‘High Entropy Alloys’ (HEA) represent a recent class of materials which constitute multiple principal elements in equiatomic or non-equiatomic proportions. The presence of a number of principal elements, particularly in equiatomic proportion, leads to a large configurational entropy and thus, high entropy of mixing. Consequently, these alloys tend to form solid solutions with simple crystal structures like FCC, BCC, or a mixture of both, rather than intermetallics or other complex phases.[1,2,3,4] A variety of HEAs have been explored till date and it has been reported that these alloys exhibit very promising properties like high strength, ductility, excellent fracture toughness at cryogenic temperatures (i.e., ≤ 77 K),[5] good resistance to wear, oxidation, and high temperature softening. These properties are directly related to the underlying crystal structure of the solid-solution phase. However, due to the absence of reliable information about the phase diagrams of multicomponent systems, the knowledge about phase formation in such systems is restrictively dependent upon the experimental determination. Lately, in order to tackle this present limitation of classical thermodynamics-based phase prediction, there have been various attempts by researchers to lay down certain thumb rules for phase formation based upon first principle calculations, Calculation of Phase Diagram (CALPHAD) principles, as well as other thermodynamic and topological parametric approaches.[6,7,8,9,10,11,12,13,14,15,16,17]

The existing phase prediction techniques range from the first principle-based DFT calculations to CALPHAD-based Phase Diagram Evaluation techniques.[18,19,20,21,22] The first principle-based calculations are very much computationally intensive; however, these are the most efficient techniques when it comes to determination of ground-state crystal structure. These basically involve the solution of the Kohn–Sham formulation[23] which is a minimization procedure of a coupled set of Schrodinger Equations for each electron in the system; being solved individually, yet self-consistently for the electron density, ρ(r). The Kohn–Sham formulation is expressed as mentioned in Eq. [1], where the first term on the left represents kinetic energy of a single electron; the second term defines electron–ion coupling potential; the third term reflects the electron–electron interaction; and the fourth term is for the exchange correlation potential[18]:

$$ \left\{ {\left\lceil {\frac{{ - h^{2} }}{{2m}}\left( {\nabla ^{2} r} \right)} \right\rceil + V_{e} (r) + {\mkern 1mu} \int {{\rm{d}}r} \frac{{\rho (r^{\prime } )}}{{|r - r^{\prime } |}} + {\mkern 1mu} V_{{xc}} \left[ {\rho \left( r \right)} \right]} \right\}\Psi _{i} \left( r \right){\mkern 1mu} = {\mkern 1mu} {\text{ }}{\epsilon}_{i}\Psi i\left( r \right) $$
(1)

The Kohn–Sham formulation assumes an initial crystal structure depending upon the symmetry of which the single-electron wave function is formulated in a plane wave basis. This is followed by the adoption of an approximation technique (like Local Density Approximation or Generalized Gradient Approximation) to obtain the exchange correlation potential and finally by selection of a technique to determine the electron–ion coupling potential. This formulation is then solved to obtain the lowest energy structure in the T ≈ 0K limit. The obtained energy can be scaled to any finite temperature using Cluster Expansion models which take care of the various entropy contributions to the total energy with increasing temperature. The greatest advantage of using DFT for the ground-state structure prediction is the ability to evaluate the total energy across a spectrum of crystal structures, some of them being even hypothetical.[19,20]

Another method for prediction of stable phases in such alloys is the Calculation of Phase Diagram technique (CALPHAD). CALPHAD makes use of the Gibbs energy minimization principle to determine the stable and metastable structures that would form in a system.[21,22,24] In this technique, the Gibbs energy of a particular phase is expressed in the form a Redlich–Kister-type equation as given in Eq. [2], where the first term depicts the contribution of the pure components present in the phase, the second term signifies the ideal mixing contribution to Gibbs energy, and the third and fourth terms are the excess Gibbs energy of mixing calculated by using the binary and ternary interactions among the components.[24,25]

$$ G_{phase} = \mathop \sum \limits_{i} X_{i} G_{i}^{0} + \mathop \sum \limits_{i} X_{i} lnX_{i} + \mathop \sum \limits_{i} \mathop \sum \limits_{j > i} X_{i} X_{j} \mathop \sum \limits_{n} \omega_{ij}^{n} \left( {X_{i} - X_{j} } \right)^{n} + \mathop \sum \limits_{i} \mathop \sum \limits_{j > i} \mathop \sum \limits_{k > j} L_{ijk} X_{i} X_{j} X_{k}. $$
(2)

A recent work by Raghavan et al.[7] in predicting the stable crystal structure using the CALPHAD approach has revealed more accurate prediction for the BCC phase than the FCC. This has been attributed to the inability of CALPHAD to deal with kinetic effects which have a more dominant role to play in the formation of latter crystal structure. Currently, there exist dedicated databases of various thermodynamic parameters for multicomponent-based HEAs which take into account all possible binary interactions and tentatively the involved ternary ones as well.[26] However, there is a lack of good-quality data on ternary interactions especially at low temperatures and therefore, the structure prediction is still dependent on various extrapolation schemes. This is the reason why in certain cases CALPHAD predictions do not match with the experimentally obtained structure of various phases. Some of the examples of such ambiguity include the CoReRuV system which forms a complex multiphase structure in contrast to the single-phase HCP structure predicted by CALPHAD.[27] Similarly, in CoCrCuFeMn and CoCrMnNiV systems, CALPHAD predictions underestimate the low-temperature thermal stability of the sigma (σ) phase, which has been observed experimentally.[28] Although the above-mentioned examples utilize the TCNI database of different versions, it has to be understood that the present HEA databases are not very different and derive much of its data from the TCNI and other similar databases, hence inherit the limitations as well. The problem would pertain until the databases contain either experimental or first principle-based data for binary and ternary interactions across the entire composition range of a given system. Thus, a combination of DFT-based approach and CALPHAD can prove to be an extremely powerful technique in predicting the stable structures in multicomponent systems. However, until this combined approach becomes an exhaustive technique, the use of various thermodynamic and topological parameters for prediction of stable crystal structure would gain popularity. These parameters are essentially one or the other form of the various metallurgical parameters that were laid down by Hume Rothery as ‘Rules for Solid Solution Formation’ which in a very broad sense may be stated as follows[29,30,31]:

  • Atomic size difference of the components—solid solubility of the solute ceases as soon as the atomic size of the solute differs from that of the solvent by more than 15 pct.

  • Electrochemical difference among the components—larger the difference in chemical affinity of solute and solvent, higher is the tendency towards formation of intermediate compounds rather than solid solutions.

  • Electron concentration change on alloying—the extent of solid solution in a solvent of lower valence is always greater than the vice versa.

All such parameters being a formulation of either one or a combination of few of the Hume Rothery rules, are applicable only for room-temperature stable phase prediction. The important thing that has to be understood with respect to various such parameters is that they represent mere thumb rules based upon certain functions which intuitively can be related to either first principle-based outcomes like kinetic energy of electrons (reflected by VEC) or CALPHAD-based outcomes like Gibbs energy of the system (reflected by Mismatch Entropy). One strong limitation of such parameters is their incapability to predict the formation of complex crystal structures in multicomponent alloys like the sigma and the Laves phases, etc. These parameters will only act as a thumb rule in case of predicting simple ground-state structures like FCC and BCC at room temperature.

In the present work, an effort has been made to reassess the phase predicting capability of various such thermodynamic and topological parameters across a wide range of HEA systems. In continuation of the present effort, two new parameters have also been designed to predict the stable crystal structure that would form in the HEA systems considered for assessment.

The already existing parameters which have been reassessed are valence electron concentration (VEC), atomic mismatch (δ), electronegativity difference (Δχ), mixing entropy (ΔS mix), entropy of fusion (ΔS f), and mismatch entropy (S σ ). The two new parameters formulated are the Modified Darken–Gurry parameter (A) and the Modified Mismatch Entropy parameter (B). Most of these parameters have been used in the previous studies for predicting whether the system will form a solid solution or an intermetallic or an amorphous phase with only a few attempts for crystal structure prediction.[10,11,29,32,33,34,35,36,37] In the present work, these parameters are computed for over 150 different HEA systems whose stable crystal structures have already been reported in literature elsewhere.[7,38,39,40,41] These parameters are then plotted individually in a 1D map. Based upon certain thresholding, the regimes for different crystal structures (FCC, BCC, and FCC + BCC) are demarcated with an attempt to explain the underlying principle.

The HEA systems under consideration consist of a combination of elements (listed in Table I) in different proportions. Among the systems chosen for study, 53 have FCC, 42 have FCC + BCC, and 58 have BCC as the stable crystal structure (experimentally proven). Some of the input values used for the computation of various parameters viz. atomic radii, electronegativity, VEC, ΔS f are also listed in Table I.

Table I List of Elements Present in the HEA Systems Considered Along with Some of Their Properties
Full size table

1: Valence Electron Concentration (VEC): This parameter is basically the integration of the density of states (DOS) of s-, p-, and d-electrons in the valence band of a particular element.[9,10,42] The VEC value in case of iron, for example is 8 as its configuration in the outer shells is 4s2 3d6. So, by adding up the 4s and 3d electrons, it gives the VEC value of 8. Thus for the overall system, this is calculated using the VEC values of individual elements as presented in Eq. [3].

$$ VEC = \mathop \sum \limits_{i = 1}^{N} C_{i} *VEC_{i} $$
(3)

C i in all cases represents the atomic fraction of the ith component.

2: Atomic mismatch (δ): This is basically the measure of difference in atomic size values of constituent elements from the mean size as presented in Eq. [4].

$$ \delta = 100*\sqrt {\mathop \sum \limits_{\text{i = 1}}^{N} C_{i} *\left( { 1- \left( {\frac{{r_{i} }}{{\bar{r}}}} \right)^{2} } \right)}, $$
(4)

where r i is the atomic radii of constituent element and \( \bar{r} = \mathop \sum \limits_{i = 1}^{N} C_{i} *r_{i}. \)

3: Electronegativity difference (Δχ): This is basically the measure of difference in electronegativity of each constituent component from the mean electronegativity of the system as presented in Eq. [5].

$$ \chi \, = \mathop \sum \limits_{i = 1}^{N} C_{i} *(\chi_{i} - \bar{\chi })^{2} $$
(5)

where \( \bar{\chi } = \mathop \sum \limits_{i = 1}^{N} C_{i} \chi_{i} \) and χ i is the Pauling Electronegativity of the ith component.

4: Mismatch Entropy (S σ ): This parameter signifies the change in entropy due to atomic size mismatch. High value of mismatch entropy means the change in overall entropy due to difference in atomic sizes is more. Hence, it can be inferred that there will be less number of total possible indistinguishable arrangements and thus, lesser overall configurational entropy. This parameter has been formulated as given in Eq. [6] by Mansoori et al.,[43] based on the model of mixture consisting of hard spheres.

$$ S_{\sigma } = k*\left\lceil {\frac{3}{2}\left( {\zeta ^{2} - 1} \right)y_{1} + \frac{3}{2}\left( {\zeta - 1} \right)^{2} y_{2} - \frac{1}{2}\left\{ {\left( {\zeta - 1} \right)\left( {\zeta - 3} \right) + {\text{ln}}\zeta } \right\}\left( {1 - y_{3} } \right)} \right\rceil $$
(6)

where \( \zeta = \frac{1}{1 - \xi } \) and ξ = 0.64 for dense pack solids. The terms, y 1 and y 2 are actually directly proportional to the square of the difference in the atomic sizes and y 3 is simply a ratio of the projected area and volume of different atoms each raised to an exponent of 3 and 2, respectively. For the calculation of y 1, y 2, and y 3, the value of atomic diameter (di) of each constituent component have been used.

5: Entropy of mixing (ΔS mix): The configurational entropy attained by the system is defined as given below in Eq. [7],

$$ \Delta S_{\text{mix}} = \, - R*\mathop \sum \limits_{i = 1}^{N} C_{i} *\ln C_{i}, $$
(7)

where R is the universal gas constant and C i is the concentration on each element present in the system.

6: Entropy of Fusion (ΔSf): This depicts the change in the entropy of the system when there is a phase transformation from liquid to solid or vice versa. For the overall alloy system, it is computed as given in Eq. [8],

$$ \Delta {\text{S}}_{\text{f}} = \mathop \sum \limits_{i = 1}^{N} C_{i} *S_{f}^{i}, $$
(8)

where \( S_{f}^{i} \) is the entropy of fusion of individual element.

Each parameter calculated using the equations given above is tabulated in Table II and 1D maps for each parameter was plotted where a particular type of crystal structure was assigned a unique representation. The analysis of the spread and the trend of each of the parameters were done to threshold it into different crystal structure regimes. Figures 1 represents the 1D maps of the computed values of VEC, δ, Δχ, and S σ . The plots for VEC, Atomic mismatch, and Mismatch entropy show a good representation of demarcation of the various phase regimes in the systems considered. As reported earlier,[9,10,11] the systems with lower VEC tend to form BCC and those with higher values tend to form FCC. Here, it can be seen that VEC values less than 7.5 favor the BCC structure and values higher than 7.9 favor the FCC. This may be attributed to the fact that the Brillouin zones of the noble metals are only partially filled with electrons; although they are touched by the Fermi surface, they are not overlapped as it happens in the case of the transition elements (or elements with higher valence). Thus, when the value of VEC is low, the presence of low valence components is higher which yields a low e/a ratio or in other words lesser number of loosely bound electrons. Therefore, the peak of DOS for the electrons in the space where the Brillouin zone and Fermi surface interact (which is basically a representation of kinetic energy of electrons), occurs at lower energy levels. Since the electrons to be accommodated already possess a lower total energy, the crystal structure can be of a more open nature, i.e., the BCC. In case of higher VEC due to the higher number of loosely bound electrons or greater extent of overlapping of the Brillouin zone and Fermi surface (due to higher fraction of high valence components), the peak of DOS for the electrons occurs at higher energy levels and thus, the electrons which have to be accommodated are present with a higher total energy. This ultimately means that the crystal structure has to be of a closed nature (i.e., FCC) so as to minimize the kinetic energy of the electrons.[10,11,30,31,42]

Table II Comparison of Base Structure of Various Systems Using VEC, Atomic Mismatch, Mismatch Entropy, Modified Darken–Gurry Parameter, and Modified Mismatch Entropy Parameter
Full size table
Fig. 1
figure 1

1D maps of the computed values of (a) VEC, (b) δ pct, (c) Δχ, and (d) ΔS σ . (Triangles represent BCC phase, rhombi represent FCC phase, and squares represent mixed FCC+BCC phase)

Full size image

The atomic mismatch, δ values indicate that higher values favor the BCC phase, whereas the FCC phase is favored by lower values (δ < 5 favors the FCC and δ > 5.4 favors the BCC). The same trend can be seen for S σ as S σ  < 0.06 favors the FCC and S σ  > 0.07 favors the BCC. These two values are misfit parameters which indicate the strain in the system and the loss in overall configurational entropy due to the difference in the size of the constituent elements, respectively. Higher value of δ means the system possesses more strain energy and to relieve itself from this it tends to attain more open structure, i.e., the BCC structure.

The plot for electronegativity difference does not show very clear demarcation of phases. All the systems have values nearly in the range of 0.09 to 0.17 as only metals are involved.

Figures 2(a) and (b) presents the 1D maps of the computed values of ΔS mix and ΔS f, respectively. The computed plots of ΔS mix and ΔS f do not shed much light upon the phase regime demarcation, as all the three phases share almost equal values for both the parameters. This is because the ΔS mix is only dependent upon the number of different elements present in the system and does not take into account the nature of the elements. Similarly, according to Richard’s rule,[21] the entropy change per mole, ΔS f, during solid-to-liquid transformation is approximately one gas constant R (magnitude wise) for all the systems irrespective of the crystal structure.

Fig. 2
figure 2

1D maps of the computed values of (a) ΔSmix and (b) ΔSf

Full size image

Two new parameters which have been introduced here are the Modified Darken–Gurry parameter (A = S σ * χ) and a Modified Mismatch Entropy parameter (B = δ* S σ ). The 1D maps of these parameters for various HEA systems have been shown in Figure 3. It can be clearly seen that for alloys with values of A > 0.0095, BCC is the stable crystal structure and those with A < 0.0060, FCC is the stable phase. The Modified Darken–Gurry parameter has actually been derived from the concept of Darken–Gurry (DG) maps[31,44] which takes into account the simultaneous effect of the difference in atomic size and electronegativity of various elements on solid solution formation. These maps basically consist of atomic size on abscissa and electronegativity on the ordinate axes where any element can be represented by a point, and if two points lie close to each other, then they tend to form solid solutions across a wide composition range. The justification for this lays in the small difference in the atomic size and electronegativities of the different components tending the system towards a solid solution. The points which are positioned far on such a plot would implicate that the corresponding components would either tend to form intermetallic or an amorphous phase in the solid state. However, in the present work, the aim is to explore within the solid solution regime of various elements on respective DG maps, whether there is any presence of crystal structure-based demarcation. For this, atomic size has been replaced by mismatch entropy so as to render more thermodynamic analogy to the parameter and its product with electronegativity difference has been considered so that an area on the DG map could be reflected. Now, as the value of A is lower for a particular multicomponent system, the difference in electronegativities of the constituent components as well as the overall mismatch entropy is lower. Thus, the system is on the lower side of residual strain (due to atomic size mismatch) and with a nearly equivalent affinity of each component for the rest, tends to form a solid solution with the closest packed structure. However, when the value of A is higher yet in the regime of solid solution for the respective components on the DG map, the system tends to opt for a rather open structure so as to accommodate the strain coming from atomic size mismatch and to reduce the difference in affinity of each component for the rest. This trend of stable structure prediction can also be thought of in terms of an energy minimization notion. The ground-state structure has to be such that it minimizes the strain energy as well as electronic kinetic energy. In case of the high value of A, where the mismatch entropy and the difference in electronegativity are high, the BCC structure would be preferred as the default presence of components with higher affinity for electrons would lower the kinetic energy and an open structure would relax the strain energy due to size mismatch. On the other hand in case of a lower value of A, there would be a requirement of a close packed structure (FCC) for an overall lowering of the electronic kinetic energy.

Fig. 3
figure 3

1D maps of the computed values of (a) A = S σ *χ and (b) B = δ* S σ

Full size image

Similarly, for alloys with values of Modified Mismatch Entropy parameter, B (=δ* S σ ) > 0.38, the BCC phase is the stable one, whereas for B < 0.2, FCC is the stable one. This parameter is again a product of a thermodynamic and a topological parameter which follows the same philosophy in phase demarcation as followed by δ and S σ individually.[45] Table III shows the prediction accuracy of the various parameters which were found to be suitable for estimating the ground-state structure.

Table III Overall Prediction Accuracy of Some of the Parameters Under Consideration Viz. VEC, Atomic Mismatch, Mismatch Entropy, Modified Darken–Gurry Paramater, and Modified Mismatch Entropy Parameter, When Calculated Across the Total Number of Systems
Full size table

It is re-emphasized here again that in the present assessment the thresholding limits of various parameters used to demarcate different crystal structure regimes have been set keeping in view the maximum containment of either single-phase FCC or single-phase BCC exhibiting systems within the specified range. However, a close analysis of the spread of mixed FCC + BCC-containing systems (in Figure 3) reveals that in most cases both the Modified Darken–Gurry as well the Modified Mismatch Entropy parameters predict either a mixed FCC + BCC or single-phase BCC crystal structure for those systems at the ground state. This can be understood by looking at the composition of those mixed FCC + BCC containing systems, where there is presence of certain elements with highly differing atomic radius or electronegativity from the rest like Al, Ti, W, Mo, or Zn (refer Table I for values). Under these circumstances the prediction of these parameters of a mixed or rather an overall open structure is as per the explanation given above. Another observation that can be drawn from Table II and which further substantiates the aforementioned argument is that in cases where these parameters predict a mixed FCC + BCC structure, however, experimental observation differs, there lies the presence of these radius/electronegativity wise different elements. The exact concurrence between the predicted structure and the observed structure would have been possible only if there could have been a possibility of performing a robust total energy minimization scheme. However, it can still be seen that the two newly introduced metrics viz. Modified Darken–Gurry Parameter (A) and the Modified Mismatch Entropy Parameter (B) are reasonably capable of predicting the stable crystal structure, given it is either FCC or BCC.

  • VEC accurately predicts phase selection in HEA systems to a large extent. HEAs with higher values of VEC tend to form FCC phase and those with lower values favor the BCC phase.

  • The new parameters namely the Modified Darken–Gurry parameter, show that FCC is the stable for values less than 0.0060 and the BCC above 0.0095. The Modified Mismatch Parameter depicts the stable FCC below 0.2 and the stable BCC phase above 0.38.