## Abstract

High entropy alloys contain multiple elements in large proportions that make them prone to phase separation. These alloys generally have shallow enthalpy of mixing which makes the entropy contributions of similar magnitude. As a result, the phase stability of these alloys is equally dependent on enthalpy and entropy of mixing and understanding the individual contribution of thermodynamic properties is critical. In the overall vision of designing high entropy alloys, in this work, using density functional theory calculations, we elucidate the contributions of various entropies, i.e., vibrational, electronic and configurational towards the phase stability of binary alloys. We show that the contribution of electronic entropy is very small compared to the vibrational and configurational entropies, and does not play a significant role in the phase stability of alloys. The configurational and vibrational entropies can either destabilize or can collectively contribute to stabilize the solid solutions. As a result, even those systems that have negative mixing enthalpy can show phase instability, revealed as a miscibility gap; conversely, systems with positive mixing enthalpy can be phase stable due to entropic contributions. We suggest that including entropic contributions are critical in the development of theoretical framework for the computational prediction of stable, single-phase high entropy alloys that have shallow mixing enthalpies, unlike ordered intermetallics.

## Introduction

The phase stability of an alloy is guided by Gibbs free energy (Δ*G*_{mix}) that comprises of enthalpy (Δ*H*_{mix}) and entropy (Δ*S*_{mix}) of mixing. Conventionally, Δ*H*_{mix} has been perceived as the dominant quantity, where a strongly negative Δ*H*_{mix} indicates formation of a stable solid solution and a positive Δ*H*_{mix} 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 Δ*H*_{mix} in identifying stable metal alloys. In the same vein, Δ*S*_{mix} contributions have also been included in the calculations, albeit selectively; including Δ*S*_{mix} 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 Δ*S*_{mix} has often been considered less important, largely due to its relatively smaller size compared to Δ*H*_{mix}, 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 Δ*S*_{mix} 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 Δ*H*_{mix}^{19,20} among the constituting elements which prevents phase separation and formation of a second sub-phase. The shallow Δ*H*_{mix} in HEAs is in contrast to a deep Δ*H*_{mix} in the ordered alloys or intermetallics. In addition, due to the random distribution of multiple elements, the configurational entropy (Δ*S*_{conf}) is higher in HEAs than in the ordered alloys, which further contributes to the HEA phase stability. Thus, due to a shallow Δ*H*_{mix} and an unusually high Δ*S*_{conf}, 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 Δ*S*_{conf} 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 Δ*G*_{mix}. While Δ*S*_{conf} always contributes towards the stability by lowering Δ*G*_{mix}, the contribution of ΔS_{vib} is not so straightforward; Δ*S*_{vib} 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 Δ*H*_{mix} (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 Δ*H*_{mix}, or it can even destabilize those that have a negative Δ*H*_{mix}. 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 Δ*H*_{mix} is small and of either sign. In such cases phase stability can be equally controlled by Δ*H*_{mix} and Δ*S*_{mix}. We do not consider cases that have a strong negative or positive Δ*H*_{mix}, because they will form ordered alloys or will phase separate, respectively, and the effect of entropy will not be important. In the weakly negative Δ*H*_{mix} solid solutions, formation of a random solid solution has a higher probability than an ordered structure. In contrast, in positive Δ*H*_{mix} 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 Δ*H*_{mix}, 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 Δ*H*_{mix}, for cases 1 and 2, where Δ*H*_{mix} ≤ 0 and Δ*H*_{mix} ≥ 0 respectively, the absence (✗) and presence (✓) of a miscibility gap is intuitively expected. However, when Δ*S*_{mix} contributions are included, the trends can reverse. As shown by case 3, even when Δ*H*_{mix} ≤ 0, due to the overall negative contribution of Δ*S*_{mix}, the miscibility gap can appear. Similarly, due to the positive contribution of Δ*S*_{mix}, the miscibility gap can disappear even when Δ*H*_{mix} ≥ 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 Δ*S*_{mix} 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 Δ*H*_{mix} where entropy contributions are critical.

## 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 ∆*G*_{mix} 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*∆*S*_{elec}, -*T*∆*S*_{vib} and -*T*∆*S*_{conf} 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 L1_{0} ordered to disordered phase. Table 2 shows the comparison with the experimental and previous DFT predictions in Cu_{0.5}–Au_{0.5}, Ni_{0.5}–Fe_{0.5} and Ni_{0.5}–Pt_{0.5} systems. The temperature is computed by comparing the ∆*G*_{mix} (i.e., Δ*H*_{mix} – TΔ*S*_{conf} – TΔ*S*_{vib} – TΔ*S*_{elec}) 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 K^{4} and 755 K^{29} in Cu–Au and Ni–Pt, respectively as shown in Table 2. We also compare ∆*G*_{mix} over a range of temperatures. Figure 1 shows the comparison of relative Gibbs free energy (∆*G*_{mix}) and relative entropy (∆*S*_{mix}) of Ni_{0.5}–Pt_{0.5} 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 ∆*G*_{mix} (with and without configurational entropy) are in very close agreement. The close agreement with the previous works provides a validation to our calculations.

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 ∆*H*_{mix} < 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 ∆*H*_{mix} > 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 ∆*H*_{mix}, 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, ∆*H*_{mix} 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 ∆*H*_{mix} < 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.

Figure 2b shows the ∆*G*_{mix} vs *T* plot for Ag_{0.5}–Au_{0.5} system. ∆*G*_{mix} 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*∆*S*_{elec}, -*T*∆*S*_{vib}, and -*T*∆*S*_{conf} 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 ∆*G*_{mix} 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 ∆*G*_{mix} vs *T* plot for the Nb_{0.5}–Ta_{0.5} composition. The ∆*G*_{mix} 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 ∆*H*_{mix} and ∆*S*_{conf}, while ∆*S*_{elec} and ∆*S*_{vib} make no important contributions.

Figure 2d shows the ∆*G*_{mix} vs T plot for Pt_{0.8}–Ru_{0.2} 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, Pt_{0.8}–Ru_{0.2} also shows negative ∆*G*_{mix}, 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 ∆*S*_{conf} contribution is the same as in the two other systems, unlike these two systems, the Pt_{0.8}–Ru_{0.2} system has a substantial positive ∆*S*_{vib} slope, as shown in Fig. 2d. In this case, the vibrational entropy disfavors the phase stability of the alloy. The contribution of ∆*S*_{vib} 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 ∆*H*_{mix} ≥ 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 ∆*G*_{mix} vs *T* plot for Pt_{0.5}–Ru_{0.5} system where ∆*G*_{mix}, -*T*∆*S*_{elec}, -*T*∆*S*_{vib}, and -*T*∆*S*_{conf} are shown. From DFT calculations, ∆*H*_{mix} is found to be +14.2 meV/atom. Since at *T* = 0 K, the ∆*H*_{mix} is positive, the solid solution is unstable. When the entropy contributions are added, the ∆G_{mix} starts to decrease with temperature. The negative slope of ∆*G*_{mix} indicates that the solid solution will eventually become stable (i.e., ∆*G*_{mix} < 0), at higher temperatures. The contribution of electronic entropy is negligible for this system. However, the importance of the contribution of the two entropies (∆*S*_{vib} and ∆*S*_{conf}) towards the stability of the solid solution is highlighted in this alloy. Figure 3a shows that ∆*S*_{vib} contributes unfavorably to the overall ∆*G*_{mix} as illustrated by the positive slope of the -*T*∆*S*_{vib} plot.

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 diagram^{32,33} 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 ∆*G*_{mix} 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 ∆*G*_{mix} vs *T* plots are presented for Ni_{0.5}–Cu_{0.5} and Ni_{0.5}–Rh_{0.5} systems in Fig. 3c, d, respectively. The -*T*∆*S*_{elec} lines are flat for both systems showing the negligible contribution of the electronic entropy. The predicted transition temperature of Ni_{0.5}–Cu_{0.5} is 400 K shown in Fig. 3c which is in qualitative agreement with the phase diagram (i.e., ~620 K).^{34} The Ni_{0.5}–Cu_{0.5} has similar positive ∆*S*_{vib} slope compared to Pt_{0.5}–Ru_{0.5} indicating that the contribution of the vibrational entropy in Ni_{0.5}–Cu_{0.5} is same as that in Pt_{0.5}–Ru_{0.5}. Consequently, the slope of the total Gibbs free energy of mixing is also same between the two alloys. In contrast, we find that the ∆S_{vib} in Ni_{0.5}–Rh_{0.5} has negative slope; this means, the contribution of ∆*S*_{vib} is favorable to its phase stability as shown in Fig. 3d. The predicted miscibility gap transition temperature for Ni_{0.5}–Rh_{0.5} is 575 K as shown in Fig. 3d; here, ∆*S*_{vib} actually favors the formation of a single phase as shown by its negative slope, thereby adding to the contribution of ∆*S*_{conf} towards the overall ∆*G*_{mix}. The ∆*S*_{vib} contribution in Ni_{0.5}–Rh_{0.5} is +21% whereas in other two systems (Pt_{0.5}–Ru_{0.5} and Ni_{0.5}–Cu_{0.5}) it is −18% and −14%, as shown in Fig. 6b. With respect to the phase diagram, the transition temperature is ~350 K for Ni_{0.5}–Rh_{0.5},^{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 ∆*H*_{mix} is < 0, a stable phase is expected, whereas when ∆*H*_{mix} > 0, an unstable phase or a miscibility gap is expected. However, cases 3 and 4 can seem unintuitive, i.e., even when the ∆*H*_{mix} < 0, the phase diagram can show a miscibility gap. Similarly, when ∆*H*_{mix} > 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, Pd_{0.8}–Ag_{0.2} is expected to be unstable with respect to the two pure elements. However, we find that DFT predicts ∆*H*_{mix} < 0, indicating a stable solid solution. This ‘pseudo’ stability is shown in Fig. 4a, where ∆*G*_{mix} decreases with the increase in temperature. The contribution of ∆*S*_{elec} is negligible and ∆*S*_{vib} slightly disfavors phase stability shown by the positive slope in Fig. 4a. The ∆*S*_{conf} provides the pseudo stability that leads to the decrease in ∆*G*_{mix} with temperature. Here, the ∆*G*_{mix} vs *T* plot is rather misleading as it does not show the evidence of the miscibility gap.

The correct phase behavior is revealed in the convex-hull diagram. Figure 4b shows the convex-hull for Pd–Ag system, where ∆*G*_{mix} 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 ∆*H*_{mix} may not be sufficient in shallow ∆H_{mix} solid solutions and including entropy contributions could be crucial.

Since the phase stability of Pd_{0.8}–Ag_{0.2} 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 Pd_{0.5}–Ag_{0.5} and Pd_{0.8}–Ag_{0.2} compositions at 150 K, 500 K, 600 K, and 800 K. We find that the total entropy contribution is significantly higher for Pd_{0.8}–Ag_{0.2} than Pd_{0.5}–Ag_{0.5}, whereas in Pd_{0.8}–Ag_{0.2} it is over 75%, that in Pd_{0.5}–Ag_{0.5} is only ~40 %. This enhanced entropy contribution in Pd_{0.8}–Ag_{0.2} is the reason that drives the stability of Pd_{0.8}–Ag_{0.2} 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 ∆*H*_{mix} is predicted to be positive. While ∆*H*_{mix} 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 ∆*H*_{mix} 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 ∆*H*_{mix} 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 ∆*H*_{mix} of multiple structures for three compositions: Ni_{0.25}Pd_{0.75}, Ni_{0.5}Pd_{0.5} and Ni_{0.75}Pd_{0.25}. Our calculations predict ∆*H*_{mix} > 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}

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 ∆*H*_{mix} 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. Ni_{0.25}–Pd_{0.75} and Ni_{0.5}–Pd_{0.5} are stable at 300 K, whereas Ni_{0.75}–Pd_{0.25} becomes stable above 300 K. By 500 K, all three compositions are stable, as shown.

Figure 5c shows the ∆*G*_{mix} vs *T* plot for Ni_{0.5}–Pd_{0.5} where the individual contributions of -*T*∆*S*_{elec}, -*T∆*S_{vib} and -*T*∆*S*_{conf} 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 ∆*G*_{mix} along with ∆*S*_{conf}. It is important to point out that while ∆G_{mix} becomes negative only above ~240 K, the positive value of ∆*G*_{mix} below 240 K does not indicate a miscibility gap, unlike Fig. 3. The positive ∆*G*_{mix} originates from ∆*H*_{mix} 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 Pt_{0.8}–Ru_{0.2} system, the vibrational entropy has an opposite contribution to the configurational entropy by over 40% as shown in Fig. 6b. In contrast, in Ni_{0.5}–Pd_{0.5} and Ni_{0.5}–Rh_{0.5} 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.

## Discussion

These results show that in the narrow-range ∆*H*_{mix} solid solutions, the contribution of ∆*S*_{mix} is important, and cannot be ignored, unlike that in the ordered structures. For example, while ∆*H*_{mix} in Ni_{0.5}–Rh_{0.5} at 575 K is +40 meV/atom, the total ∆*S*_{mix} is almost the same, i.e., 40 meV/atom (Fig. 3d), illustrating that the contribution of ∆*S*_{mix} is large and quantitatively similar to ∆*H*_{mix}. The important contribution of ∆*S*_{mix} is further demonstrated in a quaternary alloy, Fe–Ni–Co–Cr. The calculated ∆*H*_{mix} 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 ∆*H*_{mix}, this alloy is expected to be unstable, however, experimentally it is found to be single phase stable.^{41} The positive ∆*H*_{mix} is flipped to negative (and stable) ∆*G*_{mix} via entropy contributions. Our calculations (in agreement with Gao et al.) show that ∆*S*_{vib} 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 ∆*S*_{vib} can be significantly different among the systems. While in some systems ∆*S*_{vib} disfavors the phase stability (such as in Pt_{0.8}–Ru_{0.2}, Pt_{0.5}–Ru_{0.5} and Ni_{0.5}–Cu_{0.5}) in others it favors the phase stability (e.g., Ni_{0.5}–Rh_{0.5} and Ni_{0.5}–Pd_{0.5}); ∆*S*_{vib} can have very little contribution as well, as observed in Nb_{0.5}–Ta_{0.5}. 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 ∆*S*_{elec} 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 ∆*H*_{mix} 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 A_{0.25}–B_{0.75} and A_{0.5}–B_{0.5} 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 PHONOPY^{48} 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. ∆G_{mix} is the difference of Gibbs free energy between an alloy and the individual elements. The expression for change in Gibbs free energy is, ∆*G*_{mix} = ∆*H*_{mix} – *T* (∆*S*_{conf} + ∆*S*_{vib} + ∆*S*_{elec}), where ∆*H*_{mix} is the enthalpy of mixing, *T* is the absolute temperature, ∆*S*_{conf}, ∆*S*_{vib} and ∆*S*_{elec} are the difference of configurational, vibrational and electronic entropy between alloy and individual elements. ∆*S*_{conf} is calculated using the standard equation, i.e., ∆*S*_{conf} = 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).

## Data availability

The authors declare that all data supporting the findings of this study are available within the paper and its supplementary information files.

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## Acknowledgements

This work was supported as part of the Energy Dissipation to Defect Evolution (EDDE), an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Basic Energy Sciences. We acknowledge the support of computational resources from Advanced Research Computing Center (ARCC) at the University of Wyoming. We also thank Richard Hennig for fruitful discussions on spin–orbit coupling calculations.

## Author information

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### Contributions

A.M. and D.S.A. developed the idea, and A.M. performed the calculations. S.P. and S.R.P provided the SQS structures. D.C. contributed to the DFT calculations. A.M. and D.S.A drafted the manuscript, which was edited by all authors. All authors collectively discussed the work and provided critical insights to the direction of the work.

### Corresponding author

Correspondence to Dilpuneet S. Aidhy.

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Manzoor, A., Pandey, S., Chakraborty, D. *et al.* Entropy contributions to phase stability in binary random solid solutions.
*npj Comput Mater* **4, **47 (2018) doi:10.1038/s41524-018-0102-y

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