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Study of ordering in (MoCrTi) $_{100-x}$ Al $_x$ refractory high-entropy alloys using machine learning interatomic potential

This study employs machine learning interatomic potentials combined with hybrid Monte Carlo and molecular dynamics simulations to reveal that chemical ordering in (MoCrTi) $_{100-x}$ Al $_x$ refractory high-entropy alloys exhibits distinct temperature-dependent transition behaviors and significantly enhances mechanical stiffness through an optimized population of stiff atomic pairs, offering critical insights for alloy design.

Original authors: Jiyao Zhang, Klemens Lechner, Markus Maßwohl, Petra Spoerk-Erdely, David Holec

Published 2026-07-21

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CC BY 4.0

Original authors: Jiyao Zhang, Klemens Lechner, Markus Maßwohl, Petra Spoerk-Erdely, David Holec

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the world of materials science as a giant, high-stakes kitchen where chefs are trying to invent the ultimate super-pan. This pan needs to survive the heat of a jet engine or a rocket nozzle without melting, warping, or falling apart. For decades, the best tools for this job were "superalloys," but scientists are now hunting for something even tougher: Refractory High-Entropy Alloys (RHEAs). Think of these not as simple mixtures, but as chaotic crowds of different metal atoms (like Molybdenum, Chromium, Titanium, and Aluminum) all jammed into the same tiny space. The big mystery is how these atoms behave when they get hot. Do they stay in a messy, random pile, or do they suddenly decide to line up in neat, organized rows? This "ordering" is like a crowd at a concert suddenly forming a synchronized dance; it changes how strong the material is. Understanding this dance is crucial because if we can control it, we can build engines that fly higher and last longer.

In this study, a team of researchers acted as digital detectives, using a powerful computer tool called a "machine learning interatomic potential" to watch these metal atoms dance. Instead of melting real metal in a furnace (which is expensive and hard to control), they built a virtual world where they could simulate temperatures ranging from a chilly 200 Kelvin up to a scorching 2000 Kelvin. They tested four different recipes of the (MoCrTi)100−xAlx alloy, each with a slightly different amount of Aluminum.

The researchers discovered that the atoms don't just behave the same way in every recipe. It's like a party where the music changes depending on how many people are wearing red shirts. For the alloys with the most Aluminum (25%) and the least (4%), the atoms go through a single, dramatic transition: they all switch from a messy pile to an organized pattern at roughly the same time. However, for the middle recipes (16% and 10% Aluminum), the party splits into two acts. First, a small group of specific atoms (like Molybdenum and Aluminum, or just Aluminum and Aluminum) decides to pair up and organize at lower temperatures. Then, much later, the rest of the crowd joins in at a higher temperature.

The most exciting finding is how this "dancing" affects the strength of the metal. The team found that when the atoms are in a messy, random state (like a disordered crowd), the metal gets stiffer as you remove Aluminum, following a predictable, straight-line rule. But when the atoms are organized (ordered), the story changes completely. The stiffness doesn't just go up; it peaks unexpectedly at the 10% Aluminum recipe. It's as if the organized atoms found a "sweet spot" where their specific pairings create a super-strong structure that random mixing can't achieve. The simulations suggest that this happens because the ordered state creates a perfect mix of "stiff" atomic pairs, much like building a bridge with the strongest possible combination of steel beams.

While these results come from computer simulations rather than physical experiments, the patterns they found align with what scientists have seen in real-world tests. The study suggests that by carefully tuning the Aluminum content to hit that "sweet spot" of atomic ordering, engineers could design next-generation alloys that are significantly stronger and more heat-resistant than current materials, paving the way for more advanced aerospace technology.

Technical Summary: Ordering in (MoCrTi)100−xAlx Refractory High-Entropy Alloys

Problem Statement
Refractory high-entropy alloys (RHEAs) are promising candidates for high-temperature aerospace applications due to their superior mechanical properties compared to traditional Ni-based superalloys. However, optimizing their performance requires a fundamental understanding of the thermodynamic mechanisms governing chemical ordering and phase stability in these complex, multi-principal element systems. Traditional theoretical modeling approaches, such as cluster expansion and the embedded atom method (EAM), often suffer from overfitting or lack transferability, while first-principles Density Functional Theory (DFT) is computationally prohibitive for the large-scale simulations required to capture order-disorder transitions and short-range order (SRO) in RHEAs. Specifically, the atomic-scale mechanisms driving order-disorder transitions and their impact on mechanical stiffness in the (MoCrTi)100−xAlx system remain insufficiently understood.

Methodology
This study employs a hybrid simulation approach combining Molecular Dynamics (MD) and Monte Carlo (MC) methods, driven by a universal machine learning interatomic potential (uMLIP).

Potential: The GRACE-1L-OAM model, a uMLIP developed by Bochkarev et al., was utilized for its balance of accuracy and computational efficiency. Its reliability was validated against DFT calculations for specific B2 configurations.
Simulation Protocol: A hybrid MC/MD workflow was implemented using the Atomic Simulation Environment (ASE). The process involved three stages for four specific alloy compositions (Mo25Cr25Ti25Al25, Mo28Cr28Ti28Al16, Mo30Cr30Ti30Al10, and Mo32Cr32Ti32Al4):
1. MC Equilibration: 2,000,000 steps of Metropolis MC simulations on a fixed lattice to achieve chemical equilibrium.
2. MD Thermal Expansion: NPT MD simulations (50,000 steps) to relax cell volumes and account for thermal expansion.
3. Production MC: An additional 2,000,000 MC steps on the thermally expanded structures to collect data for property calculation.
Analysis:
- Thermodynamics: Heat capacity ( $C_V$ ) was calculated from energy fluctuations in the NVT ensemble to identify order-disorder transitions.
- Structural Characterization: Warren-Cowley short-range order (SRO) parameters were computed for first and second nearest-neighbor shells to quantify atomic ordering.
- Mechanical Properties: Elastic constants ( $C_{11}, C_{12}, C_{44}$ ) and moduli (Bulk, Young's, Shear) were derived from stress-strain calculations on equilibrated snapshots at 200 K (ordered) and 2000 K (disordered).
- Validation: DFT calculations (VASP) were performed on specific B2 configurations to validate the uMLIP energetics and investigate sublattice preferences.

Key Results

Order-Disorder Transitions: The heat capacity curves reveal distinct transition behaviors dependent on Aluminum content.
- Single Transition: The equiatomic Al25 and the low-Al Al4 alloys exhibit a single, dominant order-disorder transition.
- Dual Transitions: The intermediate compositions (Al16 and Al10) display bimodal behavior with two separate transitions.
  - In Al16, the low-temperature transition is driven by Mo–Al ordering, while the high-temperature transition is governed by other pairs.
  - In Al10, the low-temperature peak is driven by Al–Al ordering (clustering), with the high-temperature transition driven by remaining pairs.
Structural Ordering: Structural analysis of the low-temperature ordered phase (B2-type) indicates a specific sublattice arrangement: Mo and Al atoms share one sublattice, while Cr and Ti atoms share the other. This configuration is slightly higher in energy than the (Mo, Ti)/(Cr, Al) arrangement predicted by 0 K DFT but is favored at finite temperatures due to entropic effects and the small energy difference (~0.005 eV/atom) between competing states.
Mechanical Stiffness and SRO:
- Disordered State: In random solid solutions (2000 K), mechanical stiffness increases monotonically as Al content decreases, following the rule of mixtures.
- Ordered State: In ordered configurations (200 K), stiffness exhibits a non-monotonic trend, peaking at the Al10 composition. The Al10 alloy shows a massive enhancement in moduli compared to its disordered counterpart (e.g., ~58% increase in Bulk modulus).
- Mechanism: This enhancement is attributed to the specific population of atomic pairs induced by SRO. The Al10 composition maximizes the contribution of stiff atomic pairs (specifically Mo–Mo and Mo–Cr), whereas the random distribution does not optimize this pairing.

Significance and Claims
The paper claims to provide fundamental insights into the interplay between chemical composition, short-range order, and mechanical performance in refractory high-entropy alloys. By utilizing a universal machine learning potential, the study successfully models the complex thermodynamics of the (MoCrTi)100−xAlx system, revealing that:

Order-disorder transitions in these systems are not uniform but depend critically on specific atomic pair interactions driven by Al concentration.
Chemical ordering significantly enhances mechanical stiffness, but this enhancement is non-monotonic with respect to composition, contradicting simple rule-of-mixtures predictions.
The optimization of mechanical properties in RHEAs can be guided by understanding and controlling the population of specific stiff atomic pairs through short-range ordering.

These findings offer atomistic guidance for the design of high-performance RHEAs, suggesting that balancing composition to maximize beneficial short-range order (as seen in the Al10 alloy) is a viable strategy for optimizing strength and stiffness.

Technical Summary: Ordering in (MoCrTi)100−xAlx Refractory High-Entropy Alloys

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