The research, carried out by Thi-Nhai Vu of Nha Trang University and Van-Trung Pham and colleagues at the University of Danang – University of Science and Technology in Vietnam, relies on molecular dynamics simulations, a technique that tracks the motion of every individual atom in a material by solving the classical equations of motion under an interatomic potential. Rather than testing bulk samples in a mechanical testing frame, the team simulated nanoindentation, the process of pushing a hard, sharply pointed tip into a material and measuring its resistance. Nanoindentation is the standard method for probing hardness and elastic modulus at very small length scales, and by reproducing it atom by atom, the researchers could watch dislocations nucleate, grain boundaries slide, and stacking faults spread in real time, something no laboratory microscope can capture with such completeness.

The first major finding concerns the celebrated Hall-Petch relationship, one of the oldest rules in metallurgy. For more than half a century, engineers have known that shrinking the grains of a polycrystalline metal makes it harder, because grain boundaries act as obstacles that block the glide of dislocations, the line defects responsible for plastic flow. The simulations confirmed this classic strengthening trend in CoCuFeNiPd alloys with larger grains, where plastic deformation was dominated by dislocation activity nucleated near the indenter tip and propagating through the crystal interior. But when the grain size was reduced into the ultrafine nanocrystalline regime, the trend reversed dramatically. Hardness and the reduced Young’s modulus began to fall rather than rise as grains became smaller, a phenomenon known as inverse Hall-Petch behavior.

The reason for this reversal, revealed directly by the atomic-scale trajectories, is a change in the dominant deformation mechanism. In the inverse Hall-Petch regime, grains have become so small that dislocations can no longer pile up effectively against boundaries; instead, the material deforms by grains sliding past one another along their shared boundaries and by whole grains rotating under the indenter’s pressure. Grain boundary sliding and grain rotation allow the material to accommodate strain without the intense dislocation activity that characterizes conventional plasticity. This mechanistic crossover has been observed in other nanocrystalline metals and ceramics, but the present study pins down how it plays out in a chemically complex, five-element high-entropy system, where the rugged energy landscape created by chemical disorder adds further obstacles to dislocation motion.

The second variable the team explored was the palladium concentration, denoted by the subscript x in CoCuFeNiPdx. Palladium is the largest and heaviest of the five elements in this alloy family, and adding it increases what metallurgists call lattice distortion: the local warping of the crystal lattice caused by atoms of different sizes trying to share the same lattice sites. The simulations showed that increasing Pd content enhances both hardness and indentation resistance. The mechanism is twofold. First, stronger lattice distortion raises the frictional resistance that dislocations experience as they move through the lattice, effectively making the entire crystal a rougher terrain for defect motion. Second, higher palladium levels promote short-range ordering, a subtle chemical phenomenon in which atoms of particular species preferentially occupy neighboring sites, creating local patterns within the nominally random solid solution. These ordered domains act as additional barriers that significantly impede dislocation glide, a strengthening route that has attracted intense interest since recent experimental and computational work demonstrated that short-range order can simultaneously boost the strength and ductility of high-entropy alloys.

However, the study also uncovered a cautionary limit. When the palladium content becomes excessive, the alloy begins to undergo local phase separation, with palladium-rich regions segregating from the rest of the matrix. This chemical inhomogeneity suppresses the material’s ability to accommodate strain, undermining the very ductility and damage tolerance that make high-entropy alloys attractive in the first place. The message for alloy designers is that palladium is a powerful strengthening agent only up to an optimum concentration; beyond that point, the benefits of lattice distortion and short-range ordering are outweighed by the embrittling effects of compositional segregation. This kind of composition-property trade-off is exactly the information needed to guide the rational design of Pd-containing HEAs for real applications.

Temperature emerged as the third decisive factor. As the simulation temperature increased, the alloys exhibited pronounced thermal softening, with both hardness and elastic modulus declining. Elevated temperature promotes structural disorder, thermally shaking atoms out of their ideal lattice positions and weakening the short-range order that had helped block dislocations. At the same time, the increased atomic disorder interferes with the propagation of dislocations in a different way, scattering and absorbing them before they can carry plastic deformation efficiently across grains. The net result is a material that yields more easily and deforms more diffusely at high temperature, an important consideration for any application in which nanoscale contact occurs under thermal load, such as microelectromechanical systems, high-temperature coatings, or cutting tools.

Perhaps the most valuable contribution of the work is its unified atomic-scale picture of how these mechanisms couple. The researchers demonstrate that nanoindentation-induced deformation in CoCuFeNiPdx alloys is never governed by a single process but by the simultaneous evolution of grain boundary activity, dislocation nucleation, stacking fault formation, and amorphous transformation, the latter being the local loss of crystallinity under the extreme contact pressures beneath the indenter tip. Stacking faults, which are planar packing defects in the face-centered cubic lattice, appear as intermediate carriers of plasticity between perfect dislocation slip and full amorphization. Which of these channels dominates depends on grain size, composition, and temperature, and the simulations show how shifts in one variable can redirect the entire deformation pathway. This mechanistic map gives experimentalists concrete signatures to look for, for example in transmission electron microscopy of indented regions, and gives modelers benchmarks against which to validate larger-scale simulations.

The study also situates itself within a rapidly growing literature on high-entropy alloys, a field launched in 2004 when independent groups led by Jien-Wei Yeh and Brian Cantor first reported the surprising stability and properties of multi-principal-element alloys. In the two decades since, HEAs have been explored for structural, cryogenic, irradiation-resistant, and high-temperature applications, and concepts such as the four core effects of high-entropy materials, including severe lattice distortion and sluggish diffusion, have been progressively clarified. Recent work has highlighted chemical short-range order as a tuning knob of exceptional power, and the present results extend that theme to the specific case of palladium-bearing CoCuFeNiPd, an alloy system that earlier simulations had already flagged as one where short-range ordering and grain boundary segregation can inhibit the onset of inverse Hall-Petch softening.

For engineers thinking about applications, the relevance is direct. Nanoscale contact governs the performance and reliability of micro- and nanoelectromechanical devices, thin-film coatings, wear-resistant surfaces, and precision machining processes. A material that maintains high hardness and elastic resilience while tolerating strain at elevated temperature is the ideal candidate for such roles, and the new results indicate how to move toward that target: choose grain sizes on the Hall-Petch side of the crossover, tune palladium content high enough to maximize distortion and ordering but low enough to avoid phase separation, and account for thermal softening in any environment where heat is present. The Vietnamese team’s atomistic tour of CoCuFeNiPd thus delivers both fundamental insight into how chemically complex crystals deform under contact and practical guidance for the next generation of high-entropy alloys designed for nanoscale service. As simulation tools and interatomic potentials continue to improve, studies of this kind are expected to play an ever larger role in screening alloy compositions before a single ingot is melted, accelerating the path from atomic-scale understanding to real-world materials.

Subject of Research: Atomic-scale simulation of grain size, palladium composition, and temperature effects on the nanoindentation behavior of CoCuFeNiPdx high-entropy alloys

Article Title: Atomic-scale investigation of grain-size, Pd composition, and temperature effects on the nanoindentation behavior of CoCuFeNiPdx high-entropy alloys

Article References: Vu, T.-N., Pham, V.-T., Tran, N.-H., Luu, D.-B., Tao, Q.-B., & Nguyen, T.-T. (2026). Atomic-scale investigation of grain-size, Pd composition, and temperature effects on the nanoindentation behavior of CoCuFeNiPdx high-entropy alloys. Journal of Materials Science. https://doi.org/10.1007/s10853-026-13891-6

Image Credits: AI Generated

DOI: 10.1007/s10853-026-13891-6

Keywords: high-entropy alloys, nanoindentation, molecular dynamics, Hall-Petch, grain boundary sliding, palladium, short-range ordering, lattice distortion, dislocations, thermal softening, hardness, CoCuFeNiPd

The alloy in question, Mg-8wt.%Li, sits in a sweet spot of composition. Above roughly 8 percent lithium by weight, the material becomes dual-phase, containing both the hexagonal close-packed alpha phase typical of magnesium and the body-centered cubic beta phase that lithium stabilizes. That b.c.c. beta phase is the secret ingredient: it deforms far more readily than the hexagonal phase, endowing the alloy with ductility that no other magnesium alloy system can match. The Mg–Li family holds the record for elongation among all magnesium alloys, although its absolute strength remains modest, which is precisely why understanding and controlling its deformation mechanisms matters so much for lightweight structural applications.

To push the alloy into its most deformable state, the team turned to high-pressure torsion, a severe plastic deformation technique in which a disk of material is squeezed under a nominal pressure of 4 gigapascals while being twisted through five full turns. The process shears the microstructure into an ultrafine grain structure. Electron microscopy and transmission Kikuchi diffraction revealed that the extruded material, which started with grains of about 2.2 micrometers, was refined to an average spatial grain size of roughly 0.4 micrometers, with individual subgrains resolved in the range of 0.15 to 0.40 micrometers. The two phases became elongated and aligned with the torsion direction, a signature of the intense shear the disk experienced.

Miniature tensile specimens, cut from the disk mid-radius by electric discharge machining with a gauge cross-section of just 0.8 by 0.8 millimeters, were then pulled at room temperature across strain rates spanning from 10^-5 to 10^-3 per second. The results were striking. The peak stress varied dramatically with strain rate, a hallmark of high strain rate sensitivity, and the alloy exhibited large elongations at the slowest rates of 10^-4 and 10^-5 per second. This unusual increase in ductility at low strain rates, previously reported for similar alloys, is exactly the behavior that makes dual-phase Mg–Li alloys candidates for room-temperature superplasticity, a property normally reserved for ceramics and exotic superalloys at soaring temperatures.

Creep tests added another layer of intrigue. Under constant loads between 10 and 100 megapascals at room temperature, and at 373 kelvin under 10 and 25 megapascals, the creep rates steadily decreased as strain accumulated. Such decelerating creep is normally seen in coarse-grained metals at high temperature, where dislocation substructures build up during the initial stage of deformation, but that explanation should not apply to an ultrafine-grained material whose grains are already smaller than any stable subgrain size. The authors point to a recent study on ultrafine-grained aluminum that attributed similar behavior to strengthening by dynamic recovery, and they adopted the strain rate at 2 percent strain as representative of the as-processed state for further analysis.

The most consequential number to emerge from the creep experiments was the activation energy. Using the temperature dependence of the creep rate between room temperature and 373 kelvin, the team estimated activation energies of 64 kilojoules per mole at 25 megapascals and 77 kilojoules per mole at 10 megapascals. Both values fall well below the 135 kilojoules per mole for self-diffusion and the 92 kilojoules per mole for grain boundary diffusion in pure magnesium. The authors attribute this reduction to the large fraction of the b.c.c. beta phase, which has a lower melting temperature and correspondingly lower diffusion barriers. Reported grain boundary diffusion energies for related alloys, 65 kilojoules per mole for Mg-9%Li and 61 kilojoules per mole for Mg-8%Li-1%Zn, bracket the measurements, and the team settled on 75 kilojoules per mole for their modeling. The low activation energy is a clear signal that thermally activated mechanisms are at work even at room temperature.

To interpret these observations, the researchers deployed a two-pronged computational strategy. First, they applied a recently updated model for grain boundary sliding, the mechanism in which grains slide past one another along their boundaries, which is the canonical explanation for superplastic flow. The model treats the total flow stress as the sum of a threshold stress, capturing other strengthening contributions such as solid solutions, and a grain boundary sliding contribution that depends on shear modulus, temperature, grain size, Burgers vector and the grain boundary diffusion coefficient. A rectified linear function is used as a mathematical constraint to prevent the model from predicting negative stresses when thermal activation is strong. Second, they trained machine learning algorithms on 67 carefully curated data points from the literature, restricted to dual-phase Mg–Li alloys containing no alloying elements other than lithium, with lithium contents between 8 and 12 weight percent, to keep composition effects from contaminating the analysis.

The machine learning lineup included two black-box models, support vector regression and random forest, and two white-box symbolic regression models, one unconstrained and one constrained by dimensional consistency. The black-box models, evaluated with Shapley Additive Explanations, agreed that temperature is the dominant control on flow stress, that strain rate plays an intermediate role, and that grain size matters only marginally. On the training data the algorithms performed well, with mean absolute percentage errors ranging from 11.5 percent for the random forest to 35.3 percent for support vector regression. But when the models were asked to predict the new experimental data generated in this study, the black-box approaches collapsed, producing errors exceeding 250 percent, while the grain boundary sliding model and the dimension-constrained symbolic regression held their ground at roughly 35 percent error.

The deeper story emerged when the team plotted predicted flow stress against grain size, strain rate and temperature and compared the trends with experiment. The experiments revealed a fascinating crossover: for grains larger than about half a micrometer, refining the grains strengthens the alloy in classic Hall–Petch fashion, but below that threshold, further refinement actually softens it, the inverse Hall–Petch regime. Only the grain boundary sliding model captured this transition cleanly. The black-box models failed to reproduce the steep strain rate dependence of the flow stress, with support vector regression even predicting a physically implausible negative strain rate sensitivity at high rates. The dimension-constrained symbolic regression, guided by physically meaningful features such as the Zener–Holomon parameter and dimensionless groups derived from the grain boundary sliding equations, performed far better than its unconstrained counterpart, demonstrating that embedding physical knowledge into machine learning pays real dividends.

The take-home message resonates well beyond magnesium. Machine learning models are only as trustworthy as the data behind them, and with a sparse training set of 67 points, the black-box algorithms interpolated plausibly but extrapolated badly, especially outside the temperature and strain rate ranges they had seen. The century-old physics-based model, by contrast, reproduced every experimental trend once its activation energy was adapted to the dual-phase alloy. For engineers dreaming of lightweight magnesium–lithium components that can be superplastically formed at room temperature, the study offers both a validated design tool and a cautionary tale: in materials science, the white box and the physics still matter, and the smartest models are those that know the laws they are trying to learn.

Subject of Research: Deformation mechanisms and grain boundary sliding in ultrafine-grained dual-phase Mg–Li alloys processed by high-pressure torsion

Article Title: Analysis of deformation mechanism of dual-phase Mg–Li alloy, processed by high-pressure torsion, using experiments and machine learning

Article References: Carvalho, A. P., Edalati, K., & Figueiredo, R. B. (2026). Analysis of deformation mechanism of dual-phase Mg–Li alloy, processed by high-pressure torsion, using experiments and machine learning. Journal of Materials Science. https://doi.org/10.1007/s10853-026-13355-x

Image Credits: AI Generated

DOI: 10.1007/s10853-026-13355-x

Keywords: magnesium-lithium alloy, high-pressure torsion, grain boundary sliding, severe plastic deformation, machine learning, superplasticity, creep, activation energy, Hall-Petch, symbolic regression, ultrafine grains, flow stress