibbs energies. Many CALPHAD databases have been developed for traditional alloys so the full thermodynamic assessment of the Gibbs energies is usually given only for compositions enriched with one main element. In contrast, HEAs and CCAs lie in the central regions of compositional space, far from the known boundaries delimited by the binary and ternary subsystems. Consequently, the calculations in these vast composition spaces rely on the availability and quality of the descriptions of the lower order constituent systems and require large extrapolations from these lower order systems by assigning higher order interaction parameters, which are refined using available experimental data. Reliability and accuracy of the predictions increases when using more complete, experimentally verified, thermodynamic descriptions specially designed for CCA compositions (e.g., TCHEA and PanHEA special HEA databases developed by Thermo-Calc Software AB and CompuTherm LLC, respectively). A complete thermodynamic description (full assessment) of binary and ternary systems is generally sufficient to correctly describe thermodynamic properties of higher order (quaternary, quinary, etc.) systems because the probability of the occurrence of quaternary or higher order intermetallic phases is low and decreases rapidly with an increase in the number of components. Therefore, complete thermodynamic description of binary and ternary systems is considered to be sufficient for the correct prediction of thermodynamic properties of higher order systems by extrapolation from the lower order systems. Unfortunately, even the most developed thermodynamic databases lack in the complete thermodynamic description (full assessment) of all binary and, especially, ternary systems based on the elementary components present in the database. For example, the TCHEA3 database includes 26 elements that can form 325 binary and 2600 ternary systems, but only 294 binary and 136 ternary systems are assessed in the full range of composition and temperature. Therefore, complete thermodynamic description is currently unavailable for a huge number of quaternary (or higher order) alloy systems, which may have a potential practical interest. For those reasons, it is very important to know what the minimum level of the thermodynamic assessment of binary and ternary systems is to be sufficient to qualitatively predict equilibrium phases and phase transformations in higher order CCAs.

The goal of this paper is to evaluate the quality of prediction of phases and phase compositions in a quaternary system at different levels of the assessment (thermodynamic description) of the respective ternary systems and a complete description of the respective binary systems. To achieve this goal, we used the current capabilities of a commercially specialized CALPHAD database and identified quaternary alloys systems that have a complete description of their binary and ternary sub-systems. We then calculated ternary and quaternary phase diagrams for these quaternary systems and considered them as "reference diagrams", assuming that they are 100% accurate relative to known experimental data due to a complete thermodynamic description. A parallel coordinate plots method, which is briefly described in the following section, was used to display three-dimensional data related to the extent of phase regions in the quaternary compositional space. Finally, we artificially reduced the levels of the thermodynamic description of the ternary systems, calculated the ternary and quaternary phase diagrams using this incomplete description, and compared the calculations with the reference diagrams. This analysis allowed us to identify conditions when the databases with incomplete thermodynamic descriptions can still be used to reliably predict thermodynamic properties of quaternary systems.

#### **2. Methods**

Senkov et al. [8,9] defined credibility criteria for CALPHAD calculations based on the fraction of fully thermodynamically assessed binary systems (FAB) and the fraction of fully assessed ternary systems (FAT) included in the database. By definition, a fully assessed system is one that has a complete

thermodynamic description (of the respective elements, binary systems and ternary systems) within a given thermodynamic database allowing CALPHAD software to calculate thermodynamic properties for this system, which agree with the experimentally available data within the whole composition and temperature range. Using these theoretical credibility criteria, Wertz et al. [11] have recently demonstrated the importance of (a) the number of assessed constituent binary systems and (b) the compositional distance from them, on the accuracy of the predictions for ternary systems. Their study was carried out by comparing equilibrium calculations of the same ternary systems using eight different databases developed by CompuTherm LLC (Madison, WI, USA). Each of the studied ternary systems had complete thermodynamic descriptions (FAB = 1, FAT = 1) in at least one database, while other databases calculated the ternary space by extrapolation from binary descriptions with different levels of FAB and FAT = 0 (Figure 1a).

**Figure 1.** The possible values of the fraction of assessed binaries (FAB) and fraction of assessed ternaries (FAT) quantifying the fraction of binary/ternary descriptions assessed within a database for ternary and quaternary alloys. FAB = 1 and FAT = 0 represents partially assessed conditions, while FAB = 1 and FAT = 1 represents fully assessed conditions.

Here we expand a similar approach to higher order systems to evaluate the ability to predict quaternary phase diagrams. A quaternary system has 6 constituent binaries and 4 ternaries, so the possible FAB levels are 0, 1/6, 1/3, 1/2, 2/3, 5/6 or 1 for databases that include full thermodynamic descriptions for 0, 1, 2, 3, 4, 5 or 6 of the constituent binary systems, and the FAT values are 0, 1/4, 1/2, 3/4, or 1 for databases that include full thermodynamic descriptions for 0, 1, 2, 3, or 4 of the constituent ternary system (Figure 1b). The commercial database TCHEA3 from Thermo-Calc Software AB (Stockholm, Sweden) used in the present work includes 26 elements, 294 fully assessed binary and 136 fully-assessed ternary descriptions, providing few quaternary systems with FAB = 1 and FAT = 1 (e.g., Co-Cr-Fe-Ni and Co-Cu-Fe-Ni), and FAB = 1 and FAT = 3/4 (e.g., Al-Co-Ni-Ti, Al-Cu-Fe-Ni, Al-Cu-Fe-Ti, Al-Cu-Ni-Ti, Al-Mn-Ni-Ti, Al-Fe-Ni-Ti, Al-Ni-Si-Ti). Among them, we have selected three phase diagram templates for this study:


TCHEA3 is an encrypted database which does not permit access to the values of the Gibbs free energy parameters of the phases. However, it is possible to fix the value of any parameter using the commands available in the Gibbs Energy System (GES) module implemented in Thermo-Calc® software. Using this command for all possible relevant parameters allows a comparison of the predictions for the reference systems, referred to as FAT=1 for Co-Cr-Fe-Ni and Co-Cu-Fe-Ni, and FAT = 3/4 for Al-Co-Ni-Ti (the Co-Ni-Ti ternary subcomponent is only partially assessed in TCHEA3), with the results obtained when (1) setting to zero all the ternary interaction parameters for all the relevant phases (solid solutions and a binary intermetallic having ternary solubility) but retaining ternary intermetallics and (2) additionally suspending all the ternary intermetallics. The scenarios (1) and (2) are referred to as FAT = 0 since they are equivalent to a database in which the ternary systems are not assessed for the three selected quaternary systems (the description of the quaternary relies only on the descriptions of the 6 constituent binaries). We have selected one comparison metric which is the extent of the compositional range of each phase throughout the 3D composition space. These data were retrieved from the phase point coordinates calculated at a fixed temperature for 12,341 equally spaced system points (2.5 at.% compositional steps) throughout the quaternary diagram. It is worth mentioning that our methodology uses the same commercial database for calculation of thermodynamic properties of a quaternary alloy at different levels of assessment of the respective ternary systems, which is achieved by artificial removal of the ternary interaction parameters. At the same time, the binary interaction parameters in the database containing the description of ternary systems can slightly differ from the published binary assessment to better reproduce the ternary phase diagram. However, these modifications are made in such a way that the calculated thermodynamic properties of the respective binary systems still agree with the accepted experimental data. Therefore, setting all ternary terms to zero (FAT=0) can still be treated as a condition with FAB = 1.

As previously discussed by Miracle and Senkov [12], and Wertz et al. [11], visualizing the extension and topology of binary and ternary terminal solid solution phase fields into a higher order dimensional composition space remains difficult because the properties of high dimensional objects diverge our two or three-dimensional intuition. An alternative approach to get an insight into the phase points (representative points of a phase, i.e., single-phase region and its boundaries) in high dimension is to use parallel coordinate plots [3,13]. In such a plot, the axes for the variables (e.g., composition of each alloying element and temperature) lie parallel to each other and span from the minimum to maximum value of each variable. The coordinates of a phase (i.e., phase points) or a phase field with two or more phases in equilibrium (i.e., system points) are visualized by segmented lines connecting each variable (molar composition and temperature), so the number of lines connecting the composition coordinates of a selected phase reflects the extent of that phase in the compositional space. It is then possible to evaluate the extent of a phase (or a phase field) in the n-dimensional space of a phase diagram as the ratio of the number of phase points over the number of system points. The ratio represents the (hyper)-volume fraction occupied by a phase in the n-dimensional composition space.

#### **3. Results and Discussion**

Figure 2a shows an unfolded tetrahedral quaternary phase diagram for the Co-Cr-Fe-Ni system showing the four isothermal sections of the ternary subsystems calculated at 800 ◦C under the condition FAT = 1. The single-phase fields are highlighted by red (*fcc*) green (*bcc*) gray (*hcp*) and blue (sigma) color codes, while the adjacent two- or three-phase regions are left un-colored (white). Figure 2b shows the difference between the extents of the phase fields in the subcomponent ternaries calculated for FAT = 1 and FAT = 0. The phase field regions where the calculations do not match are shown in white. It can be seen that the disagreements in the calculations appear mainly in the Co-Cr-Ni and Cr-Fe-Ni, while perfect match occurs in Co-Cr-Fe and Co-Fe-Ni. The disagreements of the FAT=1 and FAT=0 calculations for Co-Cr-Ni mainly appear in the regions associated with *hcp* and δ phases and at phase-field boundaries. In Cr-Fe-Ni, these entirely occur at the phase-field boundaries. In Figure 2c–d, the parallel coordinate plots show the *fcc* phase points in the Co-Cr-Fe-Ni quaternary system for FAT = 1 and FAT = 0, respectively. There is no significant difference between both predictions. As summarized in Figure 3, the phase points within the *fcc* and *bcc* phase fields calculated using FAT = 0 match 98% of the FAT = 1 predictions. These results suggest that the phase equilibria and locations of the phase boundaries (solubility limits) delimiting the single solid solution regions in the Co-Cr-Fe-Ni quaternary diagram are weakly sensitive to the number of assessed ternary subsystems. In other words, ternary contributions are negligible, and the accuracy remains when extrapolating the Co-Cr-Fe-Ni quaternary system exclusively from the six fully-assessed binary subsystems. Correct prediction of miscibility gap and/or intermetallic phases may however require more complete description of ternary interactions.

**Figure 2.** Co-Cr-Fe-Ni system: (**a**) Unfolded tetrahedral quaternary phase diagram showing the four isothermal sections of the ternary subsystems calculated at 800◦C for FAT = 1, highlighting the single-phase fields. (**b**) Difference (appearing in white) between the extent of the various phase fields calculated for FAT = 1 and FAT = 0. (**c,d**) Parallel coordinate plots showing the predicted range of existence (4D compositional coordinates) of the *fcc* phase in the quaternary Co-Cu-Fe-Ni phase diagram at 800 ◦C for (c) FAT = 1 and (d) FAT = 0.

The effects from canceling ternary interactions are slightly different for the Co-Cu-Fe-Ni (Figure 4). This system contains disordered *fcc* and *bcc* phases and no intermetallic phases (Figure 4a). The calculations with FAT = 0 correctly predict the presence and location of these phases. Perfect match of FAT = 0 and FAT = 1 calculations occurs for the Co-Fe-Ni ternary, similar to the previous case with the Co-Cr-Fe-Ni system. However, the main disagreement between the FAT = 0 and FAT = 1 calculations arises from incorrect predictions of the miscibility gap boundaries for the high-temperature fcc phase in the Co-Cu-Ni and Cu-Fe-Ni ternary subcomponents (Figure 4b) and in the quaternary volume (Figure 4c,d). The absence of ternary interactions incorrectly shifts the miscibility gap boundaries toward higher concentrations of Ni, while the presence of these Ni-associated ternary interactions expands the high-temperature *fcc* phase to higher concentrations of Cu and Fe or Co and Cu (Figure 4b–d). Due to this, only 63% of the *fcc* phase points calculated from FAT = 0 match the FAT = 1 calculations, while almost 100% agreement occurs for the bcc phase fields (Figure 3).

**Figure 3.** Percentage of predictions matching FAT = 1 for the Co-Cr-Fe-Ni and Co-Cu-Fe-Ni quaternary systems, for the *fcc*/L12 and *bcc*/B2 phase points, respectively, when neglecting ternary interactions (FAT = 0). For the Al-Co-Ni-Ti system, the first condition corresponds to a special case of FAT = 0, when all ternary interactions are canceled but the ternary phases are artificially retained, while the second condition corresponds to the actual FAT = 0 condition, when both ternary interactions and ternary phases are suspended.

**Figure 4.** Co-Cu-Fe-Ni system: (**a**) Unfolded tetrahedral quaternary phase diagram showing the four isothermal sections of the ternary subsystems calculated at 800◦C for FAT = 1, highlighting the single-phase fields. (**b**) Difference (appearing in white) between the extent of the various phase fields calculated for FAT = 1 and FAT = 0. Parallel coordinate plots showing the predicted range of existence (4D compositional coordinates) of the *fcc* phase in the quaternary Co-Cu-Fe-Ni phase diagram at 800 ◦C for (**c**) FAT = 1 and (**d**) FAT = 0.

The discrepancies between predictions performed based on ternary descriptions and those without are even more severe for the Al-Co-Ni-Ti system (Figure 5). In addition to the disordered fcc and bcc solid solutions, this system also contains several ordered (L12, B2) binary intermetallic (Al5Co2, Al3Co, Al13Co4, Al9Co2, Al3Ni, Al3Ni2, BCT, Al2Ti, AlTi, AlTi3, Ni3Ti, (Co,Ni)Ti2, Laves-C15 and Laves-C16) and several ternary intermetallic (H\_L21, Laves\_C14) phases (Figure 5a). The presence of many different phase field regions in this quaternary system results in high density of boundaries between these regions, where the main disagreements between FAT = 0 and FAT = 1 calculations occur. In addition, FAT = 0 calculations cannot predict composition regions for ternary intermetallic phases. Instead, they fill these regions with available binary phases (mainly the bcc and B2 phases). This can be well visualized by comparing the bcc and B2 phase points represented in parallel coordinate plots for FAT = 3/4 (Figure 5c) to FAB = 0 (Figure 5d).

**Figure 5.** Al-Co-Ni-Ti system: (**a**) Unfolded tetrahedral quaternary phase diagram showing the four isothermal sections of the ternary subsystems calculated at 800◦C (FAT = 3/4), highlighting the *single-*phase fields. In addition to the solid solution phases, the quaternary system includes several ordered solid solutions (L12, B2), binary intermetallics (Al5Co2, Al3Co, Al13Co4, Al9Co2, Al3Ni, Al3Ni2, BCT, Al2Ti, AlTi, AlTi3, Ni3Ti, NiTi2, Laves-C15 and Laves-C16) and several ternary intermetallics (H\_L21, Laves\_C14). (**b**) Difference (appearing in white) between the extent of the various phase fields calculated for FAT = 3/4 and FAT = 0 (the case when all ternary intermetallics are suspended, see Figure 3). Parallel coordinate plots showing the predicted range of existence (4D compositional coordinates) of the *bcc* phase in the quaternary Al-Co-Ni-Ti phase diagram at 800 ◦C for (**c**) FAT = 3/4 and (**d**) FAT = 0.

The different sensitivities to FAT of the calculated phase diagrams for Co-Cr-Fe-Ni, Co-Cu-Fe-Ni and Al-Co-Ni-Ti systems result from the different thermodynamic properties and topologies of these systems. In all the cases, the number of solid solution phases and solid-solution phase identifications match perfectly between FAT = 1 (or <sup>3</sup> <sup>4</sup> ) and FAT = 0 calculations. The main differences in the calculations occur at or near the boundaries between different phase fields. Therefore, higher density of these boundaries should result in lower accuracy of the FAT = 0 calculations. This observation is in agreement with earlier results by Wertz et al. [11]. The presence of ternary phases in the

ternary systems, which the evaluated higher order system contains, will result in incorrect FAT = 0 calculations, as the extrapolations from the binary interactions cannot predict formation of ternary phases. Unfortunately, the locations of the phase boundaries and possible presence of ternary phases are not known when evaluating novel composition space. Therefore, a higher credibility database is still preferable, while the calculations using lower credibility databases may be questionable and require experimental verification.

The Co-Cr-Fe, Co-Cr-Ni, Co-Fe-Ni and Cr-Fe-Ni behave like near-ideal mixtures of the constituent elements so the Co-Cr-Fe-Ni quaternary does too, which results in the stabilization of the disordered solid solutions and the absence of ternary intermetallic phases. In such cases, neglecting ternary interaction parameters when considering FAT = 0 predictions gives approximatively the same results as for FAT = 1 because their contributions to the Gibbs free energy are also negligible. The Co-Cu-Fe-Ni quaternary system deviates to the near-ideal behavior due to a tendency for the demixing of the Cu-Co and Cu-Fe constituent binaries. In this case, the extent of the miscibility gap is sensitive to the ternary interactions and neglecting them causes incorrect predictions of the phase boundaries involving the solid solutions with the miscibility gap.

In contrast, the Al-Co-Ni-Ti system exhibits several ordered solid solutions and binary and ternary intermetallics. In this case, the most significant factor controlling the location of the phase boundaries arises from the phase assemblage, i.e., the presence and proximity of ternary intermetallics, whereas the removal of the ternary interaction parameters does not change significantly the number of phase points matching the reference predictions when the ternary phases are maintained (Figure 3). This sensitivity depends upon the neighboring phase regions. In particular, the assemblage around the *fcc* and L12 regions consists of various binary and ternary phase fields involving B2, Ni3Ti and not containing ternary intermetallics, so the extent of the *fcc* region is not constrained when ternary intermetallics are removed. In consequence, using an incomplete database (FAT = 0) does not substantially affect the composition and boundaries of the *fcc* and L12 phases. On the other hand, *bcc* and B2 regions are surrounded by several ternary intermetallics (γ and H\_L21), so neglecting them (FAT = 0) shifts significantly the *bcc* and B2 composition and boundaries, thus providing significant disagreement between FAT = 3/4 and FAT = 0 calculations (Figure 3).

#### **4. Conclusions**

The ability of a commercial CALPHAD database for high entropy alloys to predict quaternary phase diagrams has been evaluated through comparisons of predictions made from full binary system descriptions with the results from full ternary system descriptions. This point is of prime interest because even the most comprehensive databases (such as TCHEA3 from Thermo-Calc) include a limited number (about 5%) of fully assessed ternary systems. This entails that quaternary and higher order predictions are systematically obtained from incomplete thermodynamic descriptions and rely almost entirely on extrapolations from binary subsystem descriptions. The analysis has led to the following conclusions.


Considering that quaternary and higher order interactions are negligibly weak, and the formation of novel quaternary intermetallics very unlikely, the present conclusions for quaternary alloys can be extended to higher order systems. Because the locations of the phase boundaries and possibility of formation of ternary phases are not known when evaluating novel composition space, a higher credibility database is still preferable, while the calculations using lower credibility databases may be questionable and require additional experimental verification.

**Author Contributions:** Conceptualization, S.G. and O.N.S.; Methodology, S.G. and O.N.S.; Formal Analysis, S.G. and O.N.S.; Writing-Original Draft Preparation, S.G.; Writing-Review & Editing, S.G. and O.N.S.; Visualization, S.G.

**Funding:** This research received no external funding.

**Acknowledgments:** Work by O.N.S. was supported through the Air Force on-site contract FA8650-15-D-5230 managed by UES, Inc., Dayton, Ohio.

**Conflicts of Interest:** The authors declare no conflicts of interest.

#### **References**


© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

*Article*
