[Paper Review] Elastic energy of multi-component solid solutions and strain origins of phase stability in high-entropy alloys

Reza Darvishi Kamachali, Lei Wang|arXiv (Cornell University)|Jul 1, 2021

High Entropy Alloys Studies41 references25 citations

TL;DR

This paper generalizes Eshelby's elastic energy model to multi-component solid solutions, introducing a strain coefficient 𝜆∗ to predict phase stability in high-entropy alloys (HEAs). It shows that 𝜆∗ < 0.16 strongly correlates with solid solution formation, outperforming the polydispersity index 𝛿, and establishes a direct link between elastic energy and 𝛿 via 𝑒 = 𝑞𝛿².

ABSTRACT

The elastic energy of mixing for multi-component solid solutions is derived by generalizing Eshelby's sphere-in-hole model for binary alloys. By surveying the dependence of the elastic energy on chemical composition and lattice misfit, we propose a lattice strain coefficient {\lambda}*. Applying to several high-entropy alloys and superalloys, we found that most solid solution alloys are stable when {\lambda}*<0.16, analogous to the Hume-Rothery atomic-size rule for binary alloys. We also reveal that the polydispersity index {\delta}, frequently used for describing strain in multi-component alloys, is directly related to the elastic energy (e) with e=q{\delta}^2, q being an elastic constant. Furthermore, the effects of (i) the number and (ii) the atomic-size distribution of constituting elements on the phase stability of high-entropy alloys were quantified. The present derivations open for richer considerations of elastic effects in high-entropy alloys, offering immediate support for quantitative assessments of their thermodynamic properties and studying related strengthening mechanisms.

Motivation & Objective

• To extend Eshelby's elastic energy model from binary to multi-component solid solutions.
• To identify a quantitative strain parameter that predicts phase stability in high-entropy alloys.
• To clarify the relationship between the polydispersity index 𝛿 and elastic energy in complex alloys.
• To quantify the effects of the number and size distribution of elements on HEA stability.
• To provide a framework for integrating elastic energy into thermodynamic and strengthening models of HEAs.

Proposed method

• Generalizes Eshelby’s sphere-in-hole model to derive the elastic energy of mixing for N-component random solid solutions.
• Derives the elastic energy equation: 𝑒 = 𝑞∑𝜆𝑖ℎ²(𝑋𝑖 − 𝑋𝑖²) − 2𝑞∑𝜆𝑖ℎ𝜆𝑗ℎ𝑋𝑖𝑋𝑗 for 𝑖,𝑗 ≠ ℎ, 𝑖 ≠ 𝑗.
• Introduces a new strain coefficient 𝜆∗ = √𝜆𝑚𝑎𝑥² − 𝜆𝑚𝑖𝑛² to quantify elastic driving force for decomposition.
• Establishes a direct proportionality between elastic energy and polydispersity: 𝑒 = 𝑞𝛿², with 𝑞 as an elastic constant.
• Uses Vegard’s law and atomic-size distribution to compute 𝜆𝑖ℎ and 𝜆∗ from elemental radii and compositions.
• Analyzes equiatomic HEAs to show that elastic energy decreases with increasing 𝑁, outpacing configurational entropy decay.

Experimental results

Research questions

• RQ1 Can the elastic energy of mixing in multi-component solid solutions be generalized beyond binary alloys?
• RQ2 Is there a single strain parameter that reliably predicts phase stability in high-entropy alloys?
• RQ3 How does the polydispersity index 𝛿 relate quantitatively to the elastic energy of mixing?
• RQ4 What is the role of the number of components 𝑁 and their atomic-size distribution in elastic stability?
• RQ5 How do elastic and configurational entropy contributions compare in determining Gibbs free energy of mixing in equiatomic HEAs?

Key findings

• A threshold of 𝜆∗ < 0.16 effectively distinguishes solid solution HEAs from intermetallic or amorphous phases, outperforming the polydispersity index 𝛿.
• The elastic energy of mixing is directly proportional to the square of the polydispersity index: 𝑒 = 𝑞𝛿², with 𝑞 as an elastic constant.
• For equiatomic HEAs, the elastic energy decreases with 𝑁 as 1/(𝑁−1)², faster than the configurational entropy’s 1/𝑁 dependence.
• At temperatures below ~400 K, the elastic energy contribution dominates over entropy in determining Gibbs free energy of mixing.
• A large 𝜉 value (hetero-disperse size distribution) promotes short-range ordering, as seen in TaNbHfZr, driven by elastic energy minimization.
• The strain coefficient 𝜆∗ remains a robust predictor even when composition-dependent terms in the decomposition driving force are considered.