Abstract
High entropy alloys (HEAs) contain near equimolar amounts of five or more elements and are a compelling space for materials design. In the design of HEAs, great emphasis is placed on identifying thermodynamic conditions for singlephase and multiphase stability regions, but this process is hindered by the difficulty of navigating stability relationships in highcomponent spaces. Traditional phase diagrams use barycentric coordinates to represent composition axes, which require (N – 1) spatial dimensions to represent an Ncomponent system, meaning that HEA systems with N > 4 components cannot be readily visualized. Here, we propose forgoing barycentric composition axes in favor of two energy axes: a formationenergy axis and a ‘reaction energy’ axis. These Inverse Hull Webs offer an informationdense 2D representation that successfully captures complex phase stability relationships in N ≥ 5 component systems. We use our proposed diagrams to visualize the transition of HEA solidsolutions from hightemperature stability to metastability upon quenching, and identify important thermodynamic features that are correlated with the persistence or decomposition of metastable HEAs.
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Introduction
Traditional alloy design consists of one or two principal elements alloyed with small amounts of supplementary elements. On the other hand, high entropy alloys (HEAs) contain near equimolar amounts of five or more elements and offer promising mechanical behavior^{1,2,3,4,5}, corrosion resistance^{6,7}, and compositional tunability^{8,9}. Additionally, highcomponent ceramics^{10,11,12,13} can be semiconductors^{14,15} or used as cathodes in Liion batteries^{16,17}, illustrating the promise of entropystabilized materials. A large number of compositional degrees of freedom available in the search for highcomponent materials gives rise to an exciting and still largely unexplored materials design space^{18}. So far, significant emphasis has been placed on identifying the singlephase and multiphase equilibrium regions in an HEA phase diagram^{19,20,21,22}—where a singlephase HEA is defined by random mixing of all atoms on the same lattice in a homogeneous solidsolution. Thermodynamically, this singlephase HEA solidsolution will be stable if it is lower in free energy than any linear combination of its lowercomponent ordered intermetallics and solidsolutions.
Unfortunately, there is a lack of suitable tools to visualize thermodynamic competition in highcomponent chemical spaces, which impedes the guided exploration of HEAs. Traditional phase diagrams use barycentric composition axes, where compound compositions are given by the lever rule. A barycentric representation of an Ncomponent system requires D = (N – 1) spatial dimensions, meaning systems with N ≥ 5 components cannot be visualized in D ≤ 3 spatial dimensions. Past attempts to visualize HEA stability include projecting a fivecomponent phase diagram into three dimensions, or constructing a series of quaternary phase diagrams^{23}. However, these approaches do not scale to an arbitrary number of components, and do not quantitatively display relative thermodynamic relationships between phases—especially as a function of temperature. Developing a scheme to visualize stability relationships in highcomponent systems would help facilitate the discovery of promising HEA compositions.
Here, we propose an informationdense^{24,25} 2D representation that captures the essence of phase stability relationships in highcomponent systems while remaining easily interpretable. We choose to forgo barycentric composition axes in order to avoid the scaling relationship between components and dimensions; and instead adopt a graphbased representation of stability relationships. However, unlike previous graphbased stability representations, which do not have meaningful x and yaxes^{26,27}, we assign a formation energy axis and a reaction energy axis, capturing the absolute and relative stabilities of competing Ncomponent phases. Furthermore, we design a variety of plot features including color, linewidth, and marker shape to retain salient compositional information lost by eliminating barycentric composition axes.
We name these diagrams Inverse Hull Webs and use them to illustrate temperaturedependent phasestability during the quenching of HfMoNbTiZr and AlCrFeNi, which are experimentally reported to be a singlephase and a multiphase HEA system, respectively. Our Inverse Hull Webs capture highlevel trends in HEA solidsolution stability, as well as precise thermodynamic details regarding phase decomposition across the HEA composition space. Additionally, by animating our diagrams as a function of temperature, we can track the transition of an HEA solidsolution from stability to metastability upon cooling and identify which intermetallic compounds threaten phase separation in the HEA solidsolution phase. Finally, we identify two features from our Inverse Hull Webs that indicate the propensity of an HEA either to persist as a singlephase solidsolution at room temperature or to phase separate—and validate these features to have high classification accuracy against a dataset of 103 experimentallycharacterized HEAs.
More broadly speaking, our use of energy axes allows our Inverse Hull Webs to offer a complementary perspective to traditional phase diagrams, highlighting important thermodynamic features that are typically missing from traditional phase diagrams such as convex hull depth, reaction driving forces, metastability, and the tendency for phase separation. Our visualization is therefore useful for generally understanding materials stability, including in ordered, lowcomponent materials systems. Beyond Inverse Hull Webs, we believe that there may exist many creative ways to visualize phase stability relationships, which can serve as theoretical tools to discover and design advanced materials.
Results
Design of inverse hull webs
Phase stability is governed by the convex hull construction, as illustrated in Fig. 1a. By plotting the formation energy of materials against composition, the stable phases lie on the lower convex hull formed between compounds and their elemental reference states^{28}. Thermodynamically stable phases are vertices on the convex hull, while metastable phases have an energy above the convex hull at their associated composition^{29}. Phase coexistence occurs when a point lies on a facet, or simplex, of the convex hull. The convex hull of an Ncomponent system is plotted in a barycentric representation in (N − 1) dimensions. The scaling relationship between the number of components, N, and the number of spatial dimensions, D, is the source of the difficulty in visualizing stability relationships in highcomponent systems. Notably, a fivecomponent convex hull would need to be visualized in four spatial dimensions.
Here, we propose a 2D visualization scheme, which we name Inverse Hull Webs, that captures important information from the thermodynamic convex hull and does not use barycentric composition axes, meaning that our diagrams are not spatially constrained by the number of components in the system. Figure 1b illustrates the conversion of a traditional convex hull to our Inverse Hull Webs for the binary AlFe system at 1250 K. We construct Fig. 1 using DFTcalculated 0 K formation enthalpies of pure metals and intermetallic phases and the 1250 K regular solution free energy of the AlFe solidsolution, as described in the “Methods” section.
We use formation energy on the yaxis and inverse hull energy on the xaxis of our Inverse Hull Webs. Formation energy is the free energy difference between a phase and the linear combination of its corresponding pure elements, as noted in Fig. 1a by an arrow for AlFe_{3}. In the computational materials design community, it is conventional to define energies relative to the convex hull; a stable phase has ΔE_{hull} = 0 and metastable phases have an energy ‘above the hull’, which is the difference in formation energy between a metastable phase and its stable decomposition products^{29}. Here, we generalize this concept—defining the inverse hull energy to be the reaction energy to a phase from its stable neighbors in composition space—which we call the hull reactants. In other words, the inverse hull energy for a metastable phase is simply its ‘energy above the hull’, but for a stable phase, it is the energy below a hypothetical convex hull where the stable phase was removed, as illustrated for Al_{6}Fe in Fig. 1a. Stable and metastable phases can be quickly identified on an Inverse Hull Web, as they fall to the left or right of zero inverse hull energy (shown on the figure as 0 on the xaxis), respectively. We note that metastable phases are not shown on traditional triangular and tetrahedral phase diagrams. Additionally, our Inverse Hull Webs provide the same information regarding metastable phases as binary and ternary convex hulls, namely the energy of the metastable phase along with its composition. The equations used to calculate the formation energy of AlFe_{3} and the inverse hull energy of Al_{6}Fe in Fig. 1a can be found in Eqs. 1 and 2, respectively.
By eliminating barycentric axes, we lose important information regarding the composition of the phases. We design a variety of plot features—such as arrows, arrow width, marker shape, and color—to regain this compositional information. First, we use arrows to connect hull reactants to product phases, based on the inverse hull energy definition. We emphasize that these arrows do not represent tie lines, for example as was used by Hegde et al.^{26} in their phase stability network representation. While tie lines visualize twophase coexistence, our arrows indicate reactions between compositional neighbors on the convex hull. This distinction becomes clear in N ≥ 3 component systems, as the ‘hull reactants’ are not necessarily the phases connected to a compound by its tie lines. Supplementary Fig. 1 in the Supplementary Note can be referred to for a visualization of the difference between the hull reactants of a ternary compound and the phases it is connected to by tie lines.
We retain the coefficients of a stoichiometricallybalanced reaction by using arrow widths to represent the phase fraction of each hull reactant, where thicker arrows correspond to larger phase fractions. We use text color to represent phasetype and marker shape to represent the number of elements in the compound. Circles represent pure elements, squares represent binary compounds, triangles represent ternary compounds, diamonds represent quaternary compounds, and pentagons represent quinary compounds. In our study of HEAs, blue text indicates a solidsolution, red text indicates an ordered intermetallic, and black text indicates a pure element. In general, one should reference inverse hull energies to a compound’s stable neighbors in composition space. Here, in our specific study on HEA solidsolution stability, we also show relative energies between solidsolutions and intermetallics at the same composition, for example as shown for AlFe in Fig. 1.
We use color to represent the composition of each phase^{30,31}, shown by the color bar in Fig. 1, where Al is red and Fe is blue, and AlFe is purple. Our color scheme is based off of barycentric coordinates, where each vertex of an (N – 1) dimensional simplex in an Ncomponent compound is assigned a color. Meyer et al. proposed a method to project an Ndimensional simplex into a regular Ngon, allowing us to construct a twodimensional barycentric color legend^{32}, even for highcomponent systems. This is accomplished using Eq. 3 to calculate a 2D position corresponding to a set of vertex weights.
Where w_{j} refers to the barycentric weight of the j^{th} vertex, v_{j} refers to the 2D position of the j^{th} vertex, p refers to the 2D position of the point of interest, and θ_{j±1} refers to the angle between the lines connecting p to v_{j} and p to v_{j±1}.
We assign each pure element an RGB color, corresponding to the vertices of the Ndimensional simplex that represents a chemical system. We then find the color of each phase by taking the linear combination of pure element colors at the corresponding composition. We construct a color legend for our diagrams by projecting this Ndimensional simplex into a 2D polygon, using Eq. 3. Figure 2 demonstrates this barycentric projection with the stable phases in the Al−Cu−Co−Ni−Fe system at 1500 K, where each element lies on the vertex of a pentagon and all compounds lie in between.
One drawback of the 2D Ngon barycentric projection is that compositions do not have a unique 2D location in the Ngon. For example, CoNi and Al_{2}FeCo are at similar locations in the pentagon but represent very different compositions in Fig. 2. This limitation arises from the projection of a fivedimensional composition space into twodimensions and is unfortunately unavoidable. However, oftentimes the most stable ordered intermetallics in a system are clustered in a subregion of composition space, for example, as indicated by the high density of intermetallics (red squares) near Al in Fig. 2. Additionally, we label each composition of each phase with text in our diagrams, meaning any ambiguity in the composition of a phase can be clarified by inspecting the label of the phase. This means that highlevel compositional threats to HEA solidsolution stability can often still readily be identified with this colorimetric representation.
Our proposed representation shows quantitative, energetic relationships between phases in systems with any number of components, overcoming a limitation of previous attempts to visualize phase stability in highcomponent chemical spaces^{23}. While these features are more subjective than their representation in traditional phase diagrams, our diagrams offer a clear energetic description of a chemical system, offering a complementary visualization to existing phase diagram representations and, enabling a more holistic interpretation of the thermodynamic landscape in complex highcomponent material systems.
Application of Inverse Hull Webs to HEAs
HEAs are entropicallystabilized at high temperature by the configurational entropy that arises from mixing many elements on a single disordered lattice. A major goal in HEA development is to identify compositions that can form singlephase solidsolutions. Thermodynamically, the configurational entropy and Gibbs free energy of solidsolution and intermetallic phases can be found in Eqs. 4 and 5, respectively, where x_{i} refers to the molar fraction of the i^{th} component.
Here, we calculate the formation enthalpies of solidsolution phases using DFTcalculated regular solution parameters from Bokas et al.^{33} and use the formation enthalpies of ordered intermetallics as calculated in the Materials Project^{34}. We assume that the configurational entropy of all solidsolutions is ideal and that differences in vibrational entropy per atom between solid phases becomes negligible at high temperatures, as can be expected from the law of Dulong−Petit^{35}. Here, we do not consider magnetic contributions to the free energy of phases, due to the significant additional computational expense to account for these magnetic interactions—however, it is straightforward to modify the free energies of various phases with magnetic interactions or vibrational entropy contributions within our visualization.
The ΔS for a fivecomponent solidsolution is 1.6R, meaning the TΔS_{config} contribution at 1000 K is –0.14 eV per atom. However, formation energies of ordered binary intermetallics are often more negative than –0.3 eV per atom, meaning that below 1000 K there is a high probability that intermetallics become thermodynamically favored over solidsolutions with high configurational entropy. This means that many singlephase HEAs may be metastable at room temperature relative to these ordered lowercomponent intermetallics. Thus, many singlephase HEAs should be temperaturestabilized somewhere between room temperature and the melting point of the alloy.
We will use our Inverse Hull Webs to visualize the transition of an HEA solidsolution from stability to metastability as it is quenched and illustrate how it can be used to guide the design of compelling HEA compositions. We will examine two HEAs: HfMoNbTiZr, which was experimentally reported to be singlephase HEA when quenched; and AlCrFeNi, which phaseseparates upon quenching. In our analysis, the only metastable phases we consider are solidsolution phases, and not metastable intermetallics. We display Inverse Hull Webs of these systems at a series of representative temperatures during quenching. Additionally, we have included Python code and a Jupyter notebook in Supplementary Data 1 which can be used to generate these Inverse Hull Webs. We have also included Inverse Hull Webs of 103 equimolar, ascast HEAs listed in^{36} in Supplementary Data 2. Users can zoom into the Inverse Hull Webs when using the Python code, elucidating crowded areas of the diagram.
Inverse hull web of an experimentallyobserved singlephase HEA
Here, we examine HfMoNbTiZr, an HEA that Tseng et al. found to be a single BCC solidsolution phase at room temperature after being cast through vacuum arc melting and remelted at least four times^{37}. First, we present a visualization of phase stability in this system using traditional phase diagrams, as shown in Fig. 3. This appears as a series of tetrahedral phase diagrams, one for each of the fourelement combinations found in Hf−Mo−Nb−Ti−Zr. These phase diagrams are calculated at 2896 K, the highest melting temperature of the pure elements in the system.
In this traditional phase diagram representation, the HEA solidsolution phase is not present in Fig. 3, as a fivecomponent compound cannot be shown on a tetrahedral phase diagram. There are many tie lines present on each of the tetrahedral diagrams, and the 3D aspect of the diagrams means that many of these tie lines overlap. This makes it difficult to discern the multiphase regions and obfuscates the use of barycentric coordinates to evaluate phase fractions by the lever rule.
In contrast, the Inverse Hull Web of HfMoNbTiZr is shown in Fig. 4 at four important temperatures during a quenching process; first, at the Mo melting temperature, 2896 K; and lastly, at room temperature, 300 K. Our Inverse Hull Webs offer a complement to the traditional phase diagram visualization, as we are able to clearly display the relative energy and hull reactants of each of the phases, features not found in Fig. 3. We also show the Inverse Hull Web at two critical temperatures that we have identified during quenching.
We call the first critical temperature the ‘critical solidsolution’ temperature, or the lowest temperature where the HEA solidsolution is stable, which is 563 K in this system. This temperature represents where the HEA solidsolution becomes metastable during quenching, giving insight into the likelihood the alloy will be found as a single phase at room temperature. This temperature also represents the critical miscibility temperature of the HEA solidsolution, where below the ‘critical solidsolution’ temperature, at least one element is no longer fully miscible in the HEA lattice. The next critical temperature we consider is the alloy’s ‘critical adjacent phase’ temperature, or the lowest temperature where the Ncomponent HEA solidsolution phase’s hull reactants are all of the compositionallyadjacent, (N – 1) component solidsolutions, which is 1167 K in this system. This serves as a proxy for the temperature where enthalpic effects in the competing solidsolution and ordered phases become important.
In Fig. 4, we see that intermetallics do not offer significant thermodynamic competition in the HfMoNbTiZr system; the binary intermetallics (in red text) all have formation energies more positive than −0.05 eV per atom, and there are no highercomponent intermetallics. Out of all of the solidsolutions, MoTi is the only phase likely to pose a threat to HEA stability in the Hf−Mo−Nb−Ti−Zr system. Solidsolutions are stable across much of this system, indicated by the large number of blue marker labels, and three out of the four stable intermetallics contain Mo while the fourth, HfZr, has negligibly small formation energy. HfMoNbTiZr has the lowest formation energy in the entire fivecomponent chemical space, and at the Mo melting temperature, has a sizable inverse hull energy of −0.05 eV per atom.
As the system is quenched, HfMoNbTiZr’s ‘critical adjacent phase’ temperature is 1167 K, meaning enthalpic effects do not become important until roughly 1700 K below Mo’s melting temperature. The HEA solidsolution is stable until only a few hundred Kelvin above room temperature as it has a ‘critical solidsolution’ temperature of 563 K. Additionally, the HEA solidsolution is only slightly metastable at room temperature with an energy above the hull of 12 meV per atom, a small energy value relative to the scale of metastability in quinary compounds^{29}. Altogether, these features suggest that HfMoNbTiZr is likely to persist as a metastable single solidsolution at room temperature.
In Fig. 4, we see that the HEA solidsolution initially decomposes into Zr and HfMoNbTi when it becomes unstable, indicated by the arrows extending from the pentagon at 563 K in Fig. 4. This means the quaternary HfMoNbTi solidsolution is more stable than the quinary HfMoNbTiZr alloy, suggesting that the addition of Zr to HfMoNbTi may give rise to a miscibility gap in the alloy at lower temperatures. Additionally, the quaternary HfMoNbTi solidsolution has the lowest formation energy in the system at 1167 K, further suggesting the addition of Zr to HfMoNbTi increases the energy of the solidsolution. Supplementary Video 1 can be referenced to watch the HfMoNbTiZr solid solution transition from stability to metastability at 563 K.
Our Inverse Hull Webs display the low critical temperatures, lack of energetically deep intermetallic phases, and low metastability of the HEA solidsolution at room temperature in the Hf−Mo−Nb−Ti−Zr system, allowing us to infer that this system is amenable to persist as a metastable single solidsolution HEA at ambient conditions. These stability relationships corroborate the experimental observations of Tseng et al of a room temperature solidsolution in HfMoNbTiZr, demonstrating the utility of our proposed diagrams.
Inverse hull web of an experimentallyobserved multiphase HEA
Our Inverse Hull Webs can also be used to visualize features that may inhibit single solidsolution stability in HEAs. We demonstrate this with AlCrFeNi, a derivative of the CoCrFeMnNi Cantor alloy achieved by replacing Co and Mn with Al. The Cantor alloy was one of the first HEAs found to exhibit a single solidsolution microstructure, however it was later discovered that the HEA solidsolution phase becomes metastable at around 800 °C^{18,19}. Chen et al found the AlCrFeNi alloy to be a phase separated AlNi B2 intermetallic and CrFe BCC solidsolution at room temperature after being cast through arc melting and remelted five times^{38}.
We show the AlCrFeNi Inverse Hull Web in Fig. 5 at 3853, 2180, 1650, and 300 K, corresponding to AlCrFeNi’s ‘critical adjacent phase’ temperature, Cr’s melting temperature, AlCrFeNi’s ‘critical solidsolution’ temperature, and room temperature. A visualization of phase stability in this system using traditional phase diagrams can be found in Supplementary Fig. 2, where we show a series of tetrahedral phase diagrams of the Al−Cr−Fe−Ni system at the four temperatures used in Fig. 5.
Figure 5 shows that Al−Ni intermetallics threaten HEA solidsolution stability in all alloy systems containing these elements. Al−Ni intermetallics have very negative formation energies of −0.6 to −0.7 eV, as well as small inverse hull energies of −0.01 to −0.05 eV at all castingrelevant temperatures, which is visually apparent by the large number of green−yellow markers in the lower right of every Inverse Hull Web in Fig. 5. This suggests that the Al−Ni isopleth is littered with competing lowenergy compounds.
Upon casting, the stability of the HEA solidsolution is immediately threatened by Al−Ni intermetallics. The highest melting temperature in the Al−Cr−Fe−Ni system is 2180 K, while the ‘critical adjacent phase’ temperature is 3853 K. In this alloy, the ‘critical adjacent phase’ temperature is the lowest temperature where the fourcomponent solidsolution’s hull reactants are each of the threecomponent solidsolutions, meaning below this temperature the single solidsolution begins to compete with either binary solidsolutions or ordered intermetallics. Figure 5 shows that the HEA solidsolution has a similar formation energy as Al−Ni intermetallics at 3853 K, suggesting that these intermetallics would possess a large bulk driving force to nucleate at high temperatures, which would threaten the stability of an HEA solidsolution.
The HEA solidsolution in this system then becomes metastable at 1650, 600 K higher than in the Cantor alloy and more than 1000 K higher than in HfMoNbTiZr. Additionally, the HEA solidsolution is highly metastable at room temperature with an energy above the hull of 0.15 eV per atom, a large energy value relative to the scale of metastability in quaternary compounds, and more than 0.10 eV per atom more metastable than HfMoNbTiZr at room temperature^{29}. Supplementary Video 2 can be referenced to watch the AlCrFeNi solid solution transition from stability to metastability at 1650 K down to its high inverse hull energy at room temperature.
Altogether, AlCrFeNi’s high ‘critical adjacent phase’ temperature relative to its constituents’ melting temperatures, high ‘critical solidsolution’ temperature, and high metastability at room temperature are characteristic of phaseseparating HEA systems. Even though this system is only a quaternary HEA, the threat posed by lowenergy Al−Ni binary intermetallics should threaten other HEAs that contain these two elements. Our Inverse Hull Webs rationalize the experimental observations of Chen et al. of phase separation in this system into an AlNi intermetallic and a CrFe solidsolution.
Analysis of critical temperatures in HEAs
The critical solid solution and critical adjacent phase temperatures of an HEA are features that may be more broadly indicative of HEA solidsolution stability. We next calculate these critical temperatures for 103 equimolar, ascast HEAs found in ref. ^{36} and use each temperature individually to classify the HEAs as single or multiphase. Here, a single cutoff value of the critical temperature is used to classify HEAs as either single or multiphase, where alloys with critical temperatures below the cutoff value are classified as single phase and alloys with critical temperatures above the cutoff value are classified as multiphase. We note that this is a oneparameter linear support vector classification, where a line is fit to distinguish between the two classes of HEA phase stability under consideration^{39}. Figure 6 shows the critical temperatures of each HEA divided by the highest melting temperature in the HEA system, along with the fraction of HEAs that were correctly classified as single or multiphase. Relative ‘critical solidsolution’ temperature is shown in Fig. 6a and relative ‘critical adjacent phase’ temperature is shown in Fig. 6b.
Figure 6 suggests that the single solidsolution needs to be stable at half of the casting temperature and entropic effects need to dominate down to 4/5 of the casting temperature for the HEA to be a single solidsolution at room temperature. This can be seen in Fig. 6 as the cutoff value of relative “critical solidsolution” temperature is 0.78 the cutoff value of relative “critical adjacent phase” temperature is 0.48. AlCrFeNi and HfMoNbTiZr were correctly classified with both temperatures, and the Cantor alloy was as well.
Figure 6 shows that both ‘critical solidsolution’ temperature and ‘critical adjacent phase’ temperature are effective at distinguishing between single and multiphase HEAs, achieving classification accuracies of 80.6 and 78.6%, respectively, even when used in isolation and selecting a single cutoff value. This demonstrates the effectiveness of the critical temperatures in distinguishing between single and multiphase HEA compositions, suggesting that these temperatures may be valuable when used in a machine learning model aimed at predicting HEA phase stability^{40,41,42,43,44}. Machine learning models have been applied to predict HEA phase stability in the past, often using features such as the ideal mixing entropy, regular solution enthalpy, atomic size difference, and electronegativity difference of the HEA composition; these models are able to achieve classification accuracies > 80% by using several features in complex machine learning models, such as neural networks^{45,46,47}.
Our critical temperatures, by themselves, are able to achieve comparable classification performance as these machinelearning models, which demonstrates that the use of physicallymotivated features directly relevant to HEA solidsolution stability may be more relevant in the prediction of HEA phase stability than sophisticated machine learning methods applied on lessrelevant features^{48}. By including our critical temperature features as future machinelearning features, we anticipate that machinelearned classification performance can improve even further.
Discussion
Our Inverse Hull Webs display both highlevel trends in temperaturedependent phase stability, as well as precise details regarding the thermodynamic competition between compounds. Importantly, by displaying salient thermodynamic information in 2D for any Ncomponent system, we overcome a major limitation of traditional phase diagrams for analyzing highcomponent systems—as shown here to rationalize solidsolution stability in HEAs. Regions of composition space that pose a threat to HEA solidsolution stability can be identified by compounds with very negative formation energies, such as Al−Ni intermetallics in the AlCrFeNi system. The thermodynamic incentive for an HEA to persist as a solidsolution or to decompose can be assessed by its temperaturedependent inverse hull energy. Altogether, these Inverse Hull Webs can serve as valuable tools to assist in the guided discovery of experimentallysynthesizable disordered (or ordered) highcomponent materials.
Our analysis of single solidsolution stability in these highentropy alloys further reveals two important features—the ‘critical solidsolution’ temperature and ‘critical adjacent phase’ temperature. ‘Critical solidsolution’ temperature is the lowest temperature where the HEA solidsolution is stable and thus estimates where the HEA solidsolution begins to decompose. ‘Critical adjacent phase’ temperature is the lowest temperature where the Ncomponent HEA solidsolution’s hull reactants are each of the (N−1) component solidsolutions, estimating where entropic effects no longer dominate. These features may illuminate which HEA compositions are likely to be found as a single solidsolution phase at room temperature and may be valuable features for a machinelearning model aimed at predicting singlephase HEA compositions.
Generally speaking, our Inverse Hull Webs offer a thermodynamic perspective that is complementary to the traditional representation of a convex hull. Typical phase diagrams visualize only the equilibrium phases at a given composition and do not provide formation energy or reaction energy information. Our Inverse Hull Webs reveal other important aspects of the thermodynamic landscape, such as convex hull depth, reaction driving forces, metastability, and the tendency for phase separation, all of which are difficult to ascertain from the traditional phase diagram. Even though our Inverse Hull Webs do not use barycentric composition axes, we are able to recover composition information by utilizing a variety of plot features, such as color, line width, and marker shape. We hope that by rethinking the classical approach to constructing phase diagrams, our work inspires the creation of yettobedesigned visualizations that can illuminate important trends in the synthesisstructurepropertyperformance relationships of advanced materials.
Methods
Calculation of compound free energies
Intermetallic free energies were obtained from The Materials Project database^{34} through the pymatgen Python package^{49}. Solidsolution free energies were calculated using a regular solution model as shown in Eq. 6, where N refers to the number of components, x_{i} refers to the mole fraction of the i^{th} element, and Ω_{ij} is the parameter that quantifies the interaction resulting from mixing the i^{th} and j^{th} elements.
Bokas et al. used DFT free energies of binary solidsolutions to fit a regular solution parameter Ω for each binary pair of elements out of 27 elements commonly used in HEAs, using the Special Quasirandom Structure (SQS) method to calculate the energy of solidsolution phases^{33}. A positive value of Ω corresponds to a binary alloy system that displays an isostructural miscibility gap, while a negative value of Ω corresponds to a binary system where mixing is enthalpically preferred. These regular solution parameters were used to calculate solidsolution free energies in our Inverse Hull Webs. The lower free energy of the FCC and BCC solidsolutions was found for each equimolar combination of elements in the chemical system being plotted. All compound free energies were referenced to the stable phases of the pure elements composing the compound. Formation and inverse hull energies were calculated using pymatgen^{49}. All Inverse Hull Webs were constructed using the matplotlib Python package^{50}.
Data availability
A dataset of the critical temperatures used to create Fig. 6 can be found in Supplementary Data 2 as a csv file. Inverse Hull Webs of all 103 HEAs shown in Fig. 6 can be found in an online repository, https://www.dropbox.com/sh/0blhpbhu7ol510i/AADRKBl4WYAT3bvRqCW_w0Jya?dl=0. Each Inverse Hull Web in this repository follows a similar format to Figs. 4 and 5, where a large Inverse Hull Web is shown at the highest melting temperature of the pure elements and three smaller Inverse Hull Webs are shown at the alloy’s ‘critical adjacent phase’ temperature, ‘critical solid solution’ temperature, and room temperature.
Code availability
The code used to obtain the free energy of phases and construct Inverse Hull Webs can be found in Supplementary Data 1. Python classes to build and construct Inverse Hull Webs can be found in inverse_hull_web.py in Supplementary Data 1 and ihw_tutorial.py in Supplementary Data 1 walks through the calculation of solid solution free energies and subsequent Inverse Hull Web Construction.
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Acknowledgements
This work by D.N.E., J.C., and W.S. was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under Award #DESC0021130. The work by G.B., W.C. and G.H. was funded by the Walloon Region under the agreement no. 1610154 EntroTough in the context of the 2016 WalIinnov call. Computational resources were provided by the supercomputing facilities of the UCLouvain and the Consortium des Equipements de Calcul Intensif (CECI) en Federation Wallonie Bruxelles, funded by the Fonds de la Recherche Scientifique de Belgique (FNRS).
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J.C. and W.S. conceived the original research idea. D.N.E developed and applied the visualization to HEA systems and wrote the manuscript with feedback from W.S. G.B., W.C. and G.H. calculated the regular solution parameters used in the thermodynamic model. All authors contributed to revising the paper.
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Evans, D., Chen, J., Bokas, G. et al. Visualizing temperaturedependent phase stability in high entropy alloys. npj Comput Mater 7, 151 (2021). https://doi.org/10.1038/s41524021006261
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DOI: https://doi.org/10.1038/s41524021006261
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