Introduction

The phase stability of an alloy is guided by Gibbs free energy (ΔGmix) that comprises of enthalpy (ΔHmix) and entropy (ΔSmix) of mixing. Conventionally, ΔHmix has been perceived as the dominant quantity, where a strongly negative ΔHmix indicates formation of a stable solid solution and a positive ΔHmix indicates unmixing. Theoretical advancements in alloy theory over the past three decades have enabled computational prediction of correct ordered phases and ground-state crystal structures.1,2,3,4,5,6,7,8,9 These methods have relied on analyzing ΔHmix in identifying stable metal alloys. In the same vein, ΔSmix contributions have also been included in the calculations, albeit selectively; including ΔSmix has shown to improve the accuracy of the predicted thermodynamic quantities, such as the order–disorder phase transformation temperature and miscibility gap temperature.3,4,10,11,12,13,14,15 The contribution of ΔSmix has often been considered less important, largely due to its relatively smaller size compared to ΔHmix, particularly in the prediction of ordered structures. However, in the past decade, with the development of high entropy alloys (HEAs),16,17 it is gradually becoming evident that ΔSmix might be equally important in the phase stability predictions. HEAs are materials that contain random distribution of multiple elements, conventionally five or more, in approximately equal concentrations in a single-phase solid solution. Despite the random distribution of atoms, HEAs are structurally ordered forming simple face centered cubic (fcc) and body centered cubic (bcc) crystal structures.16,18 The stability of a single-phase HEA is due to a relatively shallow negative ΔHmix19,20 among the constituting elements which prevents phase separation and formation of a second sub-phase. The shallow ΔHmix in HEAs is in contrast to a deep ΔHmix in the ordered alloys or intermetallics. In addition, due to the random distribution of multiple elements, the configurational entropy (ΔSconf) is higher in HEAs than in the ordered alloys, which further contributes to the HEA phase stability. Thus, due to a shallow ΔHmix and an unusually high ΔSconf, the magnitude of entropy contributions can be potentially equal to that of enthalpy in HEAs, thereby making entropy an equally important thermodynamic quantity in phase stability.

High ΔSconf has been perceived to be the main reason for the stability of a single phase solid solution in HEAs.16,21 It has been argued that increasing the number of elements significantly increases the configurational entropy, and leads to phase stability. However, more recently, it has been shown that the contributions of configurational entropy might only be a part of a bigger picture, and it may not be the only quantity to be considered while explaining the stability of HEAs.22,23,24 For example, using the Hume–Rothery rules as a guide, Otto et al.22 experimentally showed that the configurational entropy may not be enough to explain the stability of NiCrCoMnFe HEAs. In a series of experiments, they systematically replaced one element at a time in NiCrCoMnFe single phase-fcc HEA with a similar element (e.g., Ni with Cu, Co with Ti). While this strategy should not significantly change the configurational entropy contribution, none of the substitutions led to the formation of a single-phase solid solution HEA. The authors concluded that along with the enthalpy contributions, the non-configurational entropy contributions might be equally important towards the phase stability of HEAs.

Among the various non-configurational entropies, vibrational entropy generally has a significant contribution to ΔGmix. While ΔSconf always contributes towards the stability by lowering ΔGmix, the contribution of ΔSvib is not so straightforward; ΔSvib of a solid solution could be positive or negative which may stabilize or destabilize the system. As a result, for those solid solutions that have a narrow range of ΔHmix (say, −50 to + 50 meV/atom), the vibrational entropy can have a significant impact on their stability. Its favorable contribution can even stabilize solid solutions that have positive ΔHmix, or it can even destabilize those that have a negative ΔHmix. Similarly, the vibrational entropy can either enhance or counteract the stability provided by the configurational entropy. Since, it is still not possible to predict with certainty which combination of elements will lead to formation of a single phase solid solution,19 understanding the individual contributions of the key thermodynamic quantities (enthalpy and entropy) and relating them to the existing phase diagrams is crucial. In addition, understanding the intricate correlation of these quantities towards the overall phase stability is vital for the development of a robust thermodynamic model for multicomponent systems.

To elucidate the impact of the delicate balance between the entropy and enthalpy on the phase diagrams in the simplest possible systems, we focus on binary solid solutions in which ΔHmix is small and of either sign. In such cases phase stability can be equally controlled by ΔHmix and ΔSmix. We do not consider cases that have a strong negative or positive ΔHmix, because they will form ordered alloys or will phase separate, respectively, and the effect of entropy will not be important. In the weakly negative ΔHmix solid solutions, formation of a random solid solution has a higher probability than an ordered structure. In contrast, in positive ΔHmix solid solutions, the phase instability is likely to be evidenced in the form of a two-phase mixture or a miscibility gap. Due to weak negative or positive ΔHmix, the entropy contributions could become comparable, which could impact the phase stability. Here, by taking examples of several binary phase diagrams, we show that the appearance of a miscibility gap is controlled by the intricate interplay of the enthalpy of mixing, and of the configurational and vibrational entropies. To illustrate, Table 1 lists four possible cases that can lead to the presence or absence of a miscibility gap. Simply based on ΔHmix, for cases 1 and 2, where ΔHmix ≤ 0 and ΔHmix ≥ 0 respectively, the absence (✗) and presence (✓) of a miscibility gap is intuitively expected. However, when ΔSmix contributions are included, the trends can reverse. As shown by case 3, even when ΔHmix ≤ 0, due to the overall negative contribution of ΔSmix, the miscibility gap can appear. Similarly, due to the positive contribution of ΔSmix, the miscibility gap can disappear even when ΔHmix ≥ 0, as shown in case 4. In this paper, by performing density functional theory (DFT) calculations on multiple binary solid solutions, we provide an example for each one of the four cases, and elucidate a profound impact of ΔSmix on the overall shape of a phase diagram. We also disentangle the individual effects of electronic, vibrational and configurational entropy contributions to the presence and absence of a miscibility gap. Our calculations show that by including the three entropies (in particular vibrational and configurational), the behavior of the miscibility gap can be correctly predicted. In addition, the transformation temperature, i.e., the temperature at which the miscibility gap ends for a given composition, can also be accurately predicted. The understanding developed in this work is expected to contribute to the computational prediction of phase stability of multicomponent HEAs that have narrow ΔHmix where entropy contributions are critical.

Table 1 Four possible cases of miscibility gap based on ΔHmix ( and indicate presence and absence of miscibility gap, respectively)
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Results

The phase stability of the alloys is illustrated using the standard convex-hull approach. In this approach, the alloy is phase-stable if its free energy lies below the tie line formed by connecting the free energies of the two adjacent phases.25 In contrast, the alloy phase separates into two phases if its free energy lies above the tie line. We use ∆Gmix vs T plots to predict the miscibility-gap temperature, i.e., the temperature at which the miscibility gap ends (or the temperature at which a single phase random solid solution begins to be stable). We also show the individual plots for -T∆Selec, -T∆Svib and -T∆Sconf vs T, to disentangle the electronic, vibrational, and configurational entropy contributions. Finally, the ratios of the three entropies contributing towards the phase stabilities of each of the binary alloys are presented. Their ratio indicates the contribution of each towards the phase stability. In this work, examples from fcc and bcc solid solutions are considered; the fcc binary alloys considered are Ag–Au, Pt–Ru, Ni–Cu, Ni–Rh, Pd–Ag and Ni–Pd, whereas the bcc alloy considered is Nb–Ta. None of the compositions modeled for any of these alloys show presence of ordered phases in their respective phase diagrams.

Method validation

To validate the accuracy of our DFT calculations and the methodology used to compute the thermodynamic quantities, we compare our results with that of some alloys already available in literature. In particular, we compare the order–disorder transition temperature, i.e., the temperature at which a binary alloy transforms from L10 ordered to disordered phase. Table 2 shows the comparison with the experimental and previous DFT predictions in Cu0.5–Au0.5, Ni0.5–Fe0.5 and Ni0.5–Pt0.5 systems. The temperature is computed by comparing the ∆Gmix (i.e., ΔHmix – TΔSconf – TΔSvib – TΔSelec) between the ordered and the disordered structures for each of the alloys. The temperatures predicted for Cu–Au, Ni–Fe and Ni–Pt are 700 K, 700 K, and 730 K, which are in fairly good agreement with the experimental values of 683 K,26 618 K,27 and 900 K,28 respectively. Our calculated temperatures also compare well with the previous DFT results of 560 K4 and 755 K29 in Cu–Au and Ni–Pt, respectively as shown in Table 2. We also compare ∆Gmix over a range of temperatures. Figure 1 shows the comparison of relative Gibbs free energy (∆Gmix) and relative entropy (∆Smix) of Ni0.5–Pt0.5 between ordered and disordered structures with the DFT results from Shang et al.29 in Ni–Pt system from 0–1000 K. The overall behaviors of both ∆Gmix (with and without configurational entropy) are in very close agreement. The close agreement with the previous works provides a validation to our calculations.

Table 2 Comparison of order–disorder phase-transition temperatures (K) with literature for Cu–Au, Ni–Fe, and Ni–Pt
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Fig. 1
figure 1

∆Gmix vs T comparison with Shang et al.29 in Ni0.5–Pt0.5 system. A difference of only 25 K in the order–disorder transformation temperature is found

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In what follows, each of the four cases described in Table 1 are discussed in detail. Relevant examples of binary solid solutions for the four cases are provided, and the contribution of the entropies towards the phase stability, or the formation of a miscibility gap, is illustrated. We show that, despite ∆Hmix < 0 for a given alloy, where the composition is expected to be stable in the form of a single phase solid solution, the phase diagram can surprisingly show a miscibility gap. Similarly, despite ∆Hmix > 0, where the composition is expected to be unstable, a stable single-phase solid solution can be observed. We show that both these unexpected observations highlight significant contributions of both types of entropies, and the phase stability of alloys may not be predicted simply based on ∆Hmix, at least in some cases. While these two situations correspond to cases 3 and 4 in Table 1, we first present the results of cases 1 and 2 which behave in the manner expected on the basis of enthalpic arguments.

Case 1—Single-phase solid solution when ∆H mix ≤ 0

The Ag–Au system has an isomorphous phase diagram, which shows complete solid solubility across the whole phase diagram.30 As a result, ∆Hmix is expected to be negative across the whole composition range. Figure 2a shows the convex hull for the Ag–Au systems plotted for three alloy compositions: 0.25, 0.5, and 0.75 molar fraction of Au. The calculations indeed show that ∆Hmix < 0 for all three compositions illustrating stable Ag–Au solid solutions. The effect of temperature is added by including the entropy contributions as shown in Fig. 2a.

Fig. 2
figure 2

a Convex hull of Ag–Au binary system showing solid-solution phase stability across the composition range. ∆Gmix vs T plots for b Ag0.5–Au0.5, c Nb0.5–Ta0.5, and d Pt0.8–Ru0.2 in which -T∆Svib, -T∆Sconf and -T∆Selec are shown to highlight the contribution of vibrational, configurational, and electronic entropies, respectively. While ∆Svib contribution is negligible in c, it is unfavorable in b and d

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Figure 2b shows the ∆Gmix vs T plot for Ag0.5–Au0.5 system. ∆Gmix shows the total free energy of the system over a given temperature range. To separate the individual contributions of the electronic, vibrational, and configurational entropies, -T∆Selec, -T∆Svib, and -T∆Sconf are also shown. It is found that the electronic entropy makes no significant contributions to the total free energy, whereas vibrational entropy makes a very minor contribution. Thus, the main entropy contribution to ∆Gmix is essentially configurational.

The Nb–Ta solid solution also has an isomorphous phase diagram with complete solid solubility across the entire composition range.31 Fig. 2c shows the ∆Gmix vs T plot for the Nb0.5–Ta0.5 composition. The ∆Gmix plot shows a stable solid solution over the given temperature range. Similar to the Ag–Au system, we find that configurational entropy contributes to the phase stability, whereas the electronic and vibrational entropy make no significant contributions. From these two binary systems, it is interesting to note that the phase stability of a system could be driven only by ∆Hmix and ∆Sconf, while ∆Selec and ∆Svib make no important contributions.

Figure 2d shows the ∆Gmix vs T plot for Pt0.8–Ru0.2 system. The phase diagram of Pt–Ru shows fcc single phase solid solution at 0.2 Ru composition.32,33 As a result, similar to the above two systems, Pt0.8–Ru0.2 also shows negative ∆Gmix, indicating a stable phase. The electronic entropy contribution is negligible as observed in the above two systems. However, in contrast to the above two systems, the influence of vibrational entropy is comparatively significant. While the ∆Sconf contribution is the same as in the two other systems, unlike these two systems, the Pt0.8–Ru0.2 system has a substantial positive ∆Svib slope, as shown in Fig. 2d. In this case, the vibrational entropy disfavors the phase stability of the alloy. The contribution of ∆Svib is −43% to the total entropy, which is not negligible. From these three systems, we find that the vibrational entropy contributions could be significantly different among various alloys.

Case 2—Miscibility gap when ∆H mix ≥ 0

The solid solutions that have ∆Hmix ≥ 0 are intuitively expected to be unstable. This instability can be revealed in the form of a miscibility gap in some phase diagrams. Only when the temperature is raised high enough does the miscibility gap close and the mixture transforms into a stable single-phase random solid solution. Here, such specific solid solutions are targeted to elucidate the significance of the two entropy contributions towards the formation of a miscibility gap.

Figure 3a shows the ∆Gmix vs T plot for Pt0.5–Ru0.5 system where ∆Gmix, -T∆Selec, -T∆Svib, and -T∆Sconf are shown. From DFT calculations, ∆Hmix is found to be +14.2 meV/atom. Since at T = 0 K, the ∆Hmix is positive, the solid solution is unstable. When the entropy contributions are added, the ∆Gmix starts to decrease with temperature. The negative slope of ∆Gmix indicates that the solid solution will eventually become stable (i.e., ∆Gmix < 0), at higher temperatures. The contribution of electronic entropy is negligible for this system. However, the importance of the contribution of the two entropies (∆Svib and ∆Sconf) towards the stability of the solid solution is highlighted in this alloy. Figure 3a shows that ∆Svib contributes unfavorably to the overall ∆Gmix as illustrated by the positive slope of the -T∆Svib plot.

Fig. 3
figure 3

∆Gmix vs T plots for a Pt0.5–Ru0.5, c Ni0.5–Cu0.5 and, d Ni0.5–Rh0.5 in which -T∆Svib, -T∆Sconf, and -T∆Selec are shown to highlight the contributions of vibrational, configurational, and electronic entropies. b Convex hull of Pt–Ru binary system

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Figure 3b shows the convex–hull diagram for the Pt-Ru system. The relative stabilities of three compositions, i.e., at 0.2, 0.5 and 0.8 molar fraction Ru, predicted by DFT calculations are shown. These predictions are in agreement with the phase diagram. The Pt-Ru phase diagram32,33 shows fcc single-phase solid solution in the range up to 0.38 Ru and a miscibility gap beyond that composition at room temperature. Figure 3b shows a similar stability trend where the fcc solid solution is found to be stable at 0.2 Ru. For the 0.5 Ru composition, the fcc is expected to be unstable at room temperature, and the predicted transition temperature to stable fcc is found to be ~600 K which is in very good agreement with the phase diagram (~600 K).32,33 At 0.8 Ru the fcc phase is found to be unstable, as shown by the positive ∆Gmix data points all of which lie above the tie line for the corresponding temperatures in Fig. 3b; this fcc phase instability is again consistent with the phase diagram.

To further emphasize the importance of the contributions of the two entropies, the ∆Gmix vs T plots are presented for Ni0.5–Cu0.5 and Ni0.5–Rh0.5 systems in Fig. 3c, d, respectively. The -T∆Selec lines are flat for both systems showing the negligible contribution of the electronic entropy. The predicted transition temperature of Ni0.5–Cu0.5 is 400 K shown in Fig. 3c which is in qualitative agreement with the phase diagram (i.e., ~620 K).34 The Ni0.5–Cu0.5 has similar positive ∆Svib slope compared to Pt0.5–Ru0.5 indicating that the contribution of the vibrational entropy in Ni0.5–Cu0.5 is same as that in Pt0.5–Ru0.5. Consequently, the slope of the total Gibbs free energy of mixing is also same between the two alloys. In contrast, we find that the ∆Svib in Ni0.5–Rh0.5 has negative slope; this means, the contribution of ∆Svib is favorable to its phase stability as shown in Fig. 3d. The predicted miscibility gap transition temperature for Ni0.5–Rh0.5 is 575 K as shown in Fig. 3d; here, ∆Svib actually favors the formation of a single phase as shown by its negative slope, thereby adding to the contribution of ∆Sconf towards the overall ∆Gmix. The ∆Svib contribution in Ni0.5–Rh0.5 is +21% whereas in other two systems (Pt0.5–Ru0.5 and Ni0.5–Cu0.5) it is −18% and −14%, as shown in Fig. 6b. With respect to the phase diagram, the transition temperature is ~350 K for Ni0.5–Rh0.5,35 which is again in qualitative agreement with our calculations. Thus, in accord with the observation in case 1, we find that the entropy contributions could be different among various systems. In addition, we find that the vibrational entropy could favor or disfavor phase stability of alloys.

Case 3—Miscibility gap when ∆H mix ≤ 0

The behavior of the above two cases is generally intuitive, i.e., when ∆Hmix is < 0, a stable phase is expected, whereas when ∆Hmix > 0, an unstable phase or a miscibility gap is expected. However, cases 3 and 4 can seem unintuitive, i.e., even when the ∆Hmix < 0, the phase diagram can show a miscibility gap. Similarly, when ∆Hmix > 0, the system may be stable, and a miscibility gap can be absent. In both these cases, we show that entropy plays a crucial role in dictating the phase stability, and the contributions of each of the three entropies are disentangled.

The Pd–Ag phase diagram shows a miscibility gap in the range 0.65–0.98 molar fraction Pd.36,37 The miscibility gap transition temperature peaks at ~0.85 Pd is 620 K. Consequently, Pd0.8–Ag0.2 is expected to be unstable with respect to the two pure elements. However, we find that DFT predicts ∆Hmix < 0, indicating a stable solid solution. This ‘pseudo’ stability is shown in Fig. 4a, where ∆Gmix decreases with the increase in temperature. The contribution of ∆Selec is negligible and ∆Svib slightly disfavors phase stability shown by the positive slope in Fig. 4a. The ∆Sconf provides the pseudo stability that leads to the decrease in ∆Gmix with temperature. Here, the ∆Gmix vs T plot is rather misleading as it does not show the evidence of the miscibility gap.

Fig. 4
figure 4

a ∆Gmix vs T for Pd0.8–Ag0.2 showing -T∆Svib, -T∆Sconf, and -T∆Selec contributions. b Convex-hull diagram of Pd–Ag binary system. c Percentage entropy contribution in ∆Gmix for Pd0.5–Ag0.5 and Pd0.8–Ag0.2 at various temperatures. The entropy contribution is significantly higher in Pd0.8–Ag0.2 than Pd0.5–Ag0.5. The vibrational entropy contribution is minor in both as shown by the darker sub-bars

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The correct phase behavior is revealed in the convex-hull diagram. Figure 4b shows the convex-hull for Pd–Ag system, where ∆Gmix for three compositions, i.e., 0.2, 0.5, and 0.8 molar fraction Pd, are plotted at 0 K, 300 K, 600 K, and 800 K. We find that for all four temperatures, the free energies of 0.2 and 0.5 solid solution compositions lie on the convex-hull, indicating their phase stability, which agrees with the phase diagram. In contrast, the free energies of 0.8 Pd composition for 0 K and 300 K clearly lie above their tie lines, thereby indicating its instability. The free energy decreases with increasing temperature. By ~600 K, it lies almost on the tie line, indicating that the system is just starting to stabilize. By 800 K, the free energy lies well below the tie line, showing its complete stability at higher temperatures. The transition temperature from phase separation to the stable random solid solution is just above 600 K, which is in very good agreement with the phase diagram which also shows the transition temperature to be ~600 K.37 While the increase in phase stability with temperature is not a new observation, this example reinforces the point that simply relying on ∆Hmix may not be sufficient in shallow ∆Hmix solid solutions and including entropy contributions could be crucial.

Since the phase stability of Pd0.8–Ag0.2 is achieved by raising the temperature, the decrease in the free energy can only be due to the entropy contributions. Figure 4c shows the percentage contributions of the vibrational and configurational entropies to the total free energy for Pd0.5–Ag0.5 and Pd0.8–Ag0.2 compositions at 150 K, 500 K, 600 K, and 800 K. We find that the total entropy contribution is significantly higher for Pd0.8–Ag0.2 than Pd0.5–Ag0.5, whereas in Pd0.8–Ag0.2 it is over 75%, that in Pd0.5–Ag0.5 is only ~40 %. This enhanced entropy contribution in Pd0.8–Ag0.2 is the reason that drives the stability of Pd0.8–Ag0.2 at higher temperatures. However, by separating the contributions of the two entropies, we find that the vibrational contribution (shown by darker bars in Fig. 4c) is very small in both alloys for all four temperatures, and the primary contribution is configurational.

Case 4—Single-phase solid solution when ∆H mix ≥ 0

The final case from Table 1 is the formation of a single-phase solid solution even when ∆Hmix is predicted to be positive. While ∆Hmix is greater than 0 for both cases 2 and 4, there is presence of a miscibility gap in case 2 in contrast to formation of a solid solution in case 4. Thus, in case 4, simply relying on ∆Hmix would predict phase instability (or the presence of a miscibility gap). By taking example of Ni–Pd system, we show that incorporating the entropy contribution provides the correct phase stability in agreement with the Ni–Pd phase diagram.

In the calculation of ∆Hmix for random alloys, the total energy of a given alloy composition could significantly vary due to the multiple ways of randomly distributing atoms during the creation of initial crystal structures. Figure 5a shows ∆Hmix of multiple structures for three compositions: Ni0.25Pd0.75, Ni0.5Pd0.5 and Ni0.75Pd0.25. Our calculations predict ∆Hmix > 0 for all structures. These results are in agreement with previous DFT calculations from Teeriniemi et al.38 as shown in Fig. 5a. However, both results are in contrast to the Ni–Pd phase diagram that shows a complete solid solubility across the entire composition range.39

Fig. 5
figure 5

a Comparison of ∆Hmix with Teeriniemi et al.38 showing ∆Hmix > 0 from both studies. b Convex-hull diagram of Ni–Pd binary system. c ∆Gmix vs T plots for Ni0.5–Pd0.5 system showing contributions of vibrational, configurational and electronic entropies

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The correct structure, a stable random solid solution, is predicted when entropy contributions are included. Figure 5b shows the convex-hull diagram formed using the three compositions. The ∆Hmix structures used in Fig. 5b are the lowest energy structures from Fig. 5a for the corresponding compositions. As expected, the stability increases with the increase in temperature. Ni0.25–Pd0.75 and Ni0.5–Pd0.5 are stable at 300 K, whereas Ni0.75–Pd0.25 becomes stable above 300 K. By 500 K, all three compositions are stable, as shown.

Figure 5c shows the ∆Gmix vs T plot for Ni0.5–Pd0.5 where the individual contributions of -T∆Selec, -T∆Svib and -T∆Sconf are shown. We find that electronic entropy contributions are negligible whereas vibrational entropy has a negative slope, and it contributes favorably towards the phase stability and the overall ∆Gmix along with ∆Sconf. It is important to point out that while ∆Gmix becomes negative only above ~240 K, the positive value of ∆Gmix below 240 K does not indicate a miscibility gap, unlike Fig. 3. The positive ∆Gmix originates from ∆Hmix which is greater than 0 to begin with. Only with the addition of entropy, the complete picture emerges, as revealed by the convex-hull diagram in Fig. 5b. Thus, Pd–Ag and Ni–Pd systems discussed under cases 3 and 4 collectively highlight the importance of entropy contributions towards predicting the correct phase stability of these alloys.

Finally, we summarize the individual contributions of electronic, vibrational and configurational entropies for all the systems discussed above. Tables 3 and 4 show the electronic and vibrational entropy respectively of metals and alloys at 1000 K. Figure 6a-c shows the percentages of electronic, vibrational and configurational entropy contribution to the total entropy at 1000 K, where the negative (positive) sign indicates that electronic and vibrational entropy disfavor (favor) the phase stability or free energy. We find that the electronic entropy contributions are negligible in all alloys (of the order of 10−2 meV/atom), and they are not comparable to vibrational and configurational entropy contributions as shown in Fig. 6. The vibrational entropy is comparable to that of the configurational entropy, but it shows large variation among the alloys. For example, in Pt0.8–Ru0.2 system, the vibrational entropy has an opposite contribution to the configurational entropy by over 40% as shown in Fig. 6b. In contrast, in Ni0.5–Pd0.5 and Ni0.5–Rh0.5 it favors the phase stability and has a favorable contribution of more than 20%. While the unfavorable contribution would tend to destabilize the system, creating propensity to miscibility gap, the favorable contribution would help in phase stability.

Table 3 Electronic entropy of metals and alloys at 1000 K
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Table 4 Vibrational entropy of metals and alloys at 1000 K
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Fig. 6
figure 6

a Percentages of electronic entropy (T∆Selec), b vibrational entropy (T∆Svib), and c configurational entropy contributions (T∆Sconf) in total entropy (T∆Smix) of the binary solid solutions

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Discussion

These results show that in the narrow-range ∆Hmix solid solutions, the contribution of ∆Smix is important, and cannot be ignored, unlike that in the ordered structures. For example, while ∆Hmix in Ni0.5–Rh0.5 at 575 K is +40 meV/atom, the total ∆Smix is almost the same, i.e., 40 meV/atom (Fig. 3d), illustrating that the contribution of ∆Smix is large and quantitatively similar to ∆Hmix. The important contribution of ∆Smix is further demonstrated in a quaternary alloy, Fe–Ni–Co–Cr. The calculated ∆Hmix of this alloy is +65 meV/atom. This is consistent with previous DFT calculations from Gao et al.40 and Troparevsky et al.20 Based on such large positive ∆Hmix, this alloy is expected to be unstable, however, experimentally it is found to be single phase stable.41 The positive ∆Hmix is flipped to negative (and stable) ∆Gmix via entropy contributions. Our calculations (in agreement with Gao et al.) show that ∆Svib has a contribution of over 27% to the total entropy at room temperature (the rest being configurational entropy). Without the contribution of the vibrational entropy, the stability of this phase would occur at ~600 K, whereas experimentally it is stable at room temperature. These calculations thus illustrate the importance of including entropy contributions towards the correct prediction of phase behaviors.

Our calculations also show that the contribution of ∆Svib can be significantly different among the systems. While in some systems ∆Svib disfavors the phase stability (such as in Pt0.8–Ru0.2, Pt0.5–Ru0.5 and Ni0.5–Cu0.5) in others it favors the phase stability (e.g., Ni0.5–Rh0.5 and Ni0.5–Pd0.5); ∆Svib can have very little contribution as well, as observed in Nb0.5–Ta0.5. It has been speculated that there could be multiple reasons behind the different behaviors, such as electronegativity difference between the elements, atomic radius difference, electron-to-atom ratio, or the difference in the atomic masses.1 The vibrational entropy is the measure of stiffness of the atomic bonds; the softer the bonds, the higher the vibrational entropy.5 The effect of these factors fundamentally lies in the phonon dispersion curves which provide the frequencies of optical phonon modes. An alloy with softer bonds would have lower frequencies and higher entropy. While significant progress has been made in understanding the contributions to vibrational entropy, the contributions of each of these factors to the phonon dispersion curves is still not fully understood; our future work will focus in this direction.

Our calculations also show that electronic entropy has minor contribution to the total entropy. Similar observations were made previously on three quaternary systems, i.e., fcc Co–Cr–Fe–Ni, bcc Mo–Nb–Ta–W and hcp Co–Os–Re–Ru.40 Via DFT calculations, it was shown that the total entropy of these systems was of the order of 10−2 meV/atom or less, and the ∆Selec was negligible. Thus, these calculations seem to indicate a greater importance of vibrational and configurational entropies compared to electronic. While three entropy contributions are considered in this work, the magnetic entropy could have an impact as well.42 Previous work has shown that the Curie temperature can be tuned by controlling the Cr concentration in Fe–Co–Cr–Ni HEA.42 Understanding the individual contribution of various entropies is warranted.

In conclusion, this work shows that in the shallow-range ∆Hmix solid solutions, relying solely on enthalpy may not lead to correct prediction of phase stability, and including entropy contributions are critical. By disentangling the entropy contributions in various solid solutions, our calculations show that the two entropies (vibrational and configurational) can either destabilize or can collectively contribute to stabilize the solid solutions. Our calculations also show that electronic entropy contributions may be less significant compared to vibrational and configurational entropies towards the phase stability of binary random solid solutions. This work advances the understanding on thermodynamics of HEAs, and contributes towards building a theoretical framework for computational prediction of stable multi-elemental single-phase HEAs in future.

Methods

First-principles calculations are performed with density functional theory (DFT) using Vienna Ab-initio Software Package (VASP).43 The Projector Augmented Wave (PAW) method is used with standard GGA-PBE exchange-correlation function.44 For structure relaxation, unit cells are sampled by Brillouin zones with a 4 × 4 × 4 Monkhorst–Pack (MP)45 mesh with the 500 eV energy cut-off of the wave function. The energy convergence criterion of the electronic self-consistency is chosen as 10−8 eV for all the calculations.

Random structures are created using the special quasi-random structure (SQS) method.46 A SQS method calculates correlation functions of a finite cell and compares them to those of an infinite random system. The differences in the correlation functions can be used to quantify the randomness in the finite cell. The purpose of the SQS algorithm is to minimize the difference in the correlation functions. Thus, SQSs are believed to give a good approximation to near-randomness in solid solution alloys.

The eight-atom (SQS-8), 16-atom (SQS-16) and 32-atom (SQS-32) are used in the present study to predict the phase stability of various random binary systems. We have checked that the number of atoms in the SQS structures does not significantly affect the prediction of phase stability. The convergence is quantified by calculating the error bars as shown in Supplementary Information (see S1). The correlations in 32-atom (SQS-32) for A0.25–B0.75 and A0.5–B0.5 compositions are given in the Supplementary Information under S2. The widely used Alloy Theoretic Automated Toolkit (ATAT) is used to build the SQS structures.47

The supercell method is used for phonon calculations.5 The VASP code is used to calculate the real space force constants of supercells, and the PHONOPY48 code is used to calculate the phonon frequencies from the force constants on a supercell consisting of at least 32 atoms in all systems. In order to get the force-constant matrices for each binary system, every atom is displaced by a finite displacement of 0.01 Å in x-, y- and z-direction. For 32 atoms, 192 sets of atomic positions with displacement employed to each random alloy. Strict energy convergence criteria of (10−8 eV) and 4 × 4 × 4 k-points were used for the force constant calculations. After getting the force–constant matrices, the dynamical matrix is built for different q-vectors in the Brillouin zone. The eigenvalues of phonon frequencies and eigenvectors of phonon modes are found by solving the dynamical matrix. The thermodynamic properties require summations over the phonon eigenvectors which is implemented in the PHONOPY code. The theory on the phonon calculations is discussed in Supplementary Information (see S3). We have checked the mechanical stability of all systems, and no imaginary modes are observed in the modeled structures. The phonon band structures figures for all the studied systems have also been added to the Supplementary Information under S4. The electronic entropy has been calculated using Fermi-Dirac distribution. The theory of electronic entropy is discussed in Supplementary Information (see S5).

In order to illustrate the phase stability, the change in Gibbs free energy of mixing is calculated for all binary systems. ∆Gmix is the difference of Gibbs free energy between an alloy and the individual elements. The expression for change in Gibbs free energy is, ∆Gmix = ∆Hmix – T (∆Sconf + ∆Svib + ∆Selec), where ∆Hmix is the enthalpy of mixing, T is the absolute temperature, ∆Sconf, ∆Svib and ∆Selec are the difference of configurational, vibrational and electronic entropy between alloy and individual elements. ∆Sconf is calculated using the standard equation, i.e., ∆Sconf = Σ x ln x, where x is the number of elements in an alloy.

The relativistic effect of spin–orbit coupling (SOC) is important for heavier elements and its significance has been addressed by Xie et al.49 Their study showed that with SOC, the energies are found to be lowered by 20 meV/atom in U–Zr alloy. We also performed phase stability calculations using SOC. However, no significant difference in the transition temperature prediction was observed. A comparison of the transition temperatures between SOC and non-SOC calculations is provided in the Supplementary Information (see S6).