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

\[ \Delta H_{\text{mix}} = \sum_{i=1,\,i\neq j}^{n} 4\,\Delta H_{AB}^{\text{mix}}\,x_i x_j \quad [\text{kJ/mol}] \]

where \(x_i\) is the mole fraction of element \(i\) and \(\Delta H_{AB}^{\text{mix}}\) is the binary mixing enthalpy from Miedema's model.1

Miedema Mixing Enthalpy

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

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

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

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

Formation Enthalpy

\[ \Delta H_f = \sum_{i=1,\,i\neq j}^{n} x_i x_j\,\Delta H_{ij}^f \quad [\text{meV/atom}] \]

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

Atomic Size Difference (δ)

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

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

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

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

where \(\chi_i\) is the Allen configuration energy (CE) of element \(i\) in Pauling units and \(\bar{\chi} = \sum_i x_i \chi_i\) is the composition-weighted average.1213

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

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

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

Omega (Ω)

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

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

Gamma (γ)

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

where \(r_S\) and \(r_L\) are the radii of the smallest and largest atoms, respectively.6

Lambda (λ)

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

A combined entropy–misfit parameter.7

Valence Electron Concentration (VEC)

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

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

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

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

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

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

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

Melting Temperature

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

Solid-Solution Prediction Models

HEACalculator implements eight published criteria. Each model returns "Solid Solution", "Intermetallic", or "Multiple Phases".

Model Author(s) Criteria Reference
1 Yang & Zhang (2012) \(\Omega \geq 1.1\) and \(\delta \leq 6.6\%\) 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) \(\lambda > 0.96\) (SS); \(0.24 \leq \lambda \leq 0.96\) (SS + compound); \(\lambda < 0.24\) (compound) 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) \(\Omega(T_{\text{anneal}}) \geq k_2 \cdot \Delta S_{\text{mix}} / R\) 10
8 King et al. (2016) \(\phi = \Delta G_{\text{SS}} / (-\lvert \Delta G_{\max}\rvert) \geq 1\) 11

A microstructure prediction based on VEC is also provided:

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

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, 717–723. ↩

  16. Haynes, W.M. CRC Handbook of Chemistry and Physics, 95th ed.; CRC Press: London, 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. ↩