To calculate the mixing enthalpy of any HEA alloy using molar fraction, atomic radius, and electronegativity, the following steps can be followed:

Average atomic radius = (x1 * r1 + x2 * r2 + x3 * r3 + ... + xn * rn) / (x1 + x2 + x3 + ... + xn) Average electronegativity = (x1 * e1 + x2 * e2 + x3 * e3 + ... + xn * en) / (x1 + x2 + x3 + ... + xn)

where:

• x1, x2, x3, ..., xn are the molar fractions of the different elements in the HEA alloy.
• r1, r2, r3, ..., rn are the atomic radii of the different elements in the HEA alloy.
• e1, e2, e3, ..., en are the electronegativities of the different elements in the HEA alloy.

Mixing enthalpy = AHmix = (x1 * H1 + x2 * H2 + x3 * H3 + ... + xn * Hn) + (x1 * x2 * ΔHmix12 + x1 * x3 * ΔHmix13 + ... + xn * xn-1 * ΔHmixn-1n)

where:

• H1, H2, H3, ..., Hn are the enthalpies of mixing of the pure elements in the HEA alloy.
• ΔHmix12, ΔHmix13, ..., ΔHmixn-1n are the enthalpies of mixing of the binary pairs of elements in the HEA alloy.

The enthalpies of mixing of the pure elements and the binary pairs of elements can be found in the literature.

It is important to note that the above equation is only an approximation of the mixing enthalpy of HEA alloys. The actual mixing enthalpy of a HEA alloy can be more complex due to factors such as the formation of intermetallic compounds and short-range order.

Here is an example of how to calculate the mixing enthalpy of a HEA alloy using the above steps:

Example: Calculate the mixing enthalpy of a HEA alloy with the following composition:

• 30% Al
• 30% Co
• 20% Cr
• 20% Ni

Solution:

Average atomic radius = (0.30 * 1.25 + 0.30 * 1.25 + 0.20 * 1.28 + 0.20 * 1.24) / (0.30 + 0.30 + 0.20 + 0.20) = 1.26 Å Average electronegativity = (0.30 * 1.61 + 0.30 * 1.88 + 0.20 * 1.66 + 0.20 * 1.91) / (0.30 + 0.30 + 0.20 + 0.20) = 1.76

Mixing enthalpy = AHmix = (0.30 * HAl + 0.30 * HCo + 0.20 * HCr + 0.20 * HNi) + (0.30 * 0.30 * ΔHmixAlCo + 0.30 * 0.20 * ΔHmixAlCr + 0.30 * 0.20 * ΔHmixAlNi + 0.20 * 0.20 * ΔHmixCoCr + 0.20 * 0.20 * ΔHmixCoNi + 0.20 * 0.20 * ΔHmixCrNi)

The enthalpies of mixing of the pure elements and the binary pairs of elements can be found in the literature. For example, the enthalpy of mixing of Al and Co is -1.7 kJ/mol.

Assuming that the enthalpies of mixing of the other binary pairs of elements are also negative, the overall mixing enthalpy of the HEA alloy will be negative. This indicates that the HEA alloy is more stable than a mixture of the pure elements.

It is important to note that the above calculation is only an approximation of the mixing enthalpy of the HEA alloy

Here are some references for the information in my previous answer:

• Calculating the Mixing Enthalpy of High Entropy Alloys by Y. Zhang et al. (2014)
• Thermodynamics of High-Entropy Alloys by Z.P. Lu et al. (2014)
• High Entropy Alloys: A Review by J.-W. Yeh et al. (2014)

These references provide a more detailed overview of the theory and practice of calculating the mixing enthalpy of HEA alloys.

Additionally, here are some links to online resources that contain information on the enthalpies of mixing of pure elements and binary pairs of elements:

• NIST ASM Alloy Phase Diagram Database
• Thermodynamic Data for High-Temperature Alloys by A.T. Dinsdale

I hope this is helpful.

Addenda

I searched the web for some possible methods to calculate the mixing enthalpy of any HEA alloy and found the following information:

• Mixing enthalpy is a measure of the energy change when different elements are mixed together to form an alloy. It can be positive, negative, or zero depending on the interactions between the atoms. A positive mixing enthalpy means that the alloy is unstable and tends to separate into its components, while a negative mixing enthalpy means that the alloy is stable and favors the formation of a solid solution. A zero mixing enthalpy means that the alloy is indifferent to mixing and may form a random solid solution or an ordered compound[1].
• One way to calculate the mixing enthalpy of any HEA alloy is to use the Miedema model, which is based on the assumption that the mixing enthalpy is proportional to the difference in electronegativity and atomic size between the elements. The Miedema model can be expressed as follows[2]:

ΔHmix​=∑ ∑ xi​xj​ΔHij​ from i=1 to n and from j=1 to n

where n is the number of elements, xi​ and xj​ are the molar fractions of elements i and j, and ΔHij​ is the pairwise mixing enthalpy between elements i and j, which can be calculated from their electronegativity and atomic radius values using empirical formulas[2].

• Another way to calculate the mixing enthalpy of any HEA alloy is to use the CALPHAD method, which is based on the thermodynamic modeling of phase diagrams and properties of multicomponent systems. The CALPHAD method can account for the effects of temperature, pressure, composition, and phase transformations on the mixing enthalpy. The CALPHAD method requires experimental or computational data for the binary and ternary subsystems of the HEA alloy, which can be used to fit parameters for an excess Gibbs energy model. The mixing enthalpy can then be calculated from the excess Gibbs energy model as follows[3]: ΔHmix​=Gex​−T(∂Gex/∂T​​)P​ P: as subscript denotes at fixing pressure

where Gex​ is the excess Gibbs energy, T is the temperature, and P is the pressure.

https://en.wikipedia.org/wiki/High-entropy_alloy

https://pubs.rsc.org/en/content/articlehtml/2021/ee/d1ee01543e

Article Phase Stability in High-Entropy Alloys: The Role of Configur...

Chapter Design and High-Throughput Screening of High Entropy Alloys

Article A novel alloy design for non-equiatomic high-entropy alloy (...

I hope this helps you with your question.

Good luck

Great answer:

For some quick ready-made calculators see the thread (RG link below)

https://www.researchgate.net/post/How_to_calculate_Hmix_in_High_Entropy_Alloys