1. Iconic HEAs

The approach being explored within HEAs states that increasing the number of elements and favoring equimolar compositions increase the configurational entropy (this will be explained later), decrease the Gibbs free energy of a disordered solid solution, and stabilize it over other competing phases. Such a strategy has been largely implemented in the hope of obtaining single solid solutions in many multiprincipal element alloys. Numerous materials have been elaborated and tested according to the above rules. The following list gives examples of experimentally reported alloys exhibiting single disordered solid solution at equimolarity: AlCoCuNiTiZn,^{Reference Varalakshmi, Kamaraj and Murty14} AlCrCuFeMnNi,^{Reference Chen, Tsai, Tung, Yeh, Shun, Yang and Chen15} AlNbTaTiV,^{Reference Yang, Zhang and Liaw16} CoCrCuFeMnNi,^{Reference Cantor, Chang, Knight and Vincent9} CoCrCuFeNi,^{Reference Yeh, Chen, Lin, Gan, Chin and Shun10} CoCrFeMnNi (usually known as “Cantor alloy”),^{Reference Cantor, Chang, Knight and Vincent9} CoCuFeNiV,^{Reference Zhang, Zhou, Lin, Chen and Liaw17} CrCuFeMoNi,^{Reference Li, Li, Zhao and Jiang18} DyGdHoTbY,^{Reference Feuerbacher, Heidelmann and Thomas19} DyGdLuTbTm,^{Reference Takeuchi, Amiya, Wada, Yubuta and Zhang20} DyGdLuTbY,^{Reference Takeuchi, Amiya, Wada, Yubuta and Zhang20} HfMoNbTiZr,^{Reference Guo, Wang, Luo, Li, Su, Guo and Fu21} HfNbTaTiZr,^{Reference Senkov, Scotta, Senkova, Miracle and Woodward22} MoNbTaVW,^{Reference Senkov, Wilks, Miracle, Chuang and Liaw23} and MoNbTiVZr.^{Reference Zhang, Yang and Liaw24}

2. Deviant HEAs

Nevertheless, the scientific literature also reports several nonequimolar multiconcentrated alloys forming a single solid solution, i.e., nonequimolar HEAs. In fact, more of them have been reported than equimolar HEAs, although their configurational entropy is lower. Among others, there are a few examples where the nonequimolar alloy is single phased, whereas its equimolar counterpart is not, which was noticed early in the HEA development history, e.g., with Al_2.8CoCrCuFeNi versus AlCoCrCuFeNi,^{Reference Yeh, Chen, Lin, Gan, Chin and Shun10} although the configurational entropy of the former is lower than that of the latter. The same remark holds when comparing the equimolar HEA CoCrFeMnNi with a nonequimolar HEA of the same family, Co₂₀Cr₁₅Fe₂₆Mn₁₇Ni₂₂, the latter displaying a larger temperature range of stability of the single solid solution,^{Reference Mroz25} although having a lower configurational entropy.

3. HEAs versus CCAs

Two databases containing more than 300 multiconcentrated alloys each have recently been compiled and published by Gorsse et al.^{Reference Gorsse, Miracle and Senkov26} and Toda et al.,^{Reference Toda-Caraballo and Rivera-Díaz-del-Castillo27} respectively.

Gorsse’s database has a tree-like classification with five CCA/HEA families (3d transition metals, refractory metals, light weight, brasses & bronzes, and 4d lanthanide CCAs). Each family is expanded to show classes (one class is a unique combination of principal elements), each class contains members having variations in principal element concentrations, and finally each member is characterized by a set of attributes. This database contains more than 300 alloys with their composition, phases—i.e., solid solutions (FCC, BCC, and HCP), ordered phases, and intermetallics—and their mechanical and functional properties. Figure 1(a) displays a microstructure classification by phase types as a function of the number of components entering the alloy compositions. The occurrence of non-HEAs (multiphase microstructures) increases with the number of components, whereas the number of HEAs (single-phase disordered solid solution microstructures) decreases, which contradicts the high entropy hypothesis, i.e., entropic stabilization of a single-phase disordered solid solution when the number of components increases.

FIG. 1.

(a) Occurrence of HEAs and non-HEAs as a function of the number of elements; (b) cumulative distribution of the configurational entropy of HEAs and non-HEAs.

Toda’s database contains several tens of HEAs, i.e., forming a single solid solution, as well as many non-HEAs constituted of mixtures of solid solutions, intermetallics, mixtures of a solid solution with intermetallics, etc. These data are here analyzed by calculating the raw configurational entropy (see section II.C) of all alloys, ranking alloys of each category (HEAs and non-HEAs) by ascending order of configurational entropy, normalizing the rank in each category between 0 and 1 for comparison purposes, and finally plotting the cumulative distribution of configurational entropies as on Fig. 1(b). Obviously, for real alloys, non-HEAs exhibit overall a larger configurational entropy than HEAs.

Both these analyses show that configurational entropy, if it may contribute to the obtainment of HEAs, is clearly not the major parameter governing the formation of a single solid solution in multiconcentrated alloys. Two other reviews have come to the same conclusion.^{Reference Pickering and Jones12,Reference Miracle and Senkov28}

1. Solid solution versus all the rest

The “high entropy approach” exposed in early papers is in fact partly appropriate only, on the one hand because it would consider the sole configurational entropy (this will be discussed later), and on the other hand because it would consider only the decrease in Gibbs free energy, due to the increase in entropy, of the sole solid solution. However, the formation of a phase does not require that its free energy is low, but it needs to be lower than that of all other potential phases. When the number of elements increases, the number of potential phases increases faster, hence the number of possible constitutions concurrent to a single solid solution so that the “competition” becomes statistically ruder and ruder for the solid solution. It is impossible to calculate an exact number of possible phases or constitutions that would hold for any alloy system because it depends on each system, but an illustrative combinatorial approach can help to provide an order of magnitude of the problem. For instance, with one element, a single phase is necessarily formed. With two elements, say A and B, in equimolar proportions, a single phase AB can form or a decomposition in two phases A + B can occur (these could be solid solutions rich in A or B; besides the phase AB could well be a solid solution or an intermetallic, and several intermetallics A_mB_n could exist, which would increase the potential number of phases and constitutions, but let us make things simple: this way the model will be quite “optimistic” for HEA formation). Two constitutions can thus exist (AB or A + B) so that the probability of forming a single phase (AB) is 50%. Of course, this reasoning is purely combinatorial and not physical, but it may show trends. With three elements, say A, B, and C, one can form a single phase ABC or decompose the alloy in several phases: A + B + C, or A + BC, or B + AC, or C + AB. This makes five possible constitutions so that the probability of forming a single phase drops to 20%. And so forth when the number of elements is increased. The counting of all possible constitutions, i.e., of different manners of decomposing a set of N elements in subsets, uses Bell’s numbers.^{Reference Crepain29} The first ten Bell’s numbers, i.e., for N from 1 to 10, are 1, 2, 5, 15, 52, 203, 877, 4140, 21,147, and 115,975. The probability of forming a single phase for a given number of alloying elements can thus be thought to be equal to the inverse of these numbers. This is plotted in Fig. 2.

FIG. 2.

Probability of forming a single phase as a function of the number of alloying elements.

Once more, this model is simplistic but it gives a picture of the competition that a solid solution has to face when the number of elements increases. For instance, with five elements, which is the usual minimum number of elements considered necessary to produce a HEA, the probability is less than 2%. It becomes less than 0.5% for six elements, ∼0.1% for seven elements, and goes down to less than 0.001% for ten elements. Another consequence of this result is that single-phase HEAs may in fact be rare among multicomponent concentrated alloys; this will be discussed later. Also, this competition between the formation of a single solid solution and other constitutions will be expressed later in a rigorous manner using the CALPHAD approach.

2. Solid solution versus intermetallics

As stated earlier, the obtainment of a HEA depends on the competition between the solid solution and all other possible phases. A particularly interesting case is that of intermetallics. Indeed, their configurational entropy is very low—it is even null for a fully ordered phase since there is only one possible configuration. At the opposite, whereas in disordered solid solutions the neighborhood of a given atom is randomly made of other alloying elements so that the number of bonds between atoms of different nature its itself random, in intermetallics the closest neighbors of a given atom are not of random nature, so that only bonds between different elements are present (as long as only first neighbors are considered, as a first approximation). In this respect, let us build once more a simplistic model, based on a two-dimensional square lattice, which can be either a solid solution or an intermetallic, as depicted schematically in Fig. 3 for a binary alloy A–B.

FIG. 3.

Schematic illustration of (a) a random solid solution and (b) an ordered intermetallic on a two-dimensional square lattice.

For the sake of simplicity, let us assume that the bonding energies A–A and B–B are identical (H ₀), but different than the bonding energy between A and B atoms (H _AB). For an equimolar random solid solution, it can be shown that the total enthalpy is (H ₀ + H _AB)/2. Therefore, if the difference between H _AB and H ₀ increases (which can be regarded as a larger “scatter”, “dispersion”, or “range” of bonding energies in the alloy), the total enthalpy will vary compared to the enthalpy of pure A or pure B, H ₀. This variation is related to half of the dispersion in bonding energies. Besides, in the ordered intermetallic, there are only A–B bonds so that the total enthalpy is H _AB. As a consequence, the variation of total enthalpy is related to the difference in bonding energies between H ₀ and H _AB, not to its half as in the solid solution. Considering only configurational entropy, which is null for the intermetallic, the Gibbs free energy of the intermetallic is equal to its enthalpy. At the opposite, the solid solution possesses a configurational entropy. For a low dispersion in bonding energies between A, B, and AB, e.g., at the extreme if H _AB = H ₀, the free energy of the intermetallic is H ₀ and that of the solid solution is H ₀ – T·S _conf so that the latter is more stable than the former. When the dispersion in bonding energies increases, both free energies vary, but with a factor two between both, so that there will be a point where the intermetallic becomes more stable than the solid solution. This is illustrated schematically in Fig. 4.

FIG. 4.

Schematic variation of the Gibbs free energy of a solid solution (SS) and of an intermetallic (IM) when the dispersion in bonding energies increases in an alloy.

Let us now choose two elements A and B so that the dispersion in bonding energies is minimal. The alloy will lie on the left of Fig. 4 and a solid solution will form. Nevertheless, because the dispersion in bonding energies was minimal within the A–B system (it was selected as such), adding a third element C will necessarily increase the scatter in bonding energies: there must be one energy, among H _AC and H _BC, which differs more from H ₀ than does H _AB. The alloy A–B–C will therefore lie to the right of the alloy A–B in Fig. 4 (in a ternary system instead of a binary, but the concept still holds). Following this reasoning, adding more and more elements will necessarily result in an increase in the dispersion of bonding energies in the alloy, hence in moving further and further toward the right in Fig. 4 so that there should be a point where an intermetallic will become more stable than the solid solution.

Once again, this model is over-simplistic but it explains why, when increasing the number of elements in an alloy, the enthalpic contribution of intermetallics will always end in overcoming the entropic contribution of the solid solution, and hence why HEAs will become more and more difficult to form. This result is supported by facts: most actual HEAs contain five elements, less are made of six elements, even less contain seven elements, etc. [see also Fig. 1(a)].

1. Thermodynamics of mixtures

We will first restrict ourselves to the case of disordered substitutional solid solutions defined as crystallized phases which can vary in composition and where atoms of different components can occupy the same kind of lattice sites.

In the CALPHAD formalism, the general form of the molar Gibbs energy of such solutions can be expressed as

$${G_{\rm{m}}} = {}^{{\rm{ref}}}{G_{\rm{m}}} + {}^M{G_{\rm{m}}}\quad ,$$

where ^refG _m accounts for the mechanical mixture of the pure components and ^MG _m represents the thermodynamic effects of mixing:

$${}^{{\rm{ref}}}{G_{\rm{m}}} = \sum\limits_i {{x_i}{}^0{G_i}\quad ,}$$

$${}^M{G_{\rm{m}}} = - T \cdot {}^MS_{\rm{m}}^{{\rm{id}}} + {}^E{G_{\rm{m}}}\quad ,$$

where ⁰G _i is the molar Gibbs energy of the elemental constituent i, ^MG _m is the molar Gibbs energy of mixing which describes the behavior of the solution with respect to the reference state, ^EG _m is the excess Gibbs energy which represents the difference between the energies of a real solution and an ideal solution, and ${}^MS_{\rm{m}}^{{\rm{id}}}$ is the ideal entropy of mixing, also sometimes referred as the configurational entropy.

We now include physical considerations and the underlying statistical thermodynamics of mixtures to discuss more thoroughly the thermodynamic properties of solutions. Let us consider a binary crystal that contains N _A = N(1 − x) A atoms and N _B = Nx B atoms, with N being the total number of atoms and x being the fraction of B atoms. Assume that these two components are sufficiently alike in atomic size and electronegativity to be interchangeable in a lattice in which each site has z nearest neighbors. The number of possible arrangement of A and B on the lattice is

$${{N!} \over {\left\{ {N\left( {1 - x} \right)} \right\}!\left\{ {Nx} \right\}!}}\quad .$$

The energy of such a binary crystal depends on the arrangement of atoms and is described by three kinds of bonds, i.e., AA, BB with energies ε_AA = 2⁰g _A/z and ε_BB = 2⁰g _B/z, and AB with energy ${\varepsilon _{{\rm{AB}}}} = \left( {\nu + {1 \over 2}{\varepsilon _{{\rm{AA}}}} + {1 \over 2}{\varepsilon _{{\rm{BB}}}}} \right)$, where ⁰g _i is the energy of the i constituent (⁰g _i = ⁰G _i/N _Av with N _Av the Avogadro’s number) and ν is the exchange energy which represents the energy required to change an AA bond and a BB bond into a AB bond. For a particular configuration denoted X, the number of pairs of each kind is ${1 \over 2}z\left( {{N_{\rm{A}}} - {\rm X}} \right)$, ${1 \over 2}z\left( {{N_{\rm{B}}} - {\rm X}} \right)$, and zX, which gives the configurational energy, E, for this particular configuration:

$$E = {1 \over 2}z\left( {{N_{\rm{A}}} - X} \right){\varepsilon _{{\rm{AA}}}} + {1 \over 2}z\left( {{N_{\rm{B}}} - X} \right){\varepsilon _{{\rm{AA}}}} + zX{\varepsilon _{{\rm{AB}}}} = {N_{\rm{A}}}{}^0{g_{\rm{A}}} + {N_{\rm{B}}}{}^0{g_{\rm{B}}} + z\nu X\quad .$$

The configurational partition function is

$$\eqalign{ & \Omega = \sum {{{\rm{e}}^{{{ - E} / {kT}}}}} = \exp \left( {{{ - {N_{\rm{A}}}{}^0{g_{\rm{A}}}} / {kT}}} \right) + \exp \left( {{{ - {N_{\rm{B}}}{}^0{g_{\rm{B}}}} / {kT}}} \right) + \sum {\exp \left( {{{ - z\nu X} / {kT}}} \right)} \cr & = \exp \left( {{{ - {N_{\rm{A}}}{}^0{g_{\rm{A}}}} / {kT}}} \right) + \exp \left( {{{ - {N_{\rm{B}}}{}^0{g_{\rm{B}}}} / {kT}}} \right) + {{N!} \over {{N_{\rm{A}}}!{N_{\rm{B}}}!}}\exp \left( {{{ - z\nu \bar{X}} / {kT}}} \right)\quad , \cr}$$

where the summation extends over all arrangements of atoms on the N lattice sites of the crystal and $\bar{X}$ is the average value of the various configurations X.

We obtain that the free energy is given by

$$\eqalign{ & G = - kT\ln \Omega = {N_{\rm{A}}}{}^0{g_{\rm{A}}} + {N_{\rm{B}}}{}^0{g_{\rm{B}}} + kT\ln {{N!} \over {\left\{ {N\left( {1 - x} \right)} \right\}!\left\{ {Nx} \right\}!}} + z\nu \bar{X} \cr & = {N_{\rm{A}}}{}^0{g_{\rm{A}}} + {N_{\rm{B}}}{}^0{g_{\rm{B}}} + NkT\left[ {\left( {1 - x} \right)\ln \left( {1 - x} \right) + x\ln x} \right] + z\nu \bar{X}\quad . \cr}$$

If we assume a completely random mixture where the two components have the same probability of occupying any sites in the crystal lattice (Bragg–William approximation or zeroth approximation), the average value of $\bar{X}$ is

$$\bar{X} = {{{N_{\rm{A}}}{N_{\rm{B}}}} \over {{N_{\rm{A}}} + {N_{\rm{B}}}}} = Nx\left( {1 - x} \right)\quad .$$

We then obtain for the molar Gibbs energy of the mixture:

$${G_{\rm{m}}} = {x_{\rm{A}}}{}^0{G_{\rm{A}}} + {x_{\rm{B}}}{}^0{G_{\rm{B}}} + RT\left[ {{x_{\rm{A}}}\ln {x_{\rm{A}}} + {x_{\rm{B}}}\ln {x_{\rm{B}}}} \right] + {x_{\rm{A}}}{x_{\rm{B}}}L\quad ,$$

where L = zνN _Av is the solution parameter, which represents the interactions between the two components.

a. Ideal solutions

We now consider the simplest case of mixture in which constituents mix randomly over the lattice sites with no distinct interactions between them (ν = 0, meaning that ε_AA = ε_BB = ε_AB). In that situation, we can assume that all the possible arrangements of the two atoms have the same configurational energy:

$$\Omega = \sum {{{\rm{e}}^{{{ - E} / {kT}}}}} = {{N!} \over {{N_{\rm{A}}}!{N_{\rm{B}}}!}}{{\rm{e}}^{{{ - E} / {kT}}}}\quad ,$$

thus the molar Gibbs energy of mixing describing the behavior of the so-called ideal solution is

$${}^M{G_{\rm{m}}} = RT\left[ {{x_{\rm{A}}}\ln {x_{\rm{A}}} + {x_{\rm{B}}}\ln {x_{\rm{B}}}} \right] = - T{}^MS_{\rm{m}}^{{\rm{id}}}\quad .$$

In the case of ideal solutions, the configurational entropy is described by the ideal entropy of mixing. Figure 5(a) gives the calculated Gibbs energy of mixing curve as a function of the composition for a binary ideal substitutional solid solution. The minimum of ^MG _m is achieved when both components have the same fraction. Any deviation from the equimolarity will reduce the number of possible arrangements resulting in a decrease of the configurational entropy. Let us consider what happens when new components (ternary, quaternary, etc.) are mixed. Figure 5(b) shows that the configurational entropy increases with the number of components and that the maximum, ^cnfS _max = R ln(n), is achieved at equimolarity.

FIG. 5.

(a) Calculated Gibbs energy of mixing for an ideal binary solution and (b) calculated configurational entropy of solid solutions with n components as a function of mole fraction of the nth component.

In principle, incorporating an increasing number of components in equal fractions into a single lattice with random occupancy will promote the stability of a disordered solid solution, presumably over other possible phases, when the configurational entropy dominates the energy landscape. This approach of entropy stabilization of a single-solid solution is the funding principle being explored in multicomponent concentrated alloys, the so-called HEAs. However, the simple formula of the configurational entropy (the ideal entropy of mixing) can apply only to the ideal class of mixtures, i.e., equal energies of the different kinds of bonds and completely random distribution of the different components.

We will now extend the situation to less crude approximation.

b. Regular solutions

Logically, the next type of mixtures shares the same features describing ideal solutions except that the exchange energy is not zero, i.e., the configurational energy is no longer independent of the arrangement of the components. In this situation, the excess energy of mixing, ^EG _m = x _Ax _BL, will affect the shape of the Gibbs energy curve which will remain symmetrical with respect to x though, as shown in Fig. 6. A positive value of L (i.e., ν > 0) results in a tendency for demixing in two solid solutions (less A–B bonds than in a random arrangement because A and B do not like to mix with each other). When the temperature decreases, excess energy can counterbalance the configurational entropy. The consequence is that a miscibility gap will form below the critical temperature T _sp = x _Ax _B2L/R at x = 1/2. A negative value of L (i.e., ν < 0) means that unlike atoms prefer to be nearest neighbors (more A–B bonds than in a random arrangement). This will reduce the Gibbs free energy and stabilize the disordered solid solution, as shown in Fig. 6. However, a large negative value of ν will give a tendency of short-range and long-range ordering which will result in deviations from randomness. In this situation, we can no longer consider complete randomness (because the exchange energy influences the number of bonds of different kinds) and the single interaction parameter (regular solution parameter L) is not enough to describe the behavior of real solutions.

FIG. 6.

Configurational entropy, excess energy of mixing, and resulting Gibbs energy of mixing curves for the sign of the regular solution parameter used.

c. Deviation from random distribution (short-range order)

Neglecting of short-range order (SRO) in the zeroth approximation (the atoms mix randomly) to describe multicomponent mixtures seems questionable. Few theoretical/experimental studies evidenced noticeable SRO in HEAs.^{Reference Wang, Wang, Lin, Zou, Wu, Hu, Shang, Darling, Wang and Hui30–Reference Widom32} In the CALPHAD approach, a composition dependence of the interaction parameter is then introduced using the Redlich–Kister (RK) polynomial to better fit the experimental data:

$${}^E{G_{\rm{m}}} = {x_{\rm{A}}}{x_{\rm{B}}}L = {x_{\rm{A}}}{x_{\rm{B}}}\sum\limits_{k = 0}^n {{}^kL{{\left( {{x_{\rm{A}}} - {x_{\rm{B}}}} \right)}^k}} \quad .$$

The situation where L is independent of composition corresponds to the regular solution model (k = 0), whereas ¹L and ²L are the subregular and subsubregular solution parameters. Figure 7 illustrates the effect of these extra parameters on the shape of the excess Gibbs energy and the Gibbs energy of mixing. In case of a subregular solid solution (k = 1), the Gibbs energy curve is not symmetric and does not reach the minimum at equal amount of components even though equimolarity still provides the highest configurational entropy.

FIG. 7.

(a) Properties of the RK terms and (b) their effects on the Gibbs energy of mixing.

A temperature dependence of the fitting parameters can be used to introduce the effect of nonideal configurational entropy and nonconfigurational entropy sources:

$${}^k{L_{{\rm{AB}}}} = {}^k{a_{{\rm{AB}}}} + {}^k{b_{{\rm{AB}}}}T\quad ,$$

^ka _AB describes the excess enthalpy and ^kb _AB describes the excess entropy.

When SRO occurs, the ideal entropy of mixing does represent only the upper limit of the configurational entropy.

An alternative approach to reproduce the effect of nonrandom distribution solutions is to introduce unlike bond probabilities using the modified compound energy formalism (CEF).^{Reference Sundman and Ågren33} Higher-order approximations can also be used to model SRO such as the quasi-chemical approach^{Reference Guggenheim34} or the cluster variation method^{Reference Kikuchi35} in which fictitious constituents are introduced.

d. Long-range order

In the CALPHAD approach, long-range order (LRO) is represented by the use of several sublattices to take into account that atoms of different components can occupy different types of lattice sites. The B2 ordering of the BCC-A2 structure (the atom in the center of the unit cell is different from those at the corners) requires two sublattices with equal amounts of sites (A, B, Va)₁(A, B, Va)₁ to reproduce antisite fraction and vacancy (Va) defects. The most important ordered phase deriving from the FCC structure is L1₂ (corner atoms are of a different kind from the atoms on the faces) which can be modeled using two sublattices (A, B, Va)₃(A, B, Va)₁. More sublattices can be necessary to model ordered phases, intermetallics and metal–nonmetal phases (e.g., M₂₃C₆ or to model together L1₂ and L1₀), but the number of possible fitting parameters quickly becomes very large. The formalism for describing the Gibbs energy of such phases is the CEF which gives the following expressions:

$${}^{{\rm{ref}}}{G_{\rm{m}}} = \sum\limits_{{I_0}} {{P_{{I_0}}}\left( Y \right){}^0{G_{{I_0}}}\quad ,}$$

$${}^{{\rm{cnf}}}{S_{\rm{m}}} = - R\sum\limits_{s = 1}^n {{a_s}} \sum\limits_{i = 1}^{{n_s}} {y_i^{\left( s \right)}\ln \left( {y_i^{\left( s \right)}} \right)\quad ,}$$

$${}^E{G_{\rm{m}}} = \sum {{P_{{I_1}}}\left( Y \right){L_{{I_1}}}} + \sum {{P_{{I_2}}}\left( Y \right){L_{{I_2}}}} + \cdots \quad ,$$

where I denotes a constituent array on each sublattice s while i denotes an individual constituent, I ₀ is the constituent array of zeroth order specifying one constituent in each sublattice, ${P_{{I_0}}}\left( Y \right)$ is the product of the constituent fractions specified by I ₀, ${}^0{G_{{I_0}}}$ is the compound–energy parameter representing the Gibbs energy of formation of the compound I ₀, a _s is the number of sites on each sublattice, and $y_i^{\left( s \right)}$ denotes the constituent fraction of i on s.

Additionally, it can be noticed that the entropy is also made of other contributions, e.g., vibrational, electronic, and magnetic entropies, linked to the weight, number of electrons, and magnetic moment of each element and their interactions. As such, it is necessarily composition-dependent, contrarily to the configurational entropy. For instance, first principles calculations have been performed in alloy CoCrFeMnNi, using the density functional theory and finite-temperature ab initio methods.^{Reference Ma, Grabowski, Körmann, Neugebauer and Raabe36} Simulations considered vibrational, electronic, and magnetic as well as configurational contributions to the entropy. Among others, the results showed that at 1200 K (927 °C), the configurational entropy represents only ∼15% of the total entropy. This may also contribute to explain some of the experimental observations reported above. In CALPHAD formalism, nonideal configurational, vibrational, and electronic contributions are included in the terms ^EG _m and ^refG _m, whereas additional quantities (^physG _m), modeled separately, can be introduced to account for the contributions arising from physical phenomena which vary sharply with composition such as magnetic transitions. It is not the intention here to describe them.

2. Phase stability

In previous sections, we have examined how entropy and various physical phenomena can alter the Gibbs energy of solutions. However, it is necessary to consider the Gibbs energy of all the competing phases in the determination of phase stability since the equilibrium state is reached at the global minimum of the total energy of the system.

This concept has been reviewed by Zhang et al.^{Reference Zhang, Zhang, Chen, Zhu, Cao and Kattner37} and Miracle et al.^{Reference Miracle and Senkov28} to discuss the relative contributions of entropy and enthalpy on phase stability and to emphasize the competition among all the phases of a system. They demonstrated that the lowest energy is not necessarily reached at equimolarity, even though it represents the highest ideal entropy of mixing, and that increasing the number of components promote the formation of secondary phases.

In the following, we will further extend this analysis using hypothetical ternary systems calculated using a user-defined thermodynamic database to feature phase stability with emphasis put on the location of the disordered solid solution in two different situations: when temperature changes and for various combinations of solution parameters.

In the case of ternary systems, the excess Gibbs energy for multicomponent substitutional solution can be written as

$${}^E{G_{\rm{m}}} = {}^{{\rm{bin}} \cdot E}{G_{\rm{m}}} + {}^{{\rm{tern}} \cdot E}{G_{\rm{m}}}\quad ,$$

With ^bin·EG _m and ^tern·EG _m the binary and ternary excess Gibbs energy contributions:

$${}^{{\rm{bin}} \cdot E}{G_m} = \sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = i + 1}^n {{x_i}{x_j}{L_{ij}}} } \quad ,$$

$${}^{{\rm{tern}} \cdot E}{G_{\rm{m}}} = \sum\limits_{i = 1}^{n - 2} {\sum\limits_{j = i + 1}^{n - 1} {\sum\limits_{k = j + 1}^n {{x_i}{x_j}{x_k}{L_{ijk}}} \quad .} }$$

Let us now consider a ternary A–B–C system [Fig. 8(a)] consisting of three binaries forming a regular solid solution with a miscibility gap and having identical solution parameters (⁰L _AB = ⁰L _BC = ⁰L _AC > 0). At low temperature, where the energy landscape is dominated by the excess energy term, the equiatomic composition lies in the middle of the ternary miscibility gap and the solid solution is only stable at the three corners of the phase diagram. When the temperature increases, the solid solution becomes stable for equal amounts of the three components because its Gibbs energy reaches the minimum in the middle of the Gibbs triangle where the configurational entropy is maximum (^cnfS _m = R ln 3). At higher temperatures, the stability region of the regular solid solution further extends because the contribution of the configurational entropy (−T·^cnfS _m) overcomes the enthalpy contribution arising from positive binary interactions. The same analysis was performed with further details by Morral and Chen^{Reference Morral and Chen38} for the regular solution model with equal binary interaction parameters only (Meijering’s model). In this discussion, the Gibbs energy surface of the ternary solution was assumed symmetric resulting from identical excess Gibbs energies in all three binaries. This situation is very unlikely, and it is important to point out that contrary to what is observed in binaries, the Gibbs energy surface in a ternary and in higher-order systems will never be symmetric even though the solution behaves like a regular solution. Figure 8(b) shows the consequences of considering a more realistic case where the regular binary solution parameters are different. The stability region of the regular ternary solution deviates from equimolarity. The shift is greater with an increasing gap between the binary solution parameters. The same observation can be made when considering subregular solutions as previously depicted for binary systems [Fig. 8(c)].

FIG. 8.

Ternary isothermal sections calculated using a hypothetical thermodynamic description of an A–B–C system, highlighting the effect of temperature and binary solution parameters on the stability region of the single disordered solid solution for (a) equal regular solution parameters, (b) different regular solution parameters, and (c) subregular solution parameters. Equimolarity is represented by the red dot. Numbers given are numbers of phases in each phase field.

We now see what happens when one binary intermetallic (AB₂ line compound) is introduced in the ternary system. Figure 9 shows that the stable region of the disordered solid solution is not at equimolarity. This is because the contact point between the tangent plane representing the equilibrium and the Gibbs energy surface of the disordered solution $G_{\rm{m}}^{{\rm{SS}}}$ is not at the minimum point in the middle of the triangle. The tangent plane is tilted to pass through the stable AB₂ intermetallic, leading to a shift of the tangent point on $G_{\rm{m}}^{{\rm{SS}}}$ toward A–C richer compositions.

FIG. 9.

Ternary isothermal sections calculated using a hypothetical thermodynamic description of an A–B–C system, highlighting the effect of LRO (intermetallic AB₂) on the stability region of the single disordered solid solution. Equimolarity is represented by the red dot. Numbers given are numbers of phases in each phase field.

In summary, these simple thermodynamic calculations of hypothetical ternary diagrams illustrate that equimolarity does not represent the minimum energy of nonideal solutions and that phase stability results from a competition between phases driven by the minimization of the total Gibbs energy. Incorporating more than two components in a mixture increases the configurational entropy but shifts the minimum of the Gibbs energy away from equimolarity in the case of nonideal solutions and can lead to the formation of a more stable compound. For example, it was observed that the stability temperature range of a single solid solution in nonequimolar alloy Co₂₀Cr₁₅Fe₂₆Mn₁₇Ni₂₂ was larger than that of equimolar alloy CoCrFeMnNi,^{Reference Mroz25} confirming that maximum solid solution stability does not occur at equimolarity. The same concept is supported by the formation of a single solid solution in nonequimolar alloy Al_2.8CoCrCuFeNi whereas equimolar alloy AlCoCrCuFeNi does not form any.^{Reference Yeh, Chen, Lin, Gan, Chin and Shun10}

We can conclude that a truly entropy-stabilized single solution is an improbable event very occasionally encountered around equimolarity for near-ideal solid solutions only. This conclusion is supported by the work of Otto et al.^{Reference Otto, Yang, Bei and George39} who have analyzed the microstructure and thermodynamic properties of five quinary alloys by replacing individual elements in the so-called Cantor’s alloy (i.e., CoCrFeMnNi). They evidenced that the FCC solid solution made of ten binary pairs of Cantor’s alloy behaves closely as an ideal solid solution, which results in the formation of a true solid solution alloy, whereas all substitutional alloys exhibited multiphase microstructures. On the theoretical front, Schön et al.^{Reference Schön, Duong, Wang and Arroyave40} recently probed the different contributions to phase stability in multicomponent systems. Their work provides strong evidence that entropy has a marginal effect in stabilizing concentrated disordered solid solutions when SRO exists and that conflicting interactions and correlations (what they called configurational frustration) drive the stabilization of the disordered phase in the central region of the composition space.

1. Calculation and inspection of phase diagrams

The microstructure of an alloy results in the competition among all the possible phases in the studied system. Phase diagrams map the phase stability regions as a function of temperature, pressure, and alloy concentration. They are useful for guiding the development of new alloys because they provide the phase field of operation and detailed information on the equilibrium state, including phases present, their volume fractions, compositions, and transformation temperatures. Still only few studies highlight the application of the CALPHAD approach to analyze the microstructure of HEAs under annealed (equilibrium) conditions and to compare model calculations and experimental observations.^{Reference Zhang, Zhang, Chen and Cao41–Reference Liu, Gao, Liaw and Zhang44} They were already thoroughly reviewed by Gao et al.^{Reference Gao, Yeh, Liaw and Zhang45}

In this section, we illustrate the CALPHAD approach through an example, by analyzing the formation of the single fcc solid solution in the Cantor’s alloy from the thermodynamic properties of the Co–Cr–Fe–Mn–Ni quinary system. Zhang et al.^{Reference Zhang, Zhang, Chen, Zhu, Cao and Kattner37} used binary phase diagrams to discuss the mutual solubility of the components in this system. In the present paper, we expand the analysis to ternary and quaternary phase diagrams and introduce parallel coordinate plots to visualize the compositional relationship among the components in the fcc structure and delimitate the range of the single fcc phase. The calculations below were performed by interfacing Matlab® and ThermoCalc® software and the TCHEA2 database. In this database, all ten binaries and half of the ten ternaries were assessed in the full range of composition and temperature (Co–Cr–Fe, Co–Cr–Ni, Co–Fe–Ni, and Cr–Fe–Ni) or partially assessed (Fe–Mn–Ni). The five remaining ternaries are extrapolated from their respective constitutive binaries which may lead to inaccurate representations of the Gibbs energy for the phases in these systems. It should be noted that higher-order systems are absent in the CALPHAD database so that the contributions from quaternary and quinary interactions on the Gibbs energy model are neglected.

Figure 10(a) shows the isothermal section at 1000 °C of the Co–Fe–Ni system and the three subcomponent binary systems. Co, Fe, and Ni possess the fcc structure at 1000 °C, are mutually soluble in the fcc structure, and consequently form a continuous fcc solid solution in this ternary at 1000 °C. Adding Mn which does not have the same crystal structure leads to the formation of the σ phase in the Co–Fe–Mn ternary [Fig. 10(b)]. This phase admits Ni in solid solution, so σ can extend in the quaternary Co–Fe–Mn–Ni and limit the range of the single fcc solid solution. Visual examination of the quaternary to identify the composition range where the fcc phase is stable can be performed using a 3D plot of the fcc surface boundary. An alternative approach to visualize n-dimensional data is the parallel coordinate system that maps multidimensional data into a 2D space without losing information. In a parallel coordinate plot, the axes for the variables run vertically and span from the minimum to maximum value of each variable. Figure 11 illustrates the construction principle of such plot using simple cases. Consider the range of existence of a solid solution α in binary system A–B at constant temperature [Fig. 11(a)]. The two variables X _A and X _B (representing the molar fractions of A and B) lie parallel to each other. Capturing the relationships between these variables is done by representing segmented lines connecting the corresponding values of X _A and X _B for which the α phase exists at the considered temperature. Now assume mutual solubility between A and B (i.e., continuous solid solution α over the full composition range at the fixed temperature), then segmented lines connect all the possible values of the composition parameters from 0 to 1 with the constraint that X _A + X _B = 1. The value X _A = 1 is connected to X _B = 0, and so on every fixed increments down to X _A = 0 and X _B = 1. This gives an envelope representing the range of existence of the solid solution α [Fig. 11(a)]. Figure 11(b) extends the same representation to a hypothetical ternary A–B–C.

FIG. 10.

(a) Calculated subcomponent binaries and isothermal section at 1000 °C of the ternary phase diagram of the Co–Fe–Ni system. (b) Unfolded tetrahedron of the quaternary Co–Fe–Ni–Mn system showing the isothermal section of the 4 subcomponent ternaries calculated at 1000 °C.

FIG. 11.

Hypothetical A–B (a) binary and A–B–C (b) ternary systems illustrating the construction principle of the parallel coordinate plot in the case of the continuous solid solution α.

Applying the same construction principles to the stability range at 1000 °C for the main phases (fcc, bcc, hcp, and σ) in the quinary Co–Cr–Fe–Mn–Ni system gives the linear mapping shown in Fig. 12(a). The problem with representing a 2D answer to a multidimensional problem is that it is too static. The parallel coordinate plot is meant to be dynamic/interactive and gives different results for different questions. To make such plot readable, we have reduced the essential information with respect to the range of existence of the fcc solid solution. This is done in Figs. 12(b)–12(d) for the Co–Fe–Ni, Co–Fe–Ni–Mn, and Co–Fe–Ni–Mn–Cr systems at 1000 °C. The continuous fcc solid solution in the Co–Fe–Ni ternary is reflected by the fact that lines connect all the possible values of the component concentrations [Fig. 12(b)]. The same data set (861 composition calculations) is displayed by a scattergram of the Co–Fe–Ni system [the inset of Fig. 12(b)] that shows the absence of clustering, meaning that the fcc phase expands over the entire ternary composition space at this temperature. Figure 12(c) shows what happens when adding Mn (12,341 composition calculations). The fcc single-phase is still stable over a large range of composition in the Co–Cr–Fe–Mn system. Up to 90% of Mn can be dissolved in the fcc structure. Cr tends to have a stronger effect on the stability of the fcc single-phase, as illustrated in Fig. 12(d) (135,751 composition calculations). A maximum 45 at.% of Cr can be dissolved in fcc only when the solid solution is depleted in Fe and Mn, as highlighted by the red lines connecting the 45 and 42.5 at.% Cr compositions to the other coordinates representing the molar fractions of Co, Fe, Ni, and Mn in the fcc phase.

FIG. 12.

2D parallel coordinate plots showing the compositional relationship in the stability range of the fcc, bcc, hcp, and σ phases at 1000 °C in the quinary Co–Fe–Ni–Mn–Cr system (a). The composition relationships delimiting the range of existence of the stable phases is represented by lines connecting the component concentrations in five parallel vertical axes with a 2.5 at.% compositional increment. The range of existence of the fcc single-phase solid solution at 1000 °C is shown in (b) for Co–Fe–Ni, (c) for Co–Fe–Ni–Mn, and (d) for Co–Fe–Ni–Mn–Cr systems. The insert is a scattergram showing the existence of the fcc single solid solution in the Co–Fe composition space of the Co–Fe–Ni system (the size of the dots is proportional to the Ni content).

CALPHAD allows us to examine the underlying thermodynamic properties of mixtures. Figure 13 compares the calculated energies of mixing (Gibbs energy, enthalpy, and entropy) in the fcc disordered phase. Comparisons between the Co–Cr–Fe–Mn–Ni and Co–Cr–Fe–Mn–Cu quinary systems shed light on the ability of the Cantor’s alloy to form a model HEA (single disordered solid solution at equimolarity). Replacing Ni by Cu in the quinary alloy changes the relative importance of the enthalpy and entropy contributions. Compared to the Ni-quinary, the Cu-quinary has less entropy. In contrast with the Ni-quinary, the increase in dimensionality from 4 to 5 results in a decrease of the entropy of mixing. This behavior arises from the demixing tendency due to the pair interactions with Cu. All the binary excess enthalpies with Cu (Co–Cu, Cr–Cu, Fe–Cu, Mn–Cu) are positive which results in a positive enthalpy of mixing in the Cu-quinary while it is negative in the Ni-quinary. Both enthalpy and entropy effects lead to the destabilization of the fcc disordered solution when replacing Ni by Cu. These results show that the trend observed in the concentrated region of multicomponent systems can be derived from the properties of the constitutive binary systems if the solution behaves as an ideal or near-ideal mixture (for instance, the Ni–Cu solid solution is exploited in “Monel” or cupronickel alloys, and the formation of copper-rich solid solution clusters is used in some precipitation-hardened steels). This is the case for the Cantor’s alloy which can be regarded as a model HEA. When Ni is replaced by Cu in the concentrated quinary, the mixture deviates from ideality. In this case, increasing the number of components does not lead to an increase in the entropy, and enthalpy contributions have a large effect in destabilizing the fcc solid solution.

FIG. 13.

Calculated energies of mixing (Gibbs energy, enthalpy, and entropy) of the fcc disordered phase at 1000 °C in the Co–Cr–Fe–Mn–Ni (a) and Co–Cr–Fe–Mn–Cu (b) quinary systems and their subcomponents (each point represents the interactions in the ten binaries, ten ternaries, and five quaternaries).

2. Nonequilibrium calculations—modeling solidification

Although the formation of a single solid solution at equilibrium is a requirement to form a HEA, the microstructure of an alloy is dictated not only by composition but also by processing. To determine the resulting microstructure for a given processing route, the equilibrium phase diagrams are generally insufficient, and phase transformations must be modeled to quantitatively describe the kinetics of formation and evolution of the microstructure. The first step in alloy processing is casting so that solidification and the resulting microstructures are of prime importance.

In an attempt to explain the as-cast microstructure of HEAs, including the type and amount of phases and the compositional microsegregation, thermodynamic simulations of the solidification process can be performed based on the CALPHAD approach. Two basic models are commonly used to simulate the solidification paths: equilibrium lever rule and nonequilibrium Scheil–Gulliver.^{Reference Scheil46,Reference Gulliver47} The first condition assumes global equilibrium between liquid and solid phases, infinite diffusion in both phases, and no undercooling during nucleation or growth, while the latter condition stipulates thermodynamic equilibrium at the liquid–solid interface, infinite diffusion in liquid, and no diffusion in solid, so the liquid composition is homogeneous whereas the solid phases will retain the successive compositions from when they have been formed on cooling. The Scheil–Gulliver model has been applied successfully to some practical cases for 3d transition metal CCAs and refractory metal CCAs^{Reference Zhang, Zhang, Chen and Cao41,Reference Zhang, Gao, Zhang, Yang and Guo48,Reference Zhang, Gao, Zhang and Guo49} owing to the sluggish diffusion in these systems and the fast cooling rates resulting from the arc-melting technique generally used to produce them. However, erroneous estimations of the solidification path and significant discrepancies between simulations and experiments are often observed. They can have a thermodynamic origin, i.e., unprecise thermodynamic description reduces the reliability of the calculations, and can also arise from the assumption from the Scheil model where diffusion in the solid is neglected.

Real solidification paths range between the two above limiting cases, and certain components may redistribute within already solidified phases through diffusion. Thus, kinetic simulations are needed to give more accurate description of the mass and heat fluxes in the solidification process. In the absence of available diffusion data, different scenarios for back-diffusion can be tested to evaluate the effect of fast diffusing elements with the hope to improve the predicted solidification.

1. Classical methods for HEA design

In fact, the mostly used methods to explain the formation of a single solid solution in multiconcentrated alloys do not rely on CALPHAD. Indeed, most authors make use of rather classical approaches, based on a number of physicochemical parameters and rules. They are worth being mentioned here since they can be seen as concurrent to CALPHAD or, as will be explained later, as complementary tools.

For instance, some of the classical Hume-Rothery rules for the solubility of metals^{Reference Hume-Rothery56} have been adapted or extended to HEAs:

1. (i) The pure metals must have the same crystal structure. If this is a requirement for complete solubility, it is not one for HEAs: there are cases of multiconcentrated solid solutions of metals with different crystal structures before alloying. For example, AlCoCrCu_0.5Ni forms a single BCC phase^{Reference Yeh, Lin, Chin, Gan, Chen, Shun, Tsau and Chou57} although only Cr has a BCC structure as a pure metal: Co is either hexagonal or FCC, and the others are FCC. Consequently, this rule does not apply to HEAs.

2. (ii) Elements must have similar sizes. This rule has been applied to multiconcentrated alloys, by calculating the average atomic size mismatch, δ.^{Reference Zhang, Zhou, Lin, Chen and Liaw17,Reference Fang, Xiao, Xia, Li and Dong58} It was found that most HEAs were associated with small values of δ and that high values of δ would not allow the formation of a single solid solution.

3. (iii) To achieve complete solubility, the valence of atoms must be similar, which brings a criterion on the alloy electron concentration. Two criteria are based on this rule; on the one hand, the average number of itinerant electrons per atom, e/a, and on the other hand, the valence electron concentration, VEC. Both have been applied to multiconcentrated alloys.^{Reference Poletti and Battezzati59–Reference Guo, Ng, Lu and Liu61}

4. (iv) Atoms must have comparable electronegativities, χ, since elements with different electron affinities tend to form stronger bonds and hence favor ordered intermetallics compared to disordered solid solutions. This brought the definition of the average electronegativity mismatch, Δχ, which has been used in multiconcentrated alloys.^{Reference Fang, Xiao, Xia, Li and Dong58,Reference Guo, Ng, Lu and Liu61}

Other criteria have also been suggested to predict the formation of solid solutions, based on thermodynamic concepts (but not using the CALPHAD approach):

1. (v) Because it makes part of the total Gibbs energy, the enthalpy of mixing, ΔH _mix, calculated by a summation of interactions between pairs of elements,^{Reference Miedema, de Châtel and de Boer62} has been proposed as a criterion for the formation of HEAs.^{Reference Zhang, Zhou, Lin, Chen and Liaw17}

2. (vi) To take into account the balance between enthalpy and entropy in the stability of a phase, the parameter Ω was proposed, defined as the product of melting temperature and mixing entropy, divided by the absolute value of the mixing enthalpy.^{Reference Yang and Zhang63}

3. (vii) Finally, considering that phase separation from a solid solution could occur by spinodal decomposition, it has been proposed that a large gap between the spinodal decomposition temperature and the melting temperature would guarantee the formation of a single solid solution at high temperature. This yielded the parameter µ, defined as the ratio between the melting temperature and the critical spinodal decomposition temperature.^{Reference Poletti and Battezzati59}

However, a careful analysis of experimental results gained on several hundreds of multiconcentrated alloys^{Reference Toda-Caraballo and Rivera-Díaz-del-Castillo27} showed that none of the above-mentioned parameters, taken alone, could separate strictly HEAs from non-HEAs. In fact, most non-HEAs would lie within certain parametric ranges while most HEAs would fall within other ranges, but the latter would also cover many non-HEAs, and vice versa. Also, it was shown that “two-dimensional maps” of different couples of parameters allow us to identify regions in which many HEAs lie. Nevertheless, it was still observed that such regions containing many HEAs also cover many non-HEAs, and reciprocally. As a consequence, these criteria, taken alone or by pairs, may be seen as necessary conditions for the obtention of HEAs, but not as sufficient conditions, so that they cannot be used for alloy design. Actually, this is confirmed by the fact that although such parametric approaches have often been announced as potential tools for HEA design, it seems that the number of actual HEAs that have been designed by explicitly using them is very limited. One can find, for instance: Hf_0.5Nb_0.5Ta_0.5Ti_1.5Zr,^{Reference Sheikh, Shafeie, Hu, Ahlström, Persson, Veselý, Zýka, Klement and Guo64} HfNb_0.5TiV_0.5Zr, HfNb₂TiVZr₂, HfMo_0.5Nb_0.5TiZr, and HfNb_0.5Ta_0.5TiZr.^{Reference Chen, Li, Cheng, Wu, Cheng and Xu65}

2. CALPHAD for HEA design

Besides, since it provides fair estimates of phase formation in alloys, including in multiconcentrated ones, CALPHAD has logically been used to design new HEAs, although this seems not to have been the most common practice in the discovery of the majority of existing HEAs to date. Nevertheless, successful examples exist. In general, authors report that they pre-identified a short list of elements, based on physicochemical rules (see. Sec. IV.A.1) or considering existing binary and/or ternary phase diagrams to check for maximum solubilities of elements by pairs or triplets, or by starting with already known HEAs and modifying their composition in a limited manner. Then CALPHAD calculations are used to spot the domains of single-phase stability, to confirm presumptions and/or to refine composition. This is usually made by calculating single equilibria, pseudo-binary, or pseudo-ternary phase diagrams around given systems. Sometimes the approach fails as the predictive capacity of CALPHAD is not perfect as already stated, but several new HEAs have been successfully designed using such methods, e.g., Co₂₀Cr₁₅Fe₂₆Mn₁₇Ni₂₂,^{Reference Mroz25,Reference Fraczkiewicz66} Cr_xMoNbTaVW,^{Reference Zhang, Gao, Zhang and Guo49} MoNbTaTiVW,^{Reference Zhang, Gao, Zhang, Yang and Guo48} MoNbTaTiV,^{Reference Yao, Qiao, Hawk, Zhou, Chen and Gao67} CoCuFeMnNi, CoFeMnNiV, AlFeMnNbTi, AlCrFeMnV and CoCrFeMnV,^{Reference Tazuddin, Gurao and Biswas68} Co₁₀Cr₁₅Fe₄₀Ni₂₅V₁₀, Co₁₀Cr₁₅Fe₂₅Mn₁₀Ni₃₀V₁₀, Co₁₀Cr₁₀Fe₂₅Mn₁₂Ni₃₃V₁₀, Cr₁₅Fe₂₂Mn₁₃Ni₄₀V₁₀, and Cr₁₀Fe₃₀Mn₂₀Ni₃₀V₁₀.^{Reference Choi, Jung, Jo, Lee and Lee69}

3. Beyond thermodynamics: data mining

It was previously shown that both simple physicochemical criteria (e.g., Hume-Rothery derived rules) and CALPHAD would fail in a number of occasions to predict the occurrence of a single solid solution in multiconcentrated alloys or to predict when several phases will form, which is problematic in an alloy design perspective.

Another way that was used, either independently of these methods or coupled to them, is data mining. The latter is here understood in the general meaning of extracting knowledge from data using purely mathematical or statistical tools.

As already stated, physicochemical parameters (δ, e/a, VEC, Δχ, ΔH _mix, Ω, or µ), considered alone, cannot be used for alloy design. It is not possible either using pairs of such parameters, although using two-parameter maps can increase the predictive efficiency, which was tested on more than 320 highly concentrated multicomponent alloys.^{Reference Toda-Caraballo and Rivera-Díaz-del-Castillo27} This trend was confirmed by Asensio Dominguez et al. through a principal component analysis of data with up to five physical parameters (ΔH _mix, ΔS _mix, δ, Δχ, and VEC), over a set of 79 alloys, showing that increasing the number of parameters being taken into account simultaneously would provide an improved predictive efficiency.^{Reference Domínguez, Goodall and Todd70} They used their approach to design successfully two four-element single-phase equimolar alloys, MnFeCoNi and TiVMnNb. Besides, Nong et al.^{Reference Nong, Zhu, Cao, Yang, Lai and Liu71} used an artificial neural network to successfully predict what solid solution phase would form in a HEA, from four of the physical parameters previously mentioned (δ, ΔH _mix, Δχ, and VEC). However, the approach could not predict if a single phase would form in any multiconcentrated alloy or not since only single-phase HEAs were included in the database. The method was proposed, but not actually used, for HEA design. Also, based on an analysis of 109 equimolar or multiconcentrated alloys, it was suggested that combining computational thermodynamics with some physical parameters would bring extra information,^{Reference Raghavan, Hari Kumar and Murty72} although this proposal was not used to design new alloys. Finally, Tancret et al. performed a thorough analysis of a database of more than 320 multiconcentrated alloys, by coupling a Gaussian process statistical analysis over nine calculated physicochemical parameters (including δ, e/a, VEC, Δχ, ΔH _mix, Ω, and µ) to CALPHAD—using simultaneously four thermodynamic databases as exposed earlier.^{Reference Tancret, Toda-Caraballo, Menou and Rivera-Díaz-Del-Castillo54} Of course, using a single thermodynamic database optimized for HEAs would probably have given better predictions than several less appropriate ones, but at the time of study, no such ThermoCalc database would exist for HEAs. Nevertheless, it was shown, on pragmatic grounds through a comparison with experimental results on actual alloys, that such a combination of tools could eliminate the risk not to form a single solid solution in highly concentrated multicomponent alloys, although none of the tools or parameters could achieve it if taken separately. This was used to propose 62 new potential equimolar HEAs, although there was no experimental validation. The use of such an approach for the design of actual alloys will be detailed later.

1. CALPHAD aided design of precipitation-strengthened CCAs

Precipitation is a phenomenon that can occur in an alloy when the matrix phase is supersaturated with respect to one or more solute elements. It involves the nucleation, growth, and coarsening of one or more secondary phases. These processes depend on both thermodynamic (driving forces) and kinetic parameters (diffusivities of the solutes). They can be captured with simple continuum models similar to that developed by Deschamps and Bréchet^{Reference Deschamps and Brechet83} for the treatment of precipitation in Al and Mg alloys.^{Reference Hutchinson, Nie and Gorsse84} The nucleation and growth of each phase is described within the framework of classical nucleation theory^{Reference Aaronson, Lee and Aaronson85,Reference Russell86} and diffusional growth theory.^{Reference Zener87} The nucleation is written as

$${{{\rm{d}}N} \over {{\rm{d}}T}} \cong {N_0}Z\beta \exp \left( {{{ - \Delta {G^*}} \over {kT}}} \right)\quad ,$$

where N ₀ is the number of nucleation sites per unit volume, Z is the Zeldovich’s factor, and β is the frequency factor that describes the attachment kinetics of atoms to the nucleus. The most important parameter entering this equation is the activation barrier for nucleus formation, ΔG*. For spherical precipitates, the nucleation barrier can be expressed as

$$\Delta {G^ * } = {{16} \over 3}\pi {{{\gamma ^3}} \over {{{\left( {{{\Delta {G_{\rm{m}}}} / {{V_{\rm{m}}}}}} \right)}^2}}}\quad ,$$

where γ is the interfacial energy separating the nucleus and the matrix, ΔG _m is the onset chemical driving force for nucleation, and V _m is the molar volume of the precipitate phase. The thermodynamic quantity DGm, necessary for calculations of precipitation kinetics, can be obtained from the Gibbs energies of the involved phases when their thermodynamic description is available, as illustrated in Fig. 18.

FIG. 18.

Schematic molar Gibbs energy diagram with the curves for the α and ε phases in a binary system A–B. The tangent construction to calculate the onset driving force ΔG _m for precipitation of ε from a metastable supersaturated α matrix is illustrated.

An example of analysis which can be conducted to explain the microstructure development in CCAs is shown in Fig. 19 for the Al_0.3CoCrFeNi alloy.^{Reference Bouaziz, Allain, Scott, Cugy and Barbier88} In this alloy, a fine γ′-L1₂ precipitate distribution is developed in the γ-fcc matrix during annealing although the equilibrium phase field of operation consists of fcc + B2 + σ. The onset driving forces for nucleation, calculated using ThermoCalc coupled to a user-defined code with Matlab and the TCHEA2 thermodynamic database, highlight the competition of the different phases. As shown in Fig. 19, the nucleation of the L1₂ phase is favored over the other competing equilibrium phases. This suggests that even though the L1₂ phase does not have the highest onset driving force, it precipitates first due to the low γ/γ′ interfacial energy, hence to a lower energy barrier for nucleation.

FIG. 19.

Normalized nucleation barriers for homogenous precipitation of L1₂, B2, bcc, and σ phases from the supersaturated fcc solid solution with Al_0.3CoCrFeNi composition.

2. CALPHAD-aided design of TWIP HEAs

An exceptional balance of strength and ductility can be achieved in materials exhibiting high work-hardening capability. This results from the activation of mechanical twinning or ε or α′ martensitic transformations (in steels) induced by straining, which are competitive to dislocation glide during deformation.^{Reference Bouaziz, Allain, Scott, Cugy and Barbier88} This led to the concepts of TWIP and TRIP. The competition between these mechanisms is largely controlled by the magnitude of the stacking fault energy (SFE). Twinning is reported to occur at SFE between 18 and 45 mJ/m²,^{Reference Allain, Chateau and Bouaziz89} while dislocation glide is activated at SFE above 45 mJ/m² and martensite formation below 16 mJ/m². In other words, an alloy with a low SFE is more likely to deform by twinning or by a strain-induced martensitic transformation. To select one of these deformation mechanisms, the magnitude of the SFE has to be finely tuned to the appropriate range by adjusting the chemistry of the alloy. This approach is subject to intense research activity in the steel community since the discovery of high manganese TWIP steels in 1888.^{Reference Hadfield90} In these steels, low SFE and the segregation of C to the dislocations is responsible for mechanical twinning which acts as obstacles to dislocation glide.^{Reference Rémy91} Otto et al.^{Reference Otto, Dlouhy, Somsen, Bei, Eggeler and George92} and Gludovatz et al.^{Reference Gludovatz, Hohenwarter, Catoor, Chang, George and Ritchie93} were among the first to report the occurrence of deformation-induced twinning in the Cantor’s alloy to explain the exceptional fracture toughness at cryogenic temperatures. These studies opened up the way to new routes for the development of TWIP fcc HEAs by empirical design^{Reference Li, Pradeep, Deng, Raabe and Tasan81,Reference Deng, Tasan, Pradeep, Springer, Kostka and Raabe94} or by ab initio calculations to predict the SFE.^{Reference Beyramali Kivy and Asle Zaee95} The above-mentioned empirical approach was also successfully used to propose new TRIP CCAs.^{Reference Li, Pradeep, Deng, Raabe and Tasan81} The strategy is based on the modification of the SFE by changing the chemical composition to obtain the activation of different deformation mechanisms and promote either nanotwinning or a strain-induced martensitic transformation at room temperature.

The magnitude of SFE, γ_SFE, of γ-austenite (fcc solid solution) can be estimated on a thermodynamic basis, as a function of the temperature and solute content, which offers the mean to explore the effect of alloy components on the SFE. The example we provide below to illustrate this idea uses the model developed by Olson and Cohen^{Reference Olson and Cohen96} in which the SFE represents the Gibbs energy required for the formation of two atomic layers of ε-martensite within the dense planes:

$${\gamma _{{\rm{SFE}}}} = 2\rho \Delta {G^{\gamma \to \varepsilon }} + 2{\sigma ^{{\gamma / \varepsilon }}}\quad ,$$

where ρ is the surface density of atoms in the {111} planes, ΔG ^γ→ε is the change in the molar Gibbs energy upon phase transformation γ_fcc → ε_hcp, and σ^γ/ε is the surface energy of the interface γ/ε.

The difference of the molar Gibbs energy ΔG ^γ→ε between γ and ε martensite formation can be calculated from the Gibbs energy descriptions of the γ and ε phases from the TCHEA CALPHAD databases. Figure 20 shows an attempt to correlate the magnitude of SFE to the chemical composition. The relative influence of the level of concentration of each alloying element is predicted with respect to the Al_0.3CoCrFeNi composition. Al and Ni are shown to increase the SFE, while Co, Cr, and Fe decrease the SFE. Al has the greatest effect. Keeping in mind the lack of accuracy of current thermodynamic databases and that low SFE is a necessary but not sufficient condition for the activation of mechanical twinning, SFE predictions using CALPHAD can be regarded as a useful design guidance to select alloying elements and adjust their composition to exploit TWIP effect in HEAs or CCAs. However, the previous approaches to develop TWIP and/or TRIP HEAs used empirical concepts^{Reference Li, Pradeep, Deng, Raabe and Tasan81} or ab initio simulations.^{Reference Beyramali Kivy and Asle Zaee95} Due to their empirical nature or to the long time that is needed to run simulations, none of these approaches would be suited to an efficient computational alloy design strategy to optimize alloy properties, neither by a systematic screening of a reduced compositional space, nor by a guided/intelligent exploration as described earlier. At the opposite and as already demonstrated, CALPHAD makes fast predictions and can be advantageously used as a computational design tool. In this respect, this opens the way for multi-objective optimization of HEAs and CCAs where the SFE is an explicit criterion, among others like phase constitution, solid solution hardening, precipitation hardening, density, cost, etc., with vast perspectives in terms of potential properties.

FIG. 20.

Predicted effect of each element on the variation of the SFE with respect to the Al_0.3CoCrFeNi alloy at 300 K. Each line represents the change due to the variation in composition of one given element while the others remain constant. For Co, alloy composition varies from Al_0.3Co_0.7CrFeNi to Al_0.3Co_1.3CrFeNi. The same range is applied for Cr, Fe, and Ni, while with Al, the composition spans from Al₀Co_0.7CrFeNi to Al_0.3Co_0.7CrFeNi.