Finite Temperature Stacking Fault Stability in Random and Locally Ordered CoCrNi beyond the Harmonic Approximation

This study demonstrates that while local chemical order stabilizes positive stacking-fault energy in CoCrNi across temperatures, finite-temperature anharmonic effects fail to thermally stabilize the negative stacking-fault energy predicted for random solid solution CoCrNi, thereby resolving the discrepancy between previous harmonic DFT predictions and experimental observations.

The Gist

Imagine a metal alloy called CoCrNi (a mix of Cobalt, Chromium, and Nickel) as a giant, crowded dance floor. In this dance, the atoms are the dancers, and they usually move in a very orderly, repeating pattern called a "face-centered cubic" (FCC) structure.

Sometimes, during a dance move (deformation), a section of the floor gets a little "glitch" or a slip. In materials science, this is called a stacking fault. Think of it like a rug that has been slightly bunched up or shifted out of place.

The big question scientists have been asking is: Does this "bunching up" stay small and manageable, or does it spread out uncontrollably?

The Mystery: The "Negative Energy" Problem

For a long time, computer simulations (using a method called DFT) predicted that in a perfectly random mix of these atoms (called a Random Solid Solution or RSS), this "bunching" should be unstable.

• The Analogy: Imagine trying to hold a rubber band that has negative tension. Instead of snapping back, it wants to stretch forever.
• The Prediction: The computer said the "energy" required to create this fault was negative. This meant the atoms would want to separate endlessly, creating a massive, infinite slip.
• The Reality: Real-world experiments show the slip does happen, but it stays finite (it stops at a certain width). The rubber band doesn't stretch forever; it stops.

Scientists proposed two theories to fix this mismatch between the computer and reality:

1. Theory A (Heat): Maybe the heat of the room (temperature) acts like a stabilizer, making the rubber band stop stretching.
2. Theory B (Order): Maybe the atoms aren't actually random. Maybe they have little "friend groups" or local clusters (called Local Chemical Order or LCO) that naturally hold the slip in place.

What This Paper Did

The authors of this paper wanted to settle the debate. They used a super-accurate AI model (a "neural network potential") to simulate the atoms, but with a crucial twist: they didn't just look at the atoms as stiff balls vibrating slightly (the old "harmonic" way). They looked at them as wobbly, chaotic dancers who bump into each other hard (the "anharmonic" way). This is more like real life, where atoms get messy when they get hot.

The Findings: What Actually Happens?

1. The "Heat Stabilization" Theory is Wrong
The authors tested the Random (RSS) mix first.

• Old View: They thought heating it up would make the "rubber band" stop stretching.
• New Discovery: When they accounted for the messy, wobbly vibrations of hot atoms, they found the opposite. As the temperature went up, the "rubber band" actually wanted to stretch more.
• The Result: In a perfectly random mix, the stacking fault is not stabilized by heat. It remains unstable and wants to expand forever. The old computer models that said "heat fixes it" were missing the messy reality of how atoms vibrate.

2. The "Local Order" Theory is the Hero
Next, they looked at the mix where atoms had formed little "friend groups" (LCO).

• The Discovery: Even at high temperatures, these local groups acted like a safety net. They created a "restoring force" (like a normal rubber band) that pulled the slip back to a specific, finite size.
• The Result: The "bunching" stayed small and stable, just like in the real experiments. The local chemical order is what stops the slip from running away.

3. The Dislocation Dance (The Proof)
To be absolutely sure, they ran massive simulations with millions of atoms, watching a "dislocation" (a line of defects) move through the metal.

• In the Random Mix: The dislocation split apart and kept spreading until it hit the edge of the simulation box. It was uncontrolled chaos.
• In the Ordered Mix: The dislocation split, but then stopped. It found a comfortable, stable width and stayed there.

The Takeaway

The paper concludes that the reason we see stable, finite "bunching" in CoCrNi alloys isn't because the heat saves the day. It's because the atoms aren't actually random. They have local pockets of order that act as anchors, keeping the material stable.

In simple terms:

• Random Mix: Like a crowd of strangers pushing each other; if one person slips, the whole crowd might collapse and spread out forever.
• Ordered Mix: Like a crowd of friends holding hands in small groups; if one person slips, the group pulls them back, keeping the mess contained.

The study proves that these "friend groups" (Local Chemical Order) are the real reason this metal is so tough and stable, even when it gets hot.

Technical Summary

Technical Summary: Finite Temperature Stacking Fault Stability in Random and Locally Ordered CoCrNi beyond the Harmonic Approximation

Problem Statement
A significant discrepancy exists between theoretical predictions and experimental observations regarding the stacking-fault stability in the equiatomic CoCrNi medium-entropy alloy (MPEA). Experimental studies consistently observe finite stacking-fault widths (SFW), implying a positive intrinsic stacking-fault energy (ISFE). However, standard Density Functional Theory (DFT) calculations for random solid solution (RSS) models predict negative mean ISFE values at 0 K, which theoretically suggests continuous, unbounded separation of partial dislocations.

Two primary explanations have been proposed to resolve this:

1. Finite-Temperature Stabilization: Harmonic or quasi-harmonic approximations suggest that ISFE increases with temperature, potentially becoming positive at elevated temperatures.
2. Local Chemical Order (LCO): The presence of chemically ordered domains (ubiquitous in MPEAs) may shift the ISFE to positive values even at 0 K.

However, existing literature leaves critical gaps: previous finite-temperature calculations for RSS often neglect explicit anharmonic vibrational effects, and the temperature dependence of ISFE in LCO states remains unexplored.

Methodology
This study employs a multi-faceted computational approach to investigate the temperature-dependent generalized stacking-fault energy (GSFE) in CoCrNi for both RSS and LCO configurations:

• Interatomic Potential: A near-quantum-accuracy neural network potential (NNP) was utilized to efficiently sample diverse local atomic environments, overcoming the accuracy limitations of empirical embedded-atom method (EAM) potentials.
• Anharmonic Free Energy Calculations: The study utilizes the fully anharmonic projected average force integrator (PAFI) method. Unlike harmonic approximations, PAFI explicitly accounts for anharmonic vibrational contributions by integrating thermally averaged mean forces along the minimum energy path.
• Ensemble Sampling:
  • RSS: An ensemble of 50 independent 450-atom supercells with random equimolar distributions of Co, Cr, and Ni.
  • LCO: An ensemble of 50 independent 2,520-atom supercells containing nanometer-sized chemically ordered domains, generated via a hybrid molecular dynamics/Monte Carlo (MD/MC) protocol.
• Validation: Large-scale molecular dynamics (MD) simulations (approx. 1.67 million atoms) were performed to simulate edge dislocation structures in both chemical states, directly observing stacking-fault width evolution.
• Analysis: The Warren-Cowley (WC) parameters were calculated to quantify chemical order, and the Interval Common Neighbor Analysis (ICNA) technique was used for defect identification.

Key Results

1. Failure of Harmonic Approximation for RSS:

   • Harmonic calculations reproduce the trend of increasing ISFE with temperature.
   • In contrast, the fully anharmonic PAFI calculations reveal that the ISFE for RSS CoCrNi decreases with increasing temperature.
   • Consequently, the RSS ISFE remains negative across the studied temperature range (up to 1000 K), indicating that finite-temperature effects do not thermally stabilize stacking faults in the random state.

2. Stabilization via Local Chemical Order (LCO):

   • The presence of LCO domains shifts the mean ISFE to positive values (approx. 38 mJ/m² at 0 K).
   • While the LCO ISFE decreases monotonically with temperature due to anharmonic vibrations, it remains positive across nearly the entire range (0–1000 K), only approaching zero near 1000 K.
   • This positive ISFE provides the necessary thermodynamic restoring force to balance the repulsive elastic interaction between partial dislocations, allowing for finite SFW.

3. Dislocation Behavior:

   • RSS State: MD simulations confirm unbounded dislocation dissociation, where the stacking fault expands to the periodic boundaries, consistent with negative ISFE.
   • LCO State: Simulations show the dislocation core is confined, reaching a stable, finite stacking-fault width after initial equilibration.

4. Thermal Response of Fault Energies:

   • In the RSS state, the unstable stacking-fault energy (USFE) exhibits a significantly stronger thermal decrease (approx. 2.2 times) compared to the ISFE, indicating a decoupling of these energies due to chemical disorder.
   • In the LCO state, this ratio decreases to approx. 1.35, suggesting that local chemical order partially restores a more coupled thermal response between the unstable and intrinsic fault configurations.

Significance and Claims
The paper concludes that the origin of finite stacking-fault widths in CoCrNi at finite temperatures is local chemical order (LCO), rather than the thermal stabilization of the random alloy state. The study demonstrates that previous reliance on harmonic approximations led to an incorrect prediction of thermal stabilization for RSS CoCrNi. By explicitly including anharmonic effects, the authors show that RSS stacking faults remain thermodynamically unstable (negative ISFE) even at elevated temperatures.

The findings establish that LCO is the critical mechanism providing the thermodynamic restoring force required to stabilize finite partial dislocation separations. Furthermore, the work highlights that chemical disorder fundamentally alters the relative thermal responses of USFE and ISFE, a factor that must be considered when modeling the mechanical behavior of complex MPEAs. The results are corroborated by both free-energy calculations and large-scale dislocation simulations, providing a consistent physical picture of stacking-fault stability in these alloys.