Chemical Short-Range Ordering in Alloys

• Chemical Short-Range Ordering (CSRO) is the local deviation from randomness in atomic arrangements, quantified by metrics like the Warren–Cowley parameter.
• Modern approaches—combining simulations (Monte Carlo, DFT) and experimental techniques (APT, TEM)—enable precise quantification and control of CSRO in complex alloys.
• CSRO markedly influences thermodynamic stability, defect dynamics, mechanical strength, corrosion resistance, and radiation tolerance, guiding advanced alloy design.

Chemical short-range ordering (CSRO) describes deviations from a perfectly random atomic arrangement in alloys and intermetallics, specifically at the scale of one to a few coordination shells. Even in nominally disordered (random solid solution) or long-range ordered (LRO) systems, CSRO can profoundly influence thermodynamic, mechanical, electronic, and functional properties. Understanding, quantifying, and controlling CSRO is increasingly central to alloy design, especially for materials such as high- and medium-entropy alloys, complex concentrated alloys, and advanced intermetallics.

1. Definition and Theoretical Formulation

CSRO refers to a local preference in atomic species arrangements within one or more coordination shells, causing motifs that are over- or underrepresented relative to a random solid solution. The canonical measure is the Warren–Cowley short-range order (SRO) parameter, generally defined for a pair of species $(i,j)$ in a shell %%%%1%%%% as: $\alpha_{ij}^{(n)} = 1 - \frac{P_{ij}^{(n)}}{c_j}$ where $P_{ij}^{(n)}$ is the probability of finding a $j$-type atom at the $n$th shell from an $i$ atom, and $c_j$ the concentration of $j$. Negative $\alpha$ indicates ordering (preference for dissimilar neighbors), positive $\alpha$ indicates clustering.

For ordered systems (e.g., B2 FeAl), the classical Warren–Cowley $\alpha$ does not decay to zero at long distances, as it captures both LRO and SRO. To disambiguate genuine short-range correlations, the parameter can be decomposed (Stana et al., 2016): $\Delta \alpha_{lmn} = l_{lmn} - \bar{\alpha}_{lmn}$ where $l_{lmn}$ is the measured short-range order parameter, and $\bar{\alpha}_{lmn}$ encodes the long-range order background depending on sublattice occupations and the long-range-order parameter $\eta$. This enables rigorous extraction of local ordering in LRO backgrounds.

With increasing system complexity (multi-component alloys), the matrix of pairwise $\alpha_{ij}$ becomes prohibitively large and suffers from non-monotonicity under compositional constraints (canonical ensemble). The $\Delta$-parameter, as developed in the OPERA framework (Anand et al., 2022), serves as a scalar analog, quantifying bond-type deviations from randomness: $$\Delta = \frac{1}{K_1}\sum_{i\ne j}^{K_1}\left(1-\frac{m_{ij}}{2n}\right) + \frac{1}{K_2}\sum_{i=j}^{K_2}\left(\frac{m_{ij}}{n}-1\right)$$ with $m_{ij}$ the number of $i$–$j$ bonds and $n$ the normalization for bond counts; $K_1$, $K_2$ are the numbers of unlike and like bonds, respectively.

Recently, information-theoretic and high-dimensional motif-based metrics—such as the Kullback–Leibler divergence between motif distributions and machine-learning-based motif embeddings via E(3)-equivariant graph neural networks—have been used to more completely quantify CSRO across arbitrarily complex chemical and structural spaces (Sheriff et al., 14 May 2024, Sheriff et al., 2023).

2. Quantification, Simulation, and Experimental Characterization

Both simulation and experiment drive contemporary understanding of CSRO:

Method	Key Features	Ref.
Monte Carlo/molecular dynamics + machine learning potentials	Enables large-scale, chemically-specific sampling of realistic atomic configurations; calculates CSRO metrics from equilibrated structures.	(Zheng et al., 2022, Nitol et al., 14 Sep 2025, Liu et al., 16 Jul 2025)
DFT & DFT linear-response	Direct calculation of bond energies, formation energies, SRO parameters, and vibrational properties.	(Morris et al., 2022)
Cluster Expansion, CVM/CALPHAD (FYL)	Thermodynamic modeling that rigorously includes many-body SRO effects; FYL-CVM methods project cluster variables to site variables, scaling to multicomponent systems.	(Fu et al., 2023, Fu, 9 Aug 2025)
Atom probe tomography (APT) + machine learning	Direct 3D imaging and classification of CSRO domains and motif populations.	(Li et al., 2023)
Energy-filtered/high-resolution TEM, neutron scattering	Diffuse scattering analysis and atomic pair mapping, enabling direct real-space and reciprocal-space visualization of SRO domains.	(Zhang et al., 2019, Hsiao et al., 2022, Blades et al., 29 Feb 2024, Anber et al., 11 Jun 2025)

In all approaches, motif populations or $\alpha_{ij}$ values are extracted either via direct pair counting, statistical averages, or ML-driven motif recognition.

3. Impact on Thermodynamic, Mechanical, and Functional Properties

Thermodynamics and Phase Stability

CSRO can substantially lower the alloy's free energy by increasing preferred (often "unlike") bonds, as seen in Cu$_3$Au, where the formation energy improves from $-0.0343$ to $-0.0682$ eV/atom upon introducing thermodynamically favorable SRO (Morris et al., 2022). Controlled SRO stabilizes phases, tunes vibrational entropy (notably, vibrational entropy changes from $9k_B$ at $300$ K to $6k_B$ at $100$ K in Cu$_3$Au), and shifts order-disorder transition temperatures.

Defect Dynamics and Mechanical Response

CSRO alters defect energetics, including stacking fault energies, dislocation motion, and hardening mechanisms:

• In CrCoNi, increased SRO (via aging) nearly triples the stacking fault energy from $\approx 8.2$ to $\approx 23.3$ mJ/m$^2$ (Zhang et al., 2019), favoring planar slip and paired dislocation configurations, thus enhancing hardness.
• In refractory HEAs (e.g., MoNbTi vs. TaNbTi), increased CSRO raises the average unstable stacking fault energy (USFE), increases the required stress for dislocation glide, and reduces USFE dispersion—yielding both stronger materials and altered hardening rates (Zheng et al., 2022).
• ML potentials trained to DFT accuracy demonstrate that local chemical environments (e.g., Co-rich vs. Cr- or Fe-rich stacking fault planes in CoCrNi/CoCrFeNi) modulate stacking fault energies by $\approx 20$ mJ/m$^2$ (Nitol et al., 14 Sep 2025).

Corrosion/Passivation and Functional Properties

Passivation, especially in Cr-bearing alloys, is strongly influenced by CSRO's topology:

• Increased Cr–Cr clustering (with more positive $\alpha_{CrCr}$) lowers the percolation threshold for passive network formation, enabling robust passive film formation at reduced bulk Cr content—pivotal for developing low-Cr, corrosion-resistant "stainless" MPEAs (Roy et al., 25 Jan 2024, Blades et al., 29 Feb 2024, Anber et al., 11 Jun 2025).
• Direct links between enhanced CSRO (post-aging) and rapid, thick, Cr(III)-enriched passive film formation are established by EXAFS and electrochemical studies (Anber et al., 11 Jun 2025).

Radiation Tolerance

In MoNbTaVW HEAs, CSRO increases defect recombination efficiency by enhancing interstitial diffusion and suppressing vacancy mobility, thereby reducing long-lived radiation defects. However, CSRO is rapidly degraded under irradiation, diminishing its long-term benefits (Liu et al., 16 Jul 2025).

4. Processing Pathways and Nonequilibrium CSRO

Conventional alloy processing routes—such as heat treatment, aging, rapid solidification, or mechanical working—can tune CSRO into nonequilibrium steady states distinct from equilibrium SRO (Islam et al., 23 Sep 2024). Dislocation motion, interface-driven transformations, and rapid cooling can create or destroy ordering biases, with steady-state SRO depending on the interplay of thermally activated (Metropolis-type) and athermal (forced, randomizing) events. The evolution follows a rate law

$\frac{dD_{sro}}{dt} = \lambda D_{sro} + \Gamma$

where $D_{sro}$ quantifies SRO (via, e.g., Jensen–Shannon divergence), $\lambda < 0$ is the destruction rate, and $\Gamma > 0$ the creation or bias term. Manipulation of process variables (temperature, strain rate, undercooling) enables "dialing in" of SRO states unattainable by equilibrium annealing alone, expanding the materials design space.

5. Beyond Pairwise Metrics: Motif-Based and Information-Theoretic Formalisms

Pairwise SRO metrics (including Warren–Cowley $\alpha_{ij}$) cannot distinguish between distinct multi-element motifs of the same first-neighbor composition. Recent advances now use high-dimensional representations:

• E(3)-equivariant graph neural networks (GNNs) map each first-coordination polyhedron (motif) to a unique, symmetry-resolved fingerprint vector. The motif distribution $P(\mathcal{M},T)$ at temperature $T$ is compared to the ideal random distribution via the Kullback–Leibler divergence: $$D_{KL}[P(\mathcal{M},T)\|P(\mathcal{M},\infty)] = \sum_{i} P(\mathcal{M}_i,T) \log_2\left[\frac{P(\mathcal{M}_i,T)}{P(\mathcal{M}_i,\infty)}\right]$$ enabling quantification of SRO intensity and motif-specific physical property correlations (Sheriff et al., 14 May 2024, Sheriff et al., 2023).
• These approaches allow motif-specific quantities (e.g., local lattice strain, energy, or magnetic moment) to be mapped directly onto the motif population, elucidating structure–property trends that are otherwise hidden.

6. Computational Thermodynamics and CALPHAD Extensions

The integration of CSRO into thermodynamic models has been achieved via cluster-based frameworks such as the FYL-CVM approach, which compresses the cluster variable space by the Fowler–Yang–Li transform, making SRO modeling tractable for multicomponent alloys (Fu et al., 2023, Fu, 9 Aug 2025). In these models, the total free energy includes configurational cluster entropy, vibrational, elastic, and electronic contributions, and enables phase diagram and order–disorder boundary predictions with SRO as a fundamental variable.

This approach allows the construction of "SRO diagrams"—complementing phase diagrams—encoding not only which phases are stable, but also the temperature–composition dependence of local ordering.

7. Practical Implications and Applications

CSRO now functions as a design parameter in advanced alloy development, impacting:

• Optimization of mechanical properties (yield strength, ductility, hardening) in high- and medium-entropy alloys.
• Tailoring corrosion/passivation thresholds for low-cost, low-chromium "stainless" alloys via Cr–Cr clustering, informed by both experimental neutron scattering data and predictive percolation models (Roy et al., 25 Jan 2024, Blades et al., 29 Feb 2024, Anber et al., 11 Jun 2025).
• Enhancing radiation tolerance in refractory HEAs, where CSRO-imposed differences in defect diffusion improve resistance to swelling and embrittlement but call for strategies to stabilize SRO during service (Liu et al., 16 Jul 2025).
• Rationalization and discovery of functional properties (electrical resistivity, magnetic response) that hinge on CSRO-driven electronic structure effects (Li et al., 2023, Morris et al., 2022).

In summary, chemical short-range ordering embodies a fundamental bridge between atomic-scale interactions and macroscopic materials behavior. Rigorous quantification—by pairwise, bond-based, and motif-based methodologies—combined with advanced simulation and experimental imaging, enables predictive modeling and design of materials with tailored thermodynamic, mechanical, and functional properties by manipulating atomic-scale order as a discrete design degree of freedom.