High-Entropy Alloys: Thermodynamics & Nucleation

• High-entropy alloys (HEAs) are metallic systems with five or more principal elements in near-equiatomic ratios, generating high configurational entropy that stabilizes disordered phases.
• Advanced characterization methods like atomic electron tomography reveal atom-by-atom nucleation, mapping local chemical and structural order to challenge classical nucleation theory.
• Practical insights include tuning element selection and synthesis conditions to control twin boundary formation and optimize the strength–ductility balance in structural applications.

High-entropy alloys (HEAs) are a class of metallic materials characterized by multi-principal element compositions—typically five or more elements at near-equiatomic fractions—that place them in a unique thermodynamic regime where configurational entropy balances or exceeds the enthalpy of intermetallic formation. The resulting alloys display exceptional property combinations, including high strength-ductility trade-offs and tunable functional properties, and have created a new paradigm in alloy design for structural and catalytic applications.

1. Thermodynamic Foundations and Compositional Space

The defining feature of HEAs is their large configurational entropy of mixing:

$$\Delta S_{\rm mix} = -R \sum_{i=1}^{n} c_i \ln c_i,$$

where $R$ is the gas constant and $c_i$ the atomic fraction of element $i$ (typically $c_i = 1/n$ for equimolar $n$-component alloys). The mixing enthalpy $\Delta H_{\rm mix}$ sums pairwise interaction enthalpies, while the Gibbs free energy of mixing for the (liquid or disordered solid) phase is:

$$\Delta G_{\rm mix} = \Delta H_{\rm mix} - T\Delta S_{\rm mix}.$$

At high temperatures or for large $n$, the $T\Delta S_{\rm mix}$ term stabilizes disordered single-phase solid solutions (fcc or bcc) against ordered intermetallic compounds, vastly expanding the compositional field for alloy design. This thermodynamic stabilization is central to the distinctive properties of HEAs.

2. Structural and Chemical Order: Local Metrics and Nucleation Pathways

Atomic-scale visualization of HEA formation has recently been realized via atomic electron tomography (AET), which provides three-dimensional atom-by-atom reconstructions of nuclei across different stages of growth. In studied Co–Ni–Ru–Rh–Pd–Ag–Ir–Pt HEAs produced by carbothermal shock (rapid heating to ~2,000 K, quenching at ~$10^{5}$ K/s), thousands of nuclei were resolved and analyzed with ~20 pm precision.

Key local order parameters include:

• Bond-orientational order (BOO), $BOO(r)$: ranges from $0$ (fully disordered) to $1$ (perfect fcc), maps local structural order at radius $r$ from the nucleus center.
• Normalized structural order, $\eta(r)$:

$$\eta(r) = 1 - \frac{\sigma_{\rm local}}{\sigma_{\rm bulk}},$$

with $\sigma_{\rm local}$ the variance of local bond angles in a shell at $r$ and $\sigma_{\rm bulk}$ that of bulk fcc.

• Chemical short-range order (CSRO), $\alpha(r)$:

$\alpha(r) = \sum_{i,j} [P_{ij}(r) - c_i c_j],$

where $P_{ij}(r)$ is the probability of finding a $j$-type neighbor around an $i$-type atom at $r$ and $c_i$, $c_j$ the global concentrations.

Experimentally, the $BOO(r)$ is highest at nuclei centers and decays smoothly to boundaries, closely tracking $\alpha(r)$. Type 3 elements (Ir/Pt) locally enhance both chemical and structural order, whereas Co/Ni with Ru/Rh/Pd/Ag show suppressed order.

3. Beyond Classical Nucleation: Gradient Nucleation Pathways Model

Observation of continuous order gradients challenged the adequacy of classical nucleation theory (CNT), which assumes a sharp interface and a single activation barrier. The gradient nucleation pathways (GNP) model generalizes CNT by allowing for spatially varying order parameters $a(r)$:

$$\Delta G = -\int_V \Delta g\, a(r)\, dV + \int_V \gamma\, |\nabla a(r)|\, dV,$$

where $\Delta g$ is the volume free energy difference per unit volume, and $\gamma$ is the interfacial tension. For a step function $a(r) = H(R-r)$, this formula recovers CNT:

$$\Delta G_{\rm CNT}(R) = -\frac{4}{3}\pi R^3 \Delta g + 4\pi R^2 \gamma.$$

However, for smooth $a(r)$ (as fit to experimental $BOO(r)$ profiles), nucleation progresses through a series of intermediate states, each with its own incremental barrier $\Delta G_i$, leading to a “ladder” of energy maxima:

$\Delta G^*_{(n)} = \sum_{i=1}^n \Delta G_i.$

Fitting atomistic $BOO(r)$ profiles into this framework reveals spatially evolving interfacial terms $\gamma_{\rm eff}(r) = \gamma | \nabla a |$ and volume driving terms $\Delta g_{\rm eff}(r) = \Delta g a(r)$, resulting in continuous, rising energy barriers—not a singular critical nucleus as in CNT. This naturally explains gradual ordering from core to boundary and failures of classical or two-step models for HEA nucleation.

4. Microstructural Evolution: Coalescence and Twin Formation

Late-stage nucleation involves coalescence of individual fcc nuclei. Template matching of lattice vectors quantifies nucleus-nucleus misorientation ($\theta$) and classifies boundaries using coincidence-site-lattice ($\Sigma$) values:

• Most pairs yield $\Sigma1$ boundaries ($\theta\approx0^\circ$), indicating perfect registry.
• A fraction form coherent $\Sigma3$ twin boundaries ($\theta\approx60^\circ$ about [111]), and higher-$\Sigma$ boundaries (e.g., $\Sigma9$, $\Sigma11$, etc.) result from further twin intersections.

The prevalence of $\Sigma3$ twin boundaries is significant: such twins impede dislocation motion, contributing to work-hardening and delaying crack propagation. This mechanism underlies the remarkable strength–ductility synergy commonly observed in HEA metallurgy.

5. Implications for Rational Design and Processing of HEAs

Experimental mapping of $\eta(r)$, $\alpha(r)$, and full nucleation pathways provides actionable insight into HEA design and processing:

• Element selection controls $ΔH_{\rm mix}$ and $ΔS_{\rm mix}$, which modulate local CSRO and ability to seed structural order (e.g., enriching type 3 elements to accelerate nucleation).
• Alloying to tune $\gamma$ or $a(r)$ enables control of the nucleation pathway, affecting nucleus size distributions and the grain boundary character network.
• Engineering twin-rich microstructures leverages twin boundary formation to maximize mechanical resilience, as twins act as efficient barriers to crack/void propagation.

Modifying synthesis conditions (e.g., thermal shock, rapid quench rates) can kinetically arrest intermediate states, trapping microstructural features beneficial for targeted property optimization.

6. Analytical and Experimental Methodology Overview

The implementation of these concepts required quantitative AET measurement methodology:

• Synthesis by carbothermal shock capturing nuclei at growth stages.
• Aberration-corrected annular dark-field scanning transmission electron microscopy (ADF-STEM) with $\pm70^\circ$ tilt acquisition, followed by image reconstruction via the real-space RESIRE algorithm.
• EDX-derived classification of atomic species into three types for chemical correlation.
• Quantification of structural and chemical metrics at atomic resolution, enabling model development and validation.

The resulting data set includes $7,662$ HEA and $498$ medium-entropy alloy nuclei, over $25$ nanoparticle datasets, with atomic coordinate precision of $\sim 20$ pm and $BOO(r)$/CSRO quantification at $\sim$0.1 nm radial resolution.

7. Broader Impacts and Future Directions

The direct atomic-scale mapping of crystal nucleation and growth in HEAs provides a generalizable framework that can be applied to other multicomponent or strongly disordered systems where classical nucleation theory fails. The generality of the GNP model supports its extension to fields such as multicomponent oxides or complex intermetallics. The methodology of correlating three-dimensional local chemical and structural order to nucleation energetics suggests a path for tuning physical properties through directed microstructure engineering at the earliest stages of alloy formation, paving the way for bespoke structural and functional property design in next-generation advanced materials.