Skip to content

Overview

High Entropy Alloys

High Entropy Alloys (HEAs) are a class of metallic materials composed of five or more principal elements in near-equimolar ratios (typically 5–35 at% each). Unlike conventional alloys that center on one or two base metals, HEAs exploit the configurational entropy of mixing to stabilize single-phase disordered solid solutions over intermetallic compounds, often yielding exceptional combinations of mechanical strength, hardness, and corrosion resistance.

Predicting whether a given multi-component composition will form a solid solution (rather than segregate into intermetallic phases) is a central challenge in HEA design. HEACalculator implements a suite of published phenomenological criteria that use thermodynamic and structural parameters as proxies for phase stability.

Calculated Parameters

Mixing Enthalpy

\[ \Delta H_{\text{mix}} = \sum_{i<j} 4\,\Delta H_{ij}^{\text{mix}}\,c_i c_j \quad [\text{kJ/mol}] \]

where the sum runs over each unique pair of elements once, \(c_i\) is the atomic (mole) fraction of element \(i\), and \(\Delta H_{ij}^{\text{mix}}\) is the binary mixing enthalpy of elements \(i\) and \(j\) from Miedema's model, as tabulated by Takeuchi and Inoue.18 The formula follows Zhang et al.1

Miedema Mixing Enthalpy

\[ \Delta H_{\text{mix}}^{\text{Miedema}} = \sum_{i \neq j} c_i c_j \bigl(c_j H_{\text{chem},ij} + c_i H_{\text{chem},ji} + c_j H_{\text{el},ij} + c_i H_{\text{el},ji} + c_j H_{\text{struct},ij} + c_i H_{\text{struct},ji}\bigr) \quad [\text{kJ/mol}] \]

where the three contributions for each ordered pair \((i \to j)\) are:

  • Chemical interface term \(H_{\text{chem}}\): from the Miedema macroscopic atom model (de Boer et al. 1988)14
  • Elastic mismatch term \(H_{\text{el}}\): from Eshelby theory applied to atomic size and bulk/shear modulus mismatches
  • Structural term \(H_{\text{struct}}\): from Niessen and Miedema (1983)15 via tabulated valence-dependent energies

This three-term formula follows King et al. Supplementary Eq. S8.11 It is used for the Model 8 solid-solution criterion.

Mixing Entropy

\[ \Delta S_{\text{mix}} = -R \sum_{i=1}^{n} c_i \ln c_i \quad [\text{J/K·mol}] \]

where \(R = 8.314\,\text{J/(mol·K)}\) is the gas constant.

King Gibbs Energies and \(\varPhi\)

\[ \Delta G_{SS} = \Delta H_{\text{mix}}^{\text{Miedema}} - T_m \Delta S_{\text{mix}} \quad [\text{kJ/mol}] \]
\[ \Delta G_{\max} = \left\lfloor \frac{n}{2} \right\rfloor H^{\text{int}}_{\max} \quad [\text{kJ/mol}], \qquad \varPhi = \frac{\Delta G_{SS}}{-\lvert \Delta G_{\max} \rvert} \]

where \(\Delta G_{SS}\) is the Gibbs energy of the disordered solid solution at \(T_m\), computed with the Miedema mixing enthalpy above. \(H^{\text{int}}_{\max}\) is the binary intermetallic enthalpy \(H_{ij}^{\text{int}}\) with the largest magnitude among the element pairs in the alloy, where \(H_{ij}^{\text{int}}\) is the Miedema chemical (interface) enthalpy of an ordered intermetallic between elements \(i\) and \(j\) at their relative composition, without the elastic term (King et al. Supplementary Eq. S9). With \(n\) elements, at most \(\lfloor n/2 \rfloor\) binary intermetallics can form while keeping the alloy's stoichiometry. Model 8 predicts a solid solution when \(\varPhi \geq 1\).11

Formation Enthalpy

\[ \Delta H_f = \sum_{i<j} 4\,\Delta H_{ij}^f\,c_i c_j \quad [\text{meV/atom}] \]

Binary formation enthalpies \(\Delta H_{ij}^f\) are taken from DFT calculations by Troparevsky et al.2

Minimum and Maximum Binary Formation Enthalpy

\[ \Delta H_f^{\min} = \min_{i<j} \Delta H_{ij}^f, \qquad \Delta H_f^{\max} = \max_{i<j} \Delta H_{ij}^f \quad [\text{meV/atom}] \]

the lowest and highest binary formation enthalpies among the element pairs in the alloy. Model 6 uses both: it predicts a solid solution when \(\Delta H_f^{\min}\) is above the entropy bound \(-T_{\text{crit}}\Delta S_{\text{mix}}\) and \(\Delta H_f^{\max}\) is below 37 meV/atom.2

Atomic Size Difference (δ)

\[ \delta = \sqrt{\sum_{i=1}^{n} c_i \left(1 - \frac{r_i}{\bar{r}}\right)^2} \times 100 \quad [\%] \]

where \(r_i\) is the atomic radius of element \(i\) and \(\bar{r} = \sum_i c_i r_i\) is the average radius.4 21

Atomic Size Difference, CN12 (\(\delta_{\text{CN12}}\))

\[ \delta_{\text{CN12}} = \sqrt{\sum_{i=1}^{n} c_i \left(1 - \frac{r_i^{\text{CN12}}}{\bar{r}^{\text{CN12}}}\right)^2} \times 100 \quad [\%] \]

where \(r_i^{\text{CN12}}\) is the Goldschmidt radius of element \(i\) for 12-fold coordination and \(\bar{r}^{\text{CN12}} = \sum_i c_i r_i^{\text{CN12}}\) is the average CN12 radius. Model 1 and \(\gamma\) (Model 3) use CN12 radii, while Model 2, \(\lambda\) (Model 4), and \(S_E\) (Model 5) use the atomic radius.4 11

Allen Electronegativity Difference (\(\Delta\chi_{\text{Allen}}\))

\[ \Delta\chi_{\text{Allen}} = \sqrt{\sum_{i=1}^{n} c_i \left(1 - \frac{\chi_i}{\bar{\chi}}\right)^2} \times 100 \quad [\%] \]

where \(\chi_i\) is the Allen configuration energy (CE) of element \(i\) in Pauling units and \(\bar{\chi} = \sum_i c_i \chi_i\) is the composition-weighted average.12 13

Pauling Electronegativity Difference (\(\Delta\chi_{\text{Pauling}}\))

\[ \Delta\chi_{\text{Pauling}} = \sqrt{\sum_{i=1}^{n} c_i \left(1 - \frac{\chi_i}{\bar{\chi}}\right)^2} \times 100 \quad [\%] \]

where \(\chi_i\) is the Pauling electronegativity of element \(i\) and \(\bar{\chi} = \sum_i c_i \chi_i\) is the composition-weighted average.16

Omega (Ω)

\[ \Omega = \frac{T_m \,\Delta S_{\text{mix}}}{|\Delta H_{\text{mix}}|} \]

where \(T_m = \sum_i c_i T_{m,i}\) is the composition-weighted melting temperature.5

omega_at(T) evaluates \(\Omega\) at any temperature \(T\) (in K) by using \(T\) in place of \(T_m\) in the numerator. Model 7 uses it at the annealing temperature \(T_{\text{anneal}}\).

Gamma (γ)

\[ \gamma = \omega_S / \omega_L = \left(1 - \sqrt{\frac{(r_S + \bar{r})^2 - \bar{r}^2}{(r_S + \bar{r})^2}}\right) \Bigg/ \left(1 - \sqrt{\frac{(r_L + \bar{r})^2 - \bar{r}^2}{(r_L + \bar{r})^2}}\right) \]

where \(\omega_S\) and \(\omega_L\) are the solid angles of the smallest and largest atoms, \(r_S\) and \(r_L\) are their Goldschmidt CN12 radii, and \(\bar{r}\) is the composition-weighted average CN12 radius.6

Lambda (λ)

\[ \lambda = \frac{\Delta S_{\text{mix}}}{\delta^2} \]

A combined entropy–misfit parameter.7

Phi (ϕ)

\[ \phi = \frac{S_c - S_H}{\lvert S_E \rvert}, \qquad S_H = \frac{\lvert \Delta H_{\text{mix}} \rvert}{T_m} \]

where \(S_c = \Delta S_{\text{mix}}\) is the ideal configurational entropy of mixing, \(S_H\) is the complementary entropy derived from the mixing enthalpy, and \(S_E\) is the excess entropy of mixing caused by atomic size misfit and dense packing. \(S_E\) is computed with the Mansoori–Carnahan–Starling–Leland hard-sphere model from the atomic radii and averaged over the FCC (\(\xi = 0.74\)) and BCC (\(\xi = 0.68\)) packing fractions; phi_fcc and phi_bcc give the value at each packing fraction.9 19 20

Valence Electron Concentration (VEC)

\[ \text{VEC} = \sum_{i=1}^{n} c_i\,(\text{VEC})_i \]

Used to predict the stable crystal structure (FCC, BCC, or HCP).3

Hume-Rothery Electron-to-Atom Ratio (e/a)

\[ e/a = \sum_{i=1}^{n} c_i\,(e/a)_i \]

where \((e/a)_i\) is the number of outer s+p electrons of element \(i\); d and f electrons are not counted. This follows the Hume-Rothery convention and is distinct from VEC.17

Density

\[ \rho = \frac{\sum_i c_i M_i}{\sum_i c_i V_i} \quad [\text{g/cm}^3] \]

where \(M_i\) and \(V_i\) are the molar mass and molar volume of element \(i\).

Melting Temperature

\[ \overline{T}_m = \sum_{i=1}^{n} c_i\,T_{m,i} \quad [\text{K}] \]

Critical Temperature

\[ T_{\text{crit}} = 0.55\,T_m \quad [\text{K}] \]

Model 6 uses it in the entropy bound \(-T_{\text{crit}}\Delta S_{\text{mix}}\) (converted to meV/atom), and Model 7 uses it as the default annealing temperature \(T_{\text{anneal}}\).

Solid-Solution Prediction Models

HEACalculator implements eight published criteria. Each model returns "Solid Solution", "Intermetallic", or "Multiple Phases". A model returns "N/A" when the data it requires are unavailable.

Model Author(s) Criteria Reference
1 Yang & Zhang (2012) \(\Omega \geq 1.1\) and \(\delta_{\text{CN12}} \leq 6.6\%\) 5
2 Guo et al. (2013) \(-11.6 < \Delta H_{\text{mix}} < 3.2\,\text{kJ/mol}\) and \(\delta < 6.6\%\) 8
3 Wang et al. (2015) \(\gamma < 1.175\) 6
4 Singh et al. (2014) \(\lambda > 0.96\): Solid Solution; \(0.24 \leq \lambda \leq 0.96\): Multiple Phases; \(\lambda < 0.24\): Intermetallic 7
5 Ye et al. (2015) \(\phi = (S_c - S_H) / \lvert S_E\rvert \geq 20\) 9
6 Troparevsky et al. (2015) \(\Delta H_f^{\min} > -T_{\text{crit}}\Delta S_{\text{mix}}\) and \(\Delta H_f^{\max} < 37\,\text{meV/atom}\), \(T_{\text{crit}} = 0.55\,T_m\) 2
7 Senkov & Miracle (2016) \(k_1 = \Delta H_f / \Delta H_{\text{mix}} < 1 + \Omega(T_{\text{anneal}})(1 - k_2)\), \(T_{\text{anneal}} = 0.55\,T_m\), \(k_2 = 0.6\) 10
8 King et al. (2016) \(\varPhi = \Delta G_{SS} / (-\lvert \Delta G_{\max}\rvert) \geq 1\) 11

A microstructure prediction based on VEC is also provided. The HCP window is tested first, so a composition falling in it is reported as HCP even though it also satisfies the BCC bound:

  • 2.5 ≤ VEC ≤ 3.5: HCP
  • VEC ≥ 8: FCC
  • VEC ≤ 6.87: BCC
  • 6.87 < VEC < 8: BCC + FCC (mixed)

References

  1. Zhang, Y.; Zuo, T.T.; Tang, Z.; Gao, M.C.; Dahmen, K.A.; Liaw, P.K.; Lu, Z.P. Prog. Mater. Sci. 2014, 61, 1–93. ↩

  2. Troparevsky, M.C.; Morris, J.R.; Kent, P.R.C.; Lupini, A.R.; Stocks, G.M. Phys. Rev. X 2015, 5(1), 011041. ↩↩

  3. Guo, S.; Ng, C.; Lu, J.; Liu, C.T. J. Appl. Phys. 2011, 109, 103505. ↩

  4. Fang, S.S.; Xiao, X.S.; Xia, L.; Li, W.H.; Dong, Y.D. J. Non-Cryst. Solids 2003, 321, 120–125. ↩↩

  5. Yang, X.; Zhang, Y. Mater. Chem. Phys. 2012, 132, 233–238. ↩

  6. Wang, Z.; Huang, Y.; Yang, Y.; Wang, J.; Liu, C.T. Scr. Mater. 2015, 94, 28–31. ↩

  7. Singh, A.K.; Kumar, N.; Dwivedi, A.; Subramaniam, A. Intermetallics 2014, 53, 112–119. ↩

  8. Guo, S.; Hu, Q.; Ng, C.; Liu, C.T. Intermetallics 2013, 41, 96–103. ↩

  9. Ye, Y.F.; Wang, Q.; Lu, J.; Liu, C.T.; Yang, Y. Scr. Mater. 2015, 104, 53–55. ↩

  10. Senkov, O.N.; Miracle, D.B. J. Alloys Compd. 2016, 658, 603–607. ↩

  11. King, D.J.M.; Middleburgh, S.C.; McGregor, A.G.; Cortie, M.B. Acta Mater. 2016, 104, 172–179. ↩↩↩

  12. Mann, J.B.; Meek, T.L.; Allen, L.C. J. Am. Chem. Soc. 2000, 122, 2780–2783. ↩

  13. Mann, J.B.; Meek, T.L.; Knight, E.T.; Capitani, J.F.; Allen, L.C. J. Am. Chem. Soc. 2000, 122, 5132–5137. ↩

  14. de Boer, F.R.; Boom, R.; Mattens, W.C.M.; Miedema, A.R.; Niessen, A.K. Cohesion in Metals: Transition Metal Alloys. North-Holland, Amsterdam, 1988. ↩

  15. Niessen, A.K.; Miedema, A.R. Ber. Bunsenges. Phys. Chem. 1983, 87, 717–725. ↩

  16. Haynes, W.M. CRC Handbook of Chemistry and Physics, 95th ed.; CRC Press: Boca Raton, FL, 2014. ISBN 9781482208689. ↩

  17. Hume-Rothery, W.; Smallman, R.E.; Haworth, C.W. The Structure of Metals and Alloys, 5th ed.; Institute of Metals: London, 1969. ↩

  18. Takeuchi, A.; Inoue, A. Mater. Trans. 2005, 46(12), 2817–2829. ↩

  19. Ye, Y.F.; Wang, Q.; Lu, J.; Liu, C.T.; Yang, Y. Intermetallics 2015, 59, 75–80. ↩

  20. Mansoori, G.A.; Carnahan, N.F.; Starling, K.E.; Leland, T.W., Jr. J. Chem. Phys. 1971, 54, 1523–1525. ↩

  21. Senkov, O.N.; Miracle, D.B. Mater. Res. Bull. 2001, 36, 2183–2198. ↩