High entropy alloys for advanced electrocatalysis with computational insights and multidisciplinary design strategies

Summary

The growing demand for sustainable energy solutions drives the exploration of high-entropy alloys (HEAs) in electrocatalysis. HEAs have emerged as paradigm-shifting electrocatalysts for complex reactions, yet their mechanistic underpinnings remain underexplored. This study establishes a comprehensive computational framework integrating density functional theory (DFT), machine learning (ML), and multiscale simulations to decode the catalytic mechanisms of HEAs and guide their rational design. These insights break the “scaling relationship” bottleneck in conventional catalysts. Future efforts should bridge accuracy-efficiency trade-offs via multiscale modeling and address dynamic interface phenomena to unlock HEA potential in sustainable energy conversion and all kinds of catalytic reaction.

Graphical abstract

Catalysis; Electrochemical energy conversion; Alloys; Computational materials science

Introduction

The growing global energy demand and environmental concerns have driven research into efficient and eco-friendly energy conversion and storage technologies involving electrochemical processes.^1,2 High-performance catalysts are crucial for reducing energy barriers and accelerating reaction kinetics in these processes.³ The rapid socioeconomic development of recent decades has intensified global reliance on fossil fuels, exacerbating environmental degradation and accelerating climate change.^4,5 In response, the scientific community has prioritized the development of green technologies aimed at establishing sustainable and regenerative energy systems.^6,7 Among these efforts, electrochemical energy conversion technologies—including overall water splitting (OWS), CO₂ reduction reaction (CO₂RR), nitrogen reduction reaction (NRR) and advanced battery systems—have emerged as critical solutions for harnessing and storing renewable energy.^{8,9,10,11,12,13,14,15,16,17} While noble metal-based catalysts (e.g., Pt, Ir, Ru) currently dominate these applications, their widespread adoption faces significant barriers: prohibitive costs, limited natural abundance, and susceptibility to degradation under operational stresses (e.g., agglomeration, dissolution, and poisoning).^18,19,20,21 For instance, platinum-based catalysts, though efficient for HER, exhibit poor stability in acidic environments, while iridium oxides for OER suffer from scarcity-driven supply chain vulnerabilities.¹⁹ These limitations underscore an urgent need to reimagine catalyst design paradigms.

The adsorption/desorption behavior of intermediates, closely related to the catalyst’s electronic and geometric structures, is key to determining electrocatalytic performance.²² Alloying has emerged as a promising strategy to adjust the electronic structures of surface or near-surface atoms via strain/ligand effects, creating diverse bimetallic or trimetallic electrocatalysts.²³ Nanotechnology advances have further boosted the activities of these catalysts through geometric and electronic structure engineering, including doping²⁴ and interface engineering.²⁵ Yet, the compositional range of alloyed atoms in metal catalysts is restricted by differences in crystal structure, atomic size, electronegativity, and electron concentration between alloyed atoms.²⁶ This limits electronic structure modulation and intermediate adsorption energy optimization, hindering maximal catalytic activity.²⁷ To break these scaling relationships in electrocatalysis, continuous adjustment of the surface electronic structure of metal catalysts is essential.²⁸ High entropy alloys (HEAs) and other multicomponent materials offer new possibilities for catalyst design, potentially overcoming traditional performance limitations and advancing energy conversion technologies.²⁹ Although HEAs have demonstrated excellent catalytic potential, their practical application still faces multiple challenges. In terms of synthesis, precisely controlling the uniform mixing of five or more elements and the formation of nanostructures is extremely challenging. Existing methods (such as high-temperature smelting and laser induction) often have harsh conditions, high energy consumption, and are difficult to ensure the consistency and stability of the products. In terms of component cost, high-performance systems often rely on scarce and expensive precious metals (such as Pt, Ir, Pd) or environmentally controversial rare earth elements, which limits their large-scale economic feasibility. Environmental impact is another key bottleneck. The high-energy-consuming synthesis process leads to a significant carbon footprint. Some wet chemical methods use toxic organic reagents and produce waste liquids containing heavy metals. Moreover, the efficient and environmentally friendly recovery technology for polymetals in waste catalysts is still not mature. The complexities brought about by the “high entropy” nature-including component control, cost constraints and environmental burdens-constitute the core obstacles for the wide application of high-entropy alloy catalysts. Figure 1 provides an in-depth analysis of the structural characteristics of HEAs. Figure 1A illustrates the schematic architecture of HEAs, exposing their complex construction of five or more near-equiatomic elements. These elements blend randomly to form stable single-phase solid solutions, including face-centered cubic (FCC), body-centered cubic (BCC), hexagonal close-packed (HCP), and amorphous structures. Figure 1B delineates the trend of configurational entropy (ΔS_mix) as a function of the number of equimolar elements. It shows that at least five elements with equivalent molar ratios are needed to meet the high-entropy condition (1.61R), and ΔSmix escalates with the increasing variety of elements. Figure 1C schematically represents the properties of HEAs, underscoring their distinctive microstructures influenced by high-entropy, lattice distortion, sluggish diffusion, and cocktail effects. Figure 1D highlights the principal ingredients for synthesizing alloys of the five most typical HEA families through elemental groupings. Collectively, these analyses emphasize the structural traits of HEAs, underpinning their potential applications in electrocatalysis and beyond. Figure 1E presents the development timeline of HEAs in energy electrocatalysis. Since their introduction in 2004, HEAs have evolved from fundamental research to practical applications. Key milestones include constructing immiscible nanoparticle libraries and synthesizing single-phase HEA nanoparticles. Recently, HEAs have shown trends of size reduction and multidimensional structuring, which have greatly enhanced their electrocatalytic performance. In the field of heterogeneous catalysis, HEAs have become promising materials thanks to their varied activity sites and enhanced stability from entropy.^32,33,34,35 Their complex surfaces, with multiple randomly distributed main elements, provide a wide range of adjustable adsorption energies for intermediates, unlike uniform surfaces that offer limited adsorption types.³⁶ This creates opportunities to design active sites with optimal bonding strengths for key intermediates, following the Sabatier principle.^37,38,39 The shift toward flexible design parameters in multicomponent materials enables unprecedented control over compositional gradients and functional properties.⁴⁰ By relaxing traditional constraints such as strict stoichiometry or equilibrium phase requirements, researchers can engineer materials with tunable chemical landscapes—ranging from gradual elemental variations to localized chemical ordering.⁴¹ This compositional flexibility introduces a spectrum of microstructural phenomena absent in conventional alloys, including chemically decorated lattice defects (e.g., solute-rich dislocations, interfacial segregation) and non-equilibrium phase transitions.⁴² This ability to fine-tune the microstructure makes HEAs have broad application prospects in many fields. HEAs leverage the concerted high-entropy and severe lattice-distortion effects imparted by multiple principal elements to sustain structural integrity against sintering and phase segregation under harsh electrocatalytic environments, thereby exhibiting markedly superior thermodynamic stability compared to single-atom and two-dimensional catalysts.^43,44 Simultaneously, the inherently heterogeneous surface of HEAs generates self-cleaning active sites that are intrinsically resistant to poisoning, carbon deposition, and metal agglomeration, substantially extending their operational lifetime.^45,46 However, the rational design of HEAs for better catalytic performance is hindered by the limited understanding of the relationship between their electronic properties and reaction intermediates.

Figure 1.

Composition-property correlation and evolutionary milestones in high-entropy alloys

(A) Schematic illustration of the HEA architecture.

(B) Relationship between ΔS_mix and the number of equimolar elements.³⁰ Copyright 2022 Wiley-VCH.

(C) Schematic depiction of HEAs' properties.

(D) Element groupings showcasing the key components of the five major HEA families.³¹ Copyright 2023 Royal Society of Chemistry.

(E) Chronological evolution of high-entropy alloy milestones in electrocatalytic energy conversion systems.²⁸ Copyright 2024 American Association for the Advancement of Science.

The integration of experimental techniques, theoretical modeling, and computational simulations has greatly advanced our understanding of material behavior and design principles. This interdisciplinary approach enables researchers to explore the complex relationships between a material’s atomic structure and macroscopic properties, ultimately guiding the development of high-performance materials. MD simulations⁴⁷ provide valuable insights into atomic-scale material behavior by modeling atomic and molecular motions over time, being particularly effective for systems with complex elemental compositions (typically involving 100–10,000 atoms). However, MD faces challenges in multi-component materials due to the lack of comprehensive interatomic potentials and high computational intensity, making it difficult to accurately capture microscale electronic features critical for understanding material properties. DFT is a powerful tool for exploring material electronic structures. By focusing on electron density rather than multi-body wavefunctions, it significantly reduces computational complexity while maintaining high accuracy in predicting material properties.⁴⁸ Additionally, the introduction of machine learning (ML) technology has brought new opportunities for researching energy materials and electrocatalysts.⁴⁹ ML algorithms can quickly process massive data and uncover hidden patterns, thereby guiding the design and selection of new materials. In this process, descriptors such as Coulomb matrix,⁵⁰ bond bag (BOB),⁵¹ adsorption free energy d-band center,⁵¹ and smooth overlap of atomic positions (SOAP)⁵² play key roles, encoding material geometry, structure, and electronic information in unique ways. By combining high-throughput DFT calculations with ML frameworks, researchers can efficiently explore material structure-activity relationships and accelerate the discovery of electrocatalysts.⁵³ As computational power continues to improve and ML algorithms are optimized, multi-scale modeling and simulation of material behavior will become more accurate, significantly enhancing our ability to design materials with customized properties for specific applications. This progress highlights the importance of interdisciplinary collaboration and the strategic value of integrating experimental validation with computational insights. By adopting these integrated research methods, researchers can more efficiently navigate the complex field of materials science and accelerate the translation of theoretical knowledge into practical technological innovation.

Simulations and modeling of HEAs

Ab initio simulations based on density functional theory (DFT)

The ab initio calculation method based on the first principles of quantum mechanics, with electron motion as the core of the research, obtains the electron wave function and its energy eigenvalues by solving the multi-body Schrödinger equation^54,55; with the help of the adiabatic approximation and single-electron approximation,⁵⁶ the bonding characteristics, total energy and elastic properties of the system can be determined, and by taking electron density as the key variable, the multi-electron problem can be mapped to self-consistent solvable single-particle equations, only requiring three basic constants: Planck’s constant, electron mass and nuclear charge, can predict the material properties under zero empirical parameters,⁵⁶ thus density functional theory (DFT) has become the mainstream computational tool in the theoretical condensed matter physics field. In recent years, DFT has been widely applied in the research of high-entropy alloys (HEAs), covering component space screening,⁵⁷ electronic structure analysis,⁵⁸ phase stability assessment,⁵⁸ thermodynamic, and mechanical property prediction, but chemical and magnetic disorder pose great challenges to traditional DFT,⁵⁹ for this reason, researchers have developed a series of approximation schemes such as virtual crystal approximation (VCA),⁶⁰ generalized gradient approximation (GGA),^61,62 coherent potential approximation (CPA) [125–130,132,133], precise muffin-tin - tin orbitals (EMTO),⁶² special quasi-random structure (SQS),⁶³ small-scale ordered structure (SSOS),⁶⁴ local similar atomic environment (SLAE), maximum entropy (MaxEnt), Monte Carlo/molecular dynamics coupling (MC/MD), and quasi-harmonic approximation (QHA),⁶⁵ with the help of these strategies.

Molecular dynamics simulations

In the design of HEA catalysts, molecular dynamics (MD) simulations provide a crucial computational tool for understanding the structure-activity relationship of the catalysts and optimizing their catalytic performance through the analysis of the dynamic evolution at the atomic scale. The core computational methods focus on the surface characteristics, stability, and reaction kinetics of HEA catalysts, including the following categories:

HEA is composed of multiple-component metals, and its catalytic activity is closely related to the arrangement and distribution of surface atoms. In MD simulations, empirical potential functions such as EAM or MEAM are commonly used to analyze structural characteristics. Simulate the displacement of surface atoms during relaxation (such as surface atoms contracting into the bulk, and sub-surface atoms expanding outward), as well as surface reconstructions under reaction conditions (such as formation of atomic steps and enrichment of active elements), to determine the stable surface active site structure. By calculating the diffusion coefficients and residence times of different component atoms on the surface/sub-surface, predict the surface segregation trend of high-activity elements (such as Pt, Pd, etc.) in HEA, providing a basis for designing “surface enrichment of active components, body phase maintaining stability” catalysts.

To explore the specific reaction pathways (such as CO oxidation, hydrogen release, etc.) participated by HEA catalysts, MD simulations often use reaction force fields (ReaxFF), which have the advantage of being able to describe the breaking and formation of chemical bonds. Reactant adsorption and activation simulation: Calculate the adsorption energy, adsorption configurations (such as top, bridge, and vacancy adsorption) of reactant molecules (such as O₂, H₂, and CO) on the surface of HEA, as well as changes in bond lengths/bond angles after adsorption, to determine the activation ability of the active site for reactants (such as the dissociation energy barrier of O₂ molecules on the surface). Reaction path and energy barrier calculation: simulate the intermediate states (such as adsorbed intermediate, transition state) of catalytic reactions, combined with energy minimization algorithms, calculate the energy barriers of each elementary reaction (such as the energy barrier for the combination of CO and adsorbed O to form CO₂), and select the HEA components with higher reaction activity.

The long-term stability of HEA catalysts (such as resistance to high-temperature sintering) is a key factor for practical applications. High-temperature structure evolution simulation: Under the NPT (isothermal isobaric) ensemble, simulate the atomic diffusion behavior of HEA at reaction temperatures (such as 300–800 K), calculate the average particle size change, surface atomic migration rate, and evaluate its anti-sintering ability (such as whether the high mixing entropy of multiple components inhibits atomic aggregation). Defect evolution analysis: Simulate the formation and migration of defects such as vacancies and dislocations at high temperatures, and determine the impact of defects on the structural stability of the catalyst (such as whether vacancies promote surface segregation of active elements or inhibit particle coarsening).

Machine learning approaches

The catalytic activity of HEAs is highly dependent on the types of elements, their proportions, and the combination methods. The component space of multi-element systems exhibits an exponential growth (for example, the component proportion combinations of a 5-element alloy can reach more than 10⁴). Traditional experiments or DFT calculations are difficult to cover the entire range. Machine learning can significantly narrow the screening scope and improve efficiency by constructing a “component-catalytic activity” prediction model.

The training data usually comes from a small number of experimental tests (such as the conversion rates and Faraday efficiencies of different component HEAs in CO₂ reduction and hydrogen evolution reactions) or DFT calculation results (such as surface adsorption energies and reaction barriers). Feature parameters (descriptors) include the physical and chemical properties of elements (such as electronegativity, atomic radius, and valence electron number), the macroscopic properties of alloys (such as mixing entropy, formation energy, atomic size difference), and statistical features (such as the weighted average of element proportions). These features can quantify the potential correlations between components and performance, providing a basis for model learning.

Common models include random forests (strongly capable of handling nonlinear relationships, and able to assess feature importance), artificial neural networks (capable of capturing complex mappings in high-dimensional data), gradient boosting trees (improving the prediction accuracy of small sample data) and so on. For example, for Pt-based high-entropy alloy electrocatalysts, by inputting the proportions of Pt, Ni, Co, Fe, Cu, and the electronegativities of elements, training a neural network model, the half-wave potential of it in oxygen reduction reaction (ORR) can be quickly predicted, reducing the candidate components from thousands to dozens, and then through experimental verification, significantly reducing the R&D cost.

Machine learning not only can predict the catalytic activity of unexplored component combinations, but also can clearly identify the key influencing elements through feature importance analysis (such as in the HER reaction, the proportions of V, Mo, etc. in high-entropy alloys have a more significant effect on activity improvement), providing guidance for targeted component design.

Computational analysis of catalytic mechanism

HEA have become ideal materials for complex catalytic reactions due to their characteristics such as multi-component synergy and tunable electronic structure, and the mechanism analysis depends on DFT.⁶⁶ DFT calculations reveal that the metal sites on the surface of HEA optimize the reaction pathways through heterogeneous synergy. For instance, the “deoxygenation-hydrogenation” relay catalysis reduces the rate-determining step energy barrier of nitrate reduction, and diatomic coordination (such as Fe-Ni) promotes the cleavage of the C-N bond in urea oxidation and enhances the nitrite selectivity to over 80%.⁶⁷ The current research on the dynamic recognition of HEA active sites, electronic structure and selective regulation mechanism still needs to be deepened. In this chapter, through DFT combined with electronic structure analysis, the relay catalytic mechanism, active site response and selective origin of HEA are explored, providing a theoretical framework for the design of multi-active site catalysts.⁶⁸

Reaction pathway and energy barrier analysis

Figure 2 elucidates the relay catalysis mechanism of FL-Ag/HEA for the electrocatalytic nitrate reduction reaction (NiRR) through density functional theory (DFT) calculations. The adsorption energy distributions of intermediates (∗NO₃, ∗NO, ∗NOH, ∗NH₃) across distinct metal sites on the FL-Ag/HEA surface exhibit significant heterogeneity (Figure 2A), where Ag and Pd sites function as weak and strong adsorption centers, respectively, synergistically modulating energy barriers for multi-step reactions.⁶⁷ The NO₃ hydrogenation step (NO₃ → ∗HNO₃) dominated by Ag sites exhibits a remarkably low energy barrier of 0.36 eV, significantly lower than that of homogeneous HEA (HEA homo, 0.45 eV), indicating that FL-Ag enhances the initial deoxygenation process by increasing the proportion of Ag-Ag active sites (Figure 2B).⁷¹ However, the subsequent NO hydrogenation (NO → ∗NOH) on Ag sites requires a higher barrier (0.74 eV), whereas adjacent Pd sites accomplish this step via a spontaneous exothermic reaction (ΔG = −1.23 eV), forming a “deoxygenation-hydrogenation” relay catalysis pathway (Figure 2D). When the applied potential was −0.5 V, all the energy barriers were significantly reduced (Figure 2E), which was highly consistent with the results of the electrochemical tests. Moreover, Fe sites in the HEA substrate serve as efficient HER active centers (barrier <0.26 eV), providing essential proton sources for NiRR, while FL-Ag suppresses competing HER through localized electronic structure modulation (Figure 2C).⁷² This work establishes a theoretical framework for designing multi-active-site catalysts for complex reactions and expands the application potential of high-entropy alloys in sustainable ammonia synthesis.⁶⁸ DFT calculations revealed the energy mechanism of n-HEA for electrocatalytic nitrate-to-ammonia reduction [44]. At U = 0 V, NO₃⁻ adsorbed first on Fe, Co, and Ni sites to form ∗NO and NO₃⁻. The potential-determining step (PDS) for Ni/Fe was ∗HNO to ∗N) (1.03/0.78 eV), while for Co it was ∗NH₂ to ∗NH₃ (0.52 eV), with the lowest limiting potential consistent with Co’s earliest onset potential in experiments. At U = −0.5 V, energy barriers decreased significantly. Co’s d-band center (0.720 eV) near the Fermi level, 1.60 eV adsorption energy for NO₃⁻, 0.78 |e| charge transfer, and 2.51 |e| polarization charge indicated higher NO₃⁻ activation.

Figure 2.

Reaction energetics and relay catalysis mechanisms of nitrate reduction on high-entropy alloy surfaces

(A) Illustrates the adsorption energy distribution (E_i) of key intermediates (NO₃⁻, NO, NOH, and NH₃) on the FL-Ag/HEA surface, with the lowest adsorption energy set as 0 eV to calculate relative values.

(B) and (C) present the reaction-free energies for the NitRR and HER at various metal sites in FL-Ag/HEA and HEA homo.

(D) outlines the relay catalysis mechanism of NitRR over FL-Ag/HEA.⁶⁹ Copyright 2024 Springer Nature.

(E) and (F) depict the Gibbs energy profiles of n-HEA at different metal sites for NRA at applied potentials of U = 0 V and U = −0.57 V (vs. RHE).⁷⁰ Copyright 2024 Wiley-VCH.

Figure 3 provides a comprehensive analysis of the electronic structure and adsorption behavior of PtRuPdCoNi HEA catalysts. Figure 3A shows the electronegativity order of Pt, Ru, Pd, Ni, and Co elements. The lower electronegativity of Co and Ni compared to Pt, Ru, and Pd optimizes the alloy’s electronic structure, causing a significant downshift in the d-band center of Pt. Figure 3B presents the two-dimensional charge density distribution of surface atoms in pure metals and PtRuPdCoNi HEA systems, along with Bader charge analysis results. Figure 3C shows the d-band center positions of Pt in Pt (111), PtRuPd, and PtRuPdCoNi. As the alloy composition becomes more complex, the d-band center of Pt shifts downward, weakening the binding of reaction intermediates and enhancing catalytic efficiency. Figures 3D and 3E depict the adsorption behavior and hydrogen adsorption free energy (ΔG_H) for Pt (111) and four sizes of PtRuPdCoNi HEA. The smallest HEA (HEA-S1) exhibits a ΔG_H closest to 0, indicating optimal hydrogen binding. Figures 3F and 3G show the H adsorption behavior and reaction energy profiles for five metal sites in HEA, revealing that the Pt site has the smallest absolute ΔG_H value. Figure 3H illustrates the free energy diagram for the ORR process from ∗O₂ to ∗OOH for Pt and four sizes of HEA, with HEA-S1 showing the lowest Gibbs free energy for ∗OOH formation. Collectively, these results demonstrate that adjusting the size of HEA can significantly optimize its electronic structure and catalytic performance, offering valuable theoretical insights for designing efficient electrocatalysts.

Figure 3.

Electronic structure modulation, adsorption behavior, and ORR pathways on high-entropy alloy catalysts: A comparative study with pure metals

(A) The electronegativity sequence for Pt, Ru, Pd, Ni, and Co elements.

(B) Two-dimensional charge density distributions of surface atoms in pure metals (Pt, Ru, Pd, Co, and Ni) and PtRuPdCoNi HEA systems, along with the quantity of electrons obtained from Bader charge analysis.

(C) The d-band center positions of Pt in Pt (111), PtRuPd, and PtRuPdCoNi.

(D) Adsorption behavior on Pt (111) and HEAs with four different sizes (size order: S1 < S2 < S3 < S4).

(E) DG∗H on Pt and HEAs with four different sizes.

(F) Adsorption behavior of ∗H on HEA at another four metal sites.

(G) Reaction energy profiles of ∗H on HEA at five metal sites.

(H) Free energy diagrams for the ORR process from ∗O₂ to ∗OOH on Pt and HEAs with four different sizes.⁷³ Copyright 2025 Wiley-VCH.

Dynamic identification of active sites

Figure 4A shows that there are three adsorption configurations of NO on the catalyst surface: N-end-on, O-end-on, and NO-side-on. Energy calculations indicate that the N-end-on adsorption configuration is the most stable for all single-atom alloy catalysts and pure silver surfaces.⁷⁶ The adsorption free energy data presented in Figure 4B reveal that the adsorption free energy of NO on all catalyst surfaces is more negative than that of H, giving NO an advantage in the initial adsorption step. However, some catalysts (such as Co/Ag, Ni/Ag, etc.) adsorb NO too strongly, which may lead to difficulties in NH₃ desorption and thus affect the reaction rate.⁷⁷ Moreover, due to the positive adsorption free energy of NO on Ag and Au/Ag, these materials are thermodynamically unfavorable for NO adsorption and unsuitable for NORR.⁷⁴ Figure 4C depicts six adsorption sites of NO on the TM/Ag catalyst surface: top (t), hollow-1 (h-1), hollow-2 (h-2), bridge-1 (b-1), bridge-2 (b-2), and bridge-3 (b-3). The research shows that the optimal adsorption sites for NO on all single-atom alloy catalysts are usually located near the doping sites, such as the top, bridge-1, or hollow-1 positions. Figure 4D demonstrates that electron transfer occurs during NO adsorption on all single-atom alloy catalysts, resulting in an increase in the N=O bond length of NO (from 1.18 Å to 1.24 Å), which confirms the effective activation of NO during the adsorption process. Figures 4E and 4F provides a systematic exploration of the proposed electronic descriptor’s applicability across complex adsorption configurations and diverse reaction intermediates, highlighting its crucial role in designing HEA electrocatalysts. By introducing the generalized descriptor F_d+αχN, the study effectively extends the single-active-site model to multi-center adsorption systems, including bridge, hcp, and fcc sites. Here, F_d is constructed by weighting the d-band filling fractions of active center atoms, with weights inversely proportional to their d-band occupancy, precisely characterizing the multi-atom synergistic effects on electronic structure modulation (Figure 4E). This approach effectively addresses the limitations of traditional d-band models in complex chemical environments. Experimental results demonstrate that the descriptor maintains high precision in predicting the adsorption energies of O (ΔE_ads^O) across different sites, with a mean absolute error (MAE) only slightly higher than that of single-center configurations (Figure 4F). Despite the more dispersed distribution of adsorption energies due to the diverse local chemical environments in multi-center adsorption, the descriptor accurately captures the trends, confirming its robust adaptability to surface inhomogeneity. Furthermore, the model is extended to predict the adsorption energies of OH⁻ (ΔE_ads^OH), with results highly consistent with DFT calculations (Figure 4G). A linear scaling relationship between OH⁻ and O adsorption energies (with a slope of approximately 0.6) is observed, aligning with known patterns of hydrogen-related molecules on transition metal surfaces.⁷⁸ This finding offers new insights into the mechanisms of hydrogenation reactions on HEA surfaces. Figure 4H presents the distribution of the hydrogen adsorption Gibbs free energy (ΔG_H∗) on the surface of a HEA with 5.9% compressive strain. The plot encompasses various adsorption sites, including hcp hollow sites, fcc hollow sites, and bridge sites.⁷⁹ The green area marks the diffusion region (DR), while the red dashed circles highlight the active centers for the Volmer and Heyrovsky (or Tafel) reactions. Specifically, the active center for the Volmer reaction exhibits the minimum ΔG_H∗ of −0.099 eV, whereas the active center for the Heyrovsky or Tafel reaction displays the maximum ΔG_H∗ of 0.075 eV. This spread in ΔG_H∗ values indicates that the HEA surface comprises multiple active sites with diverse hydrogen adsorption strengths, which effectively promotes the HER.

Figure 4.

Adsorption configurations, electronic interactions, and descriptor-based activity trends in nitric oxide reduction on alloy catalysts

(A) Three NO adsorption configurations on the catalyst surface are displayed, namely N-end-on (1), O-end-on (2), and NO-side-on (3).

(B) The ΔG_H, ΔG_NO, and ΔG_∗NH3 are presented.

(C) Six NO adsorption sites on TM/Ag are illustrated, labeled as top, hollow-1 (H-1), hollow-2 (H-2), bridge-1 (B-1), bridge-2 (B-2), and bridge-3 (B-3).

(D) Electron transfer to NO and N-O bond length are shown.⁷⁴ Copyright 2025 Wiley-VCH.

(E) F_d+αχ descriptor, where F_d represents the mean d-band filling fraction of multi-center active sites, calculated as $F_{\mathrm{d}}=\frac{\sum {\omega }_i{f}_{\mathrm{d}}^{\left(i\right)}}{N_{\text{center}}}$.

(F and G) Descriptor accuracy for O∗/OH∗ on HEA bridge, hcp, and fcc sites.³⁶ Copyright 2025 Springer Nature.

(H) ΔG_H∗ distribution on 5.9%-compressed HEA across hcp, fcc, and bridge sites.⁷⁵ Copyright 2024 Springer Nature.

The origin of multi-product selectivity

Figures 5A and 5B analyze the pathways and free energy changes of the glycerol electro-oxidation reaction (GEOR) on different catalyst surfaces. Figure 5A shows possible routes for glycerol conversion to glycerate and lactate, highlighting key steps in lactate formation. Figures 5B and 5C illustrates how free energy varies with C-OH distance during dehydroxylation steps on Pt (111) and PtCu (111) surfaces. The results indicate that PtCu surfaces have higher energy barriers for forming lactate intermediates than Pt (111), suppressing lactate production and favoring glycerate selectivity.⁸⁰ This is because PtCu surfaces require more energy for hydroxyl detachment, making glycerate the dominant product. These findings show how alloying alters surface electronic structures and oxidation states to control reaction pathways and product selectivity.^82,83 Figures 5D and 5E reveal the synergistic catalysis mechanism of diatomic coordination, free energy changes, and electronic structure characteristics of the FNS/d-B catalyst in the urea oxidation reaction (UOR) through experiments and theoretical calculations.⁸⁴ The diatomic coordination mechanism shows that urea molecules are adsorbed on Fe sites via C-N bonds, while Ni sites capture OH⁻ to form a “Fe-urea-Ni-OH^-” structure, significantly reducing the energy barrier for C-N bond cleavage and promoting the generation of ∗NH intermediates.⁸⁵ Density functional theory calculations indicate that the free energy change of the rate-determining step in the bimetallic synergistic pathway (S2, 1.10 eV) is significantly lower than that in the monometallic pathway (S1, 1.73 eV), and potential regulation further optimizes the reaction kinetics. Integrated crystal orbital Hamilton population (ICOHP) analysis shows that the moderate bonding strength between Ni sites and OH⁻ (ICOHP = −0.75 eV) and the strong adsorption between Fe sites and ∗NH (ICOHP = −1.20 eV) synergistically facilitate intermediate conversion, ultimately achieving over 80% selectivity for NO₂⁻ (Figures 5F and 5G).

Figure 5.

Reaction pathways, electronic structure, and catalytic mechanisms in biomass conversion and urea synthesis on metal and alloy surfaces

(A) Possible reaction pathways to glycerate and lactate. The key steps determining lactate generation are highlighted in shadow.

(B and C) Free energy changes as a function of the C–OH bond distance during the dehydroxylation steps of 1,3-dihydroxyacetone (∗DHA), glyceraldehyde (∗GLD), and dehydrogenated glyceraldehyde (dH-GLD) over Pt (111) and (C) PtCu (111) surfaces. The energy barrier values are labeled.⁸⁰ Copyright 2025 Springer Nature.

(D) Total density of states (TDOS) of body-centered cubic intermetallic compounds (bcc-MEI) and face-centered cubic high-entropy alloys (fcc-HEA).

(E) Crystal orbital Hamilton population (COHP) of both systems.

(F) Gibbs free energy diagrams for urea synthesis on the surfaces of bcc-MEI and fcc-HEA.

(G) Corresponding structural models for urea synthesis pathways on the fcc-HEA surface.⁸¹ Copyright 2025 Wiley-VCH.

HEA are composed of five or more components with nearly equal molar amounts. They naturally form disordered and diverse active sites on the surface, which can optimize N₂ dissociation and proton hydrogenation sites separately on the same surface, and precisely control the adsorption energies of various nitrogen-containing intermediates. In contrast, single-atom catalyst sites are uniform and difficult to accommodate multiple complex pathways. Two-dimensional materials can introduce doping or defects to increase diversity, but they still fall far short of the rich and adjustable synergistic chemical environment of HEA. Figures 6A and 6B reveal the mechanism of NRR catalyzed by high-entropy oxides (HEOs): the associative-alternating pathway is optimal, where N₂ is activated at Fe sites (N≡N bond length extended from 1.098 Å to 1.278 Å), followed by stepwise hydrogenation via intermediates like ∗NNH and ∗NHNH₂. The rate-determining step (∗NHNH₂→∗NH + NH₃) has an energy barrier of 1.98 eV, significantly lower than the dissociative pathway (28.90 eV). The free energy diagram shows a total reaction free energy change of −5.54 eV, with multi-metal synergies (Fe, V, Co, Ni) and high configurational entropy reducing reaction barriers, confirming that HEOs enhance NRR efficiency by optimizing electronic structures and intermediate adsorption.⁸⁸ Figures 6D and 6E show that for PtRuRhCoNi HEA nanowires, the Ni-3d PDOS exhibits a significant peak at EV-1.20 eV, enhancing selectivity for specific intermediates. In ethanol oxidation, the initial dehydrogenation leads to energy changes favoring the reaction, especially CO oxidation to CO₂. These features indicate that the PtRuRhCoNi HEA catalyst, via its optimized electronic structure and strong self-complementary effect, improves adsorption capacity for key reactants and intermediates, enabling efficient EOR.

Figure 6.

Reaction pathways, electronic properties, and adsorption behaviors in nitrogen and ethanol electrochemical conversion

(A) The associative-alternating pathway for NRR.

(B) Free energy diagrams for NRR.

(C) Density of states (DOS) and charge difference maps for the potential-determining step of NRR from ∗NH₂NH₂ to ∗NH₂.⁸⁶ Copyright 2022 Wiley-VCH.

(D) Comparison of adsorption energies for CH₃CH₂OH, CO, and CO₂.

(E) Energy change during ethanol oxidation.⁸⁷ Copyright 2022 Elsevier B.V.

Computational driven HEAs design strategy

This chapter systematically expounds the design strategy of computationally driven HEAs, covering multi-level research methods from component screening, surface engineering to structural evolution. Firstly, through theoretical calculations combined with machine learning models, efficient screening and performance optimization of candidate materials are achieved. Based on the analysis of binding energy and electrochemical stability, thermodynamically stable and highly active single-atom alloy systems were screened out, and a descriptor-activity relationship model was established to reveal the synergistic regulation mechanism of key geometric and electronic characteristics (such as bond length, d-band center) on catalytic performance. In terms of surface engineering, high-entropy alloying significantly enhances catalytic activity by introducing local structural distortion and optimizing the adsorption energy distribution, especially demonstrating potential to surpass traditional materials in hydrogen evolution and oxygen reduction reactions. Furthermore, machine learning techniques (such as deep neural networks and graph attention networks) are applied to analyze the local chemical environment and adsorption behavior of complex multi-component systems, accurately predict adsorption energy and guide component optimization, breaking through the limitations of traditional linear scaling relationships. The research also emphasizes the significance of rotational invariant characteristics and atomic-level attention mechanisms in capturing the structure-performance correlation of materials, providing a new paradigm for the rational design of high-performance catalysts.

Component screening and optimization

Figures 7A–7F comprehensively maps out the structural stability and screening process of transition metal-doped Ag-based single-atom alloy (SAA) catalysts (TM/Ag, with TM spanning Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Ru, Rh, Pd, Pt, Au) geared toward the NO reduction reaction (NORR). The structural model in Figure 7A reveals TM atoms dispersed as isolated single atoms on the Ag (111) surface. Guided by a four-step screening strategy (Figures 7B and 7C), the stability of SAAs was primarily gauged via binding energy (E_b). All TM/Ag SAAs exhibit negative E_b values, signifying thermodynamically stable single-atom configurations. However, positive E_b−E_c values for Co/Ag, Ni/Ag, and Ru/Ag hint at a propensity for atomic aggregation, which may disrupt the uniformity of active sites.⁸⁹ Pourbaix diagrams (Figures 7D–7F) unveil the electrochemical stability of catalysts across diverse pH and potential ranges. Pure Ag remains stable in the clean state (∗) or hydrogen-covered state (∗H) between 0.30 and 0.92 V vs. RHE. SAAs such as Cu/Ag and Zn/Ag demonstrate stability across a broad potential range (0.1–0.9 V vs. RHE), while Sc/Ag and Ti/Ag are prone to dissolution or oxidation at low potentials, rendering them unsuitable for further consideration. The stability of Cu/Ag and Zn/Ag is ascribed to their electronic structure modulation of hydrogen adsorption (ΔG_∗H) and oxidation states, establishing a foundation for subsequent NORR activity evaluation. Figure 7G harnesses machine learning to unravel the intricate relationship between the NORR activity descriptor Φ_max(η) and the intrinsic properties of catalysts. Through feature engineering, 13 initial features were selected (e.g., ionization energy of the doped atom (IE), d-band center (ε_d, NO bond length D_N-O), and after applying Pearson correlation filtering, 10 independent features were retained (Figure 7H).Among these, the extreme gradient boosting regression (XGBR) model delivered exceptional performance, achieving a training set R² of 0.96 and RMSE of 0.06 eV, and a test set R² of 0.95 and RMSE of 0.07 eV, thereby confirming its high reliability in predicting Φ_max(η) (Figure 7I). Feature importance analysis (Figure 7J) highlighted the average bond length between NO and metal atoms (D_M-N, 32.2%), IE of the doped atom (16.1%), and ε_d (11.2%) as pivotal determinants of catalytic activity. Subsequent compressed sensing symbolic regression (SISSO) established a mathematical model: Φ_max(η) = 0.077 D_M−N/ε_d²-0.00029∗IE∗(1+D_M−N)+1.261 (Figure 7K). This model exhibits an RMSE of 0.11 eV and R² = 0.90, underscoring the synergistic interplay of D_M−N and ε_d. A smaller D_M−N coupled with a more negative ε_d (e.g., in Cu/Ag and Zn/Ag) reduces reaction barriers by enhancing NO adsorption and weakening intermediate binding energies.⁹⁰

Figure 7.

Integrated computational screening and stability analysis of transition metal single-atom alloys for electrochemical nitric oxide reduction

(A) Structural models of TM/Ag and the corresponding TM elements.

(B) A DFT-based four-step screening strategy diagram to obtain optimal NORR catalysts.

(C) Energy difference between binding and cohesive energy of TM/Ag. Pourbaix diagrams of (D) Ag, (E) Cu/Ag, and (F) Zn/Ag. Colored regions indicate pH-voltage windows where the catalyst is stable in the SAA state, either clean or covered by intermediates, with colors identifying ∗H, ∗, ∗OH, and ∗O states. Black dashed lines denote the redox levels of H⁺/H₂ and O₂/H₂O couples.

(G) Machine learning workflow with feature engineering.

(H) Heatmap based on Pearson’s correlation coefficient.

(I) Comparison between GC-DFT and ML predicted Φ_max (η = 0.43 V vs. RHE).

(J) Ranking of important features of XGBR.

(K) Comparison between GC-DFT and SISSO predicted Φ_max (η = 0.43 V vs. RHE).⁷⁴ Copyright 2025 Wiley-VCH.

Surface engineering and active site design

Transition metal dichalcogenides (TMDCs) have shown significant potential in the field of electrocatalysis due to their tunable electronic structure and surface activity. However, their intrinsic catalytic activity still needs to be further enhanced through rational design. In recent years, the high-entropy alloying (HEA) strategy has provided a new approach for optimizing the catalytic performance of TMDCs—by introducing local phase instability and electronic structure modulation through multi-component introduction, it is expected to break the adsorption energy limitations of traditional catalysts. Through systematic surface engineering and active site design studies, this paper reveals the unique catalytic mechanism of high-entropy pentavalent sulfides (ABCDE) S₂ in hydrogen evolution reaction (HER) and oxygen reduction reaction (ORR). Figure 8A presents the range of hydrogen adsorption energies for all stable pentanary TMDCs with equimolar concentration (ABCDE)S₂ in the 1T and 2H phases. Vertical black lines indicate adsorption energies at different sites, while blue and red colormaps represent the free energy of the 1T and 2H structures at 1000 K. Experimentally synthesized HEAs are labeled in blue and red, corresponding to their reported phases. Re is shown in black due to ReS₂’s greater stability in the distorted 1T phase. The analysis reveals that high-entropy alloying introduces local phase instability and optimizes hydrogen adsorption energies, thereby significantly improving the catalytic activity of TMDCs. Some pentanary alloys in the 1T phase exhibit hydrogen adsorption energies close to the optimal condition for HER catalysts. Figure 8B examines the hydrogen adsorption energies for two different SQS models of (MoWNbTaTi)S₂ and (CrMoWReMn)S₂ in the 2H phase. The (CrMoWReMn)S₂ composition was chosen for its unique atomic environment. The atomic environments of the most active adsorption sites are illustrated at the top of the figure. Even in the absence of ideal adsorption environments, the catalytic activity remains largely unaffected, which confirms the robustness of the high-entropy alloy strategy. Figure 8C presents a comparison of reaction currents between model predictions and experimental results, confirming the model’s accuracy. Figures 8D–8I display ORR activity maps for HEAs with different principal elements and continuously varying composition concentrations. The heatmaps indicate the reaction current relative to pure Pt metal. The results reveal that certain regions exhibit remarkable catalytic performance surpassing pure Pt. Optimal alloys are predominantly located near the edges of the compositional space, suggesting element-dependent catalytic properties in noble-metal HEAs rather than a “cocktail effect”. Ir–Pt and Ir–Au compositions show significantly larger regions with optimal activities in the compositional space, potentially serving as promising candidates to expedite ORR processes. These findings indicate promising directions for optimizing ORR catalysts using noble-metal HEAs.

Figure 8.

Hydrogen adsorption energetics and oxygen reduction activity in high-entropy transition metal dichalcogenides and alloy catalysts

(A) Distribution of hydrogen adsorption energies on equimolar pentanary TMDCs (ABCDE)S₂ in 1T and 2H phases.

(B) Hydrogen adsorption energy ranges for two SQS models, (MoWNbTaTi)S₂ and (CrMoWReMn)S₂, in the 2H phase. The experimentally synthesized (CrMoWReMn)S₂ was chosen for its unique atomic environment.⁹¹ Copyright 2025 American Chemical Society.

(C) Comparison of reaction currents between model-predicted and experimental results for Ag–Ir–Pd–Pt–Ru HEA surfaces.

(D–I) ORR activity maps on HEA surfaces with different principal elements and continuously changing composition concentrations.⁷⁵ Copyright 2024 Springer Nature.

Structural evolution analysis driven by machine learning

High-entropy alloy catalysts (HEA) demonstrate great potential in complex electrocatalytic reactions such as CO₂ reduction (CO₂RR) due to their multi-active-site synergy, tunable electronic structure, and strong resistance to poisoning. However, the large component space and complex local environment of HEA pose severe challenges to traditional theoretical simulations. In this study, a high-precision adsorption energy prediction model and rational design paradigm were established through machine learning (ML)-driven structural evolution analysis.

Figure 9 systematically presents the application and key progress of machine learning methods based on geometric features in predicting the adsorption energy of high-entropy alloys (HEA). Firstly, Figures 9A and 9B takes the IrPdPtRhRu HEA system as an example to describe in detail the local environmental feature extraction process of ∗OH adsorption at the bridge site [54]. This study constructed feature vectors by considering parameters such as the period, group, atomic radius and coordination number of 11 adjacent atoms, and input them into a deep neural network (DNN) for adsorption energy prediction, achieving a MAE of 0.090 eV and verifying the effectiveness of geometric features in complex multi-element systems. However, the neglect of rotational invariance by this method may lead to the feature being sensitive to the structural orientation. Figure 9C further proposes a rotational invariance feature extraction strategy based on regional division,³⁹ dividing the adsorption environment into different regions, statistically analyzing the distribution of elements within each region (such as Ir, Pd, Pt, etc.), and infeeding them into the linear regression model. This method achieved the adsorption energy prediction of OH-top and O-hollow sites on the IrPdPtRhRu (111) surface, with RMSE of 0.063 eV and 0.076 eV, respectively (Figure 9D), and was successfully applied to the optimization of ORR catalyst composition by combining the volcanic map model. Subsequent studies extended the geometric features to the CO₂RR and the CORR. Figures 9E and 9F shows the predicted adsorption energies of CO and H of CoCuGaNiZn and AgAuCuPdPt HEA.⁹³ Among them, the MAE of the GPR model at multiple adsorption sites (such as CO-TOP, H-fcc/hcp) was all lower than 0.065 eV. Based on the prediction results, the researchers further drew the CO₂RR/CORR selectivity and activity map (Figure 9F), revealing the efficient catalytic potential of the high-entropy alloy in a wide potential range. To verify the universality of geometric features, Figure 9G compares the prediction and DFT calculation results of various adsorbents (CO, H, O, CHO, etc.) on CuCoNiZnMg HEA.⁹⁴ The Symbolic regression (SISSO) model achieves the accuracy of RMSE = 0.11 eV and R² = 0.90. And the synergistic mechanism between the metal-nitrogen bond length (D_M−N) and the D-band center (ε_d) was revealed. Studies show that although HEA can partially break the traditional linear scaling relationship, the average adsorption energy of its configuration still shows local linear dependence, and the optimization strategy of multi-step reactions needs to be further explored. In conclusion, Figure 9 highlights the core role of geometric features in the prediction of HEA adsorption energy through various cases, while also pointing out their limitations (such as insufficient sensitivity to lattice strain). In the future, rotational invariance features (such as SOAP, MTP) and more complex graph neural networks (GNN) need to be combined to further enhance the model’s generalization ability in complex local environments.

Figure 9.

Machine learning-assisted feature engineering and adsorption energy prediction for high-entropy alloy catalysts

(A and B) Schematic illustration of feature extraction processes depicting the local environment of ∗OH adsorption on bridge sites of IrPdPtRhRu HEAs.⁹² Copyright 2020 Cell Press.

(C) Classification of adsorption environments into distinct zones for systematic feature analysis.

(D) Parity plots comparing machine learning-predicted versus DFT-calculated ∗OH adsorption energies for IrPdPtRhRu HEAs.³⁹ Copyright 2019, Cell Press.

(E) Parity plots of predicted versus DFT-derived ∗CO and ∗H adsorption energies for CuCoNiZnMg HEAs.

(F) Selectivity-activity mapping for CO₂RR/CORR on CoCuGaNiZn (left) and AgAuCuPdPt (right) HEAs, highlighting achievable performance spaces.⁹³ Copyright 2020, American Chemical Society.

(G) Parity plots validating predicted versus DFT-computed adsorption energies for diverse adsorbates on CuCoNiZnMg HEAs.⁹⁴ Copyright 2022, Springer Nature.

High-entropy alloy catalysts, with their advantages of multi-active-site synergy, high stability, flexible electronic regulation, strong resistance to poisoning, and ease of large-scale production, are particularly suitable for CO₂RR in scenarios requiring high current density, long-term operation, and synthesis of high-value C₂⁺ products. They make up for the shortcomings of single-atom catalysts in stability and two-dimensional catalysts in regulation, providing a more promising material option for the industrial application of CO₂RR. Figure 10A displays the formation energies of homodimers and heterodimers, highlighting certain elemental combinations’ superiority in forming stable DAA active sites. For instance, Ir1Ti1Cu exhibits higher stability than isolated dopants or homodimers. Scatterplot of Figure 10B reveals ML-based screening results for Cu-based DAA catalysts, showing that early and late transition metal combinations in a Cu matrix enhance CO₂ covalent binding compared to pure Cu. ML feature importance analysis identifies the maximum effective Löwdin charge between dopants as crucial for CO₂ binding. Scatterplot of Figure 10C further illustrates the correlation between CO₂ binding strength and Löwdin charge, reinforcing this finding. Figure 10D illustrates the model architecture of ACE-GCN, which is designed to consider diverse atomistic configurations. This architecture enables the model to capture the local environment and chemical information of the catalyst surface, making it suitable for predicting adsorption energies in complex electrocatalytic systems. ACE-GCN (Atom-Centered Environment Graph Convolutional Network) constructs a subgraph structure to represent the local chemical environment, thereby significantly improving computational efficiency while maintaining the accuracy of adsorption energy prediction (MAE <0.08 eV). Its core advantage lies in the selective modeling of neighboring atoms (typical truncation radius of 3–5 A), which significantly reduces the number of atoms involved in the calculation and approximately doubles the training speed compared to traditional DFT. Case studies have verified that this model accurately predicts the asymmetric adsorption configuration of NO at the Pt₃Sn (111) alloy bridge position (Figure 10E), and quantifies the weakening effect of OH adsorption on the step Pt (221) surface (ΔG_∗OH ≈ −0.3 eV) (Figure 10F), successfully capturing key mechanisms such as the electronic state reorganization induced by Sn doping and the shift of the d-band center at the step site. This “local environment truncation” strategy reduces the computational complexity to O(N), providing a new paradigm for high-throughput screening of complex surface catalysts. Performance improvements may be possible if short-range interactions are more comprehensively included in the subgraphs. These analyses highlight ACE-GCN’s potential in electrocatalysis research, especially for complex scenarios like strong adsorbate binding on low-symmetry alloy surfaces and direction-dependent adsorption on defective surface structures.⁹⁷ Figure 10G illustrates the architecture of the atomic graph attention network (AGAT), which resolves the complex local environments and phase spaces of high-entropy electrocatalysts (HEECs) through an attention mechanism. The model comprises an AGAT layer (upper panel) and an integrated AGAT framework (lower panel), where attention scores are computed via edge-based message passing to dynamically evaluate the significance of source atoms on target nodes.⁹⁹ Applied to oxygen reduction reaction (ORR) performance predictions on RuRhPdIrPt and NiCoFePdPt surfaces, the model identifies two high-activity candidates (Ni₀.₁Co₀.₁₅Fe₀.₁₅Pd₀.₁₀Pt₀.₅₀ and Ni₀.₁₀Co₀.₀₅Fe₀.₁₀Pd₀.₃₀Pt₀.₄₀), demonstrating that HEECs circumvent conventional linear scaling relationships through diverse local coordination environments. Figure 10H validates the interpretability of the AGAT model by correlating attention scores with energy/force variations. The analysis reveals that the adsorbate-substrate distance critically governs attention scores, highlighting the dominant influence of local chemical environments (e.g., atomic coordination states) on catalytic activity.⁹⁹ This finding provides atomistic insights into the design of high-entropy alloy catalysts, emphasizing the capability of attention mechanisms to decode structure-property relationships in complex material systems. The AGAT framework quantitatively links atomic-scale chemical heterogeneity to macroscopic catalytic performance, offering a paradigm for rational catalyst optimization beyond empirical trial-and-error approaches.

Figure 10.

Computational screening, machine learning-driven prediction, and interpretable modeling of dual-atom alloys and adsorption configurations

(A) Screening of homodimer and heterodimer formation energies for M₁Pd₁Ag, M₁Pd₁Au, M₁Pt₁Ag, and M₁Ti₁Cu.⁹⁵ Copyright 2022 Royal Society of Chemistry.

(B) Scatterplot of screening results for initial candidate dual-atom alloys (DAAs).

(C) Correlation scatterplot between Löwdin charge and CO₂ binding energy among 11 key ML features screened for CO₂ binding strength.⁹⁶ Copyright 2023, American Chemical Society.

(D) The model architecture of ACE-GCN. Adsorption configurations of (E) NO and (F) OH (Generated via SurfGraph using the Adsorbate Chemical Environment-based Graph Convolutional Neural Network (ACE-GCN) mode.⁹⁷ Copyright 2022 Springer Nature.

(G) Schematic of the AGAT model architecture (Upper: AGAT layer; Lower: Complete AGAT model).³⁸ Copyright 2023 Cell Press.

(H) Interpretability analysis of the AGAT model: Comparison of attention scores in energy and force models with variations in energy and forces.⁹⁸ Copyright 2025 OAE Publishing Inc.

Collaborative verification and optimization of calculation and experiment

In recent years, high-entropy materials have demonstrated broad application prospects in energy storage and catalysis due to their unique compositional diversity and tunable properties. However, their complex multi-component nature renders traditional trial-and-error approaches inefficient for guiding material design and performance optimization, necessitating a research paradigm that deeply integrates theoretical calculations with experimental validation. Current studies face multiple challenges: theoretical models are constrained by computational scale and accuracy, limiting their ability to predict behaviors of large-scale high-entropy systems; scarce experimental data hampers the generalizability of machine learning models; and technical bottlenecks persist in resolving dynamic reaction mechanisms and structural reconstruction processes through in situ characterization. To address these issues, this chapter proposes a “computation-synthesis-validation” closed-loop strategy. This approach employs DFT to elucidate free energy regulation mechanisms of critical reaction pathways, integrates machine learning and high-throughput techniques for intelligent screening in compositional space, and utilizes advanced characterization methods to validate entropy-driven optimization of electronic structures and material stability. The study demonstrates that precise modulation of entropy-driven effects can synergistically enhance the intrinsic activity of catalytic sites and structural durability, providing a theoretical framework and methodological foundation for developing next-generation high-performance energy materials. This interdisciplinary strategy not only overcomes the efficiency limitations of traditional material development but also opens new perspectives for understanding complex structure-property relationships in high-entropy systems.

Theoretical prediction guides experimental synthesis

Figure 11 systematically investigates the critical roles of noble metals (Pt, Ir, Ru, and Pd) in the formation of single-phase solid solutions in noble metal-based high-entropy alloy three-dimensional nanoframeworks (NM-HEA-3DNFs). DFT calculations reveal the Gibbs free energy profiles for H₂ adsorption (ΔG_H2-ads), dissociation (ΔG_H2-diss), and desorption (ΔG_H-des) on different noble metal surfaces (Figure 11A). For Pt (111), Ir (111), and Ru (011) planes, the free energy values exhibit a downhill trend (e.g., Pt: ΔG_H2-ads = −0.643 eV, ΔG_H-des = −0.052 eV, ΔG_H-des = −0.289 eV), indicating spontaneous generation of highly reactive H atoms (Figures 11B–11F). These H atoms facilitate the reduction of adjacent transition metals (Ni, Co, Cu, Fe) and their integration into the noble metal lattice, enabling homogeneous single-phase solid solution formation (Figure 11I).¹⁰¹ Experimental validation using XRD and STEM-EDS confirms the single-phase solid solution structure in synthesized IrNiCoCuFe and RuNiCoCuFe HEA-3DNFs (Figures 11D and 11F). Notably, Ru-based alloys adopt a hexagonal close-packed (hcp) structure due to Ru’s inherent hcp lattice preference, while Pt- and Ir-based alloys form face-centered cubic (fcc) structures.^102,103 In contrast, Pd (111) surfaces exhibit an uphill ΔG_H-des (0.031 eV), hindering H desorption and resulting in phase-separated PdNiCoCuFe alloys (Figure 11H).¹⁰⁴ Collectively, Figure 11 elucidates the mechanism by which noble metals regulate H₂ activation pathways to achieve single-phase solid solutions, providing a theoretical foundation for the rational design of high-entropy alloys.^68,105,106

Figure 11.

Autoreduction mechanism, hydrogen adsorption energetics, and phase evolution in noble metal-based high-entropy alloy formation

(A) Gibbs free energy plot for H₂ on the Pt (111) crystal surface.

(B) Schematic of Pt-induced autoreduction of neighboring metals by reactive H atoms.

(C) Gibbs free energy plot for H₂ on the Ir (111) crystal surface.

(D) XRD pattern for IrNiCoCuFe HEA at 300°C reduction.

(E) Gibbs free energy plot for H₂ on the Ru (011) crystal surface.

(F) XRD pattern for RuNiCoCuFe HEA at 300°C reduction.

(G) Gibbs free energy plot for H₂ on the Pd (111) crystal surface.

(H) XRD pattern for PdNiCoCuFe alloy at 300°C reduction and Pd_xH_y compound formation without reactive H atoms.

(I) Proposed formation mechanism for NM-HEA-3DNF.¹⁰⁰ Copyright 2025 American Association for the Advancement of Science.

Figure 12 systematically summarizes the theoretical simulation methods and their critical roles in applying high-entropy materials to rechargeable metal batteries. DFT calculations reveal the regulatory mechanisms of unique electronic structures in high-entropy materials on catalytic performance. For instance, Bader charge analysis demonstrates that Pt acts as an electron acceptor while Ni, Co, Fe, and Cu serve as electron donors in HEAs, forming significant charge transfer effects (Figures 12A–12C). This synergy significantly enhances the adsorption capacity of lithium polysulfides (LiPSs), thereby improving the cycling stability of lithium-sulfur batteries.¹¹² Additionally, Gibbs free energy diagrams (Figures 12C and 12D) indicate that high-entropy single-atom (HESA) catalysts reduce energy barriers for ORR and OER by modulating intermediate adsorption strength, exhibiting superior bifunctional activity compared to conventional monometallic catalysts.^109,113 The integration of ML and high-throughput (HT) technologies establishes a new paradigm for rational design of high-entropy materials. Neural network models successfully predict the binding energy of OH∗ intermediates on IrPdPtRhRu high-entropy catalyst surfaces (Figures 12E and 12F), elucidating the influence of coordination environments on catalytic activity.⁹² A multi-objective Bayesian optimization framework further enables efficient screening of HEAs in compositional space, balancing catalytic activity, entropy stabilization, and cost-effectiveness (Figure 12G).¹¹⁰ Moreover, an HT experimental platform based on large language models (LLMs) rapidly identifies efficient oxygen reduction catalysts among 70 quinary HEA combinations, accelerating material development (Figures 12H–12J).¹¹¹ Despite significant progress, the complex composition and local structures of high-entropy materials pose challenges to computational accuracy. DFT is limited to small-scale systems, while ML relies on high-quality datasets, with current experimental data scarcity constraining model generalizability.¹¹⁴ Future efforts should integrate in situ characterization and multi-scale simulations to unravel dynamic reconstruction mechanisms, alongside developing autonomous experimental platforms to achieve “computation-synthesis-validation” closed loops, advancing practical applications of high-entropy materials in energy storage.^115,116

Figure 12.

Multi-scale analysis and intelligent design of high-entropy alloys: from electronic structure and adsorption properties to machine learning and automated discovery

(A) Bader charge analysis for Mo1-PtPdFeCoNi. Reproduced with permission.¹⁰⁷ Copyright 2024, Springer Nature.

(B) Binding energy of Li₂S₆ on HE-LSMO. Reproduced with permission.¹⁰⁸ Copyright 2023, Elsevier B.V.

(C) Gibbs free energy plot for various samples.

(D) Colour-filled contour profiles. Reproduced with permission.¹⁰⁹ Copyright 2023, Springer Nature.

(E) Schematic of neural network-assisted machine learning.

(F) Linear regression prediction of OH∗ intermediate binding energies. Reproduced with permission.⁹² Copyright 2020, Cell Press.

(G) Schematic of multi-objective Bayesian optimization (BO) framework for ORR-active HEA catalyst discovery. Reproduced with permission.¹¹⁰ Copyright 2024, American Chemical Society.

(H) Element combination scheme based on LLM.

(I) HT SECCM-based ORR activity screening.

(J) Relative real current density (jrelative) heatmap (E: 0.1 V vs. RHE). Reproduced with permission.¹¹¹ Copyright 2024, Wiley-VCH.

Experimental data feedback correction model

In the overall water splitting reaction, the high-entropy alloy catalyst, with its multi-activity sites that work in synergy, continuously adjustable electronic structure, self-consistent lattice stress, and excellent corrosion resistance, can simultaneously efficiently drive hydrogen evolution and oxygen evolution; in contrast, single-atom catalysts are limited by a single active center, and two-dimensional materials require additional defects and have insufficient stability. Therefore, HEA demonstrates significant advantages in terms of efficiency, lifespan, and industrial scalability. Figure 13 systematically investigates the oxygen evolution reaction (OER) performance of HEOs with varying configurational entropy. The OER polarization curves (Figures 13A and 13B) demonstrate that increasing configurational entropy significantly reduces the overpotential (from 390 mV to 320 mV) and Tafel slopes (from 98 mV dec⁻¹ to 45 mV dec⁻¹), indicating enhanced reaction kinetics [70]. Electrochemical impedance spectroscopy (EIS) further corroborates these findings, showing a notable decrease in charge transfer resistance (from 250 Ω to 30 Ω) with higher entropy (Figure 13C). Turnover frequency (TOF) analysis reveals that HEOs with the highest entropy exhibit superior intrinsic activity (0.066 s⁻¹) despite comparable electrochemical surface areas (ECSAs) to lower-entropy counterparts, highlighting the critical role of entropy-driven electronic modulation (Figure 13D). X-ray photoelectron spectroscopy (XPS) and X-ray absorption near-edge structure (XANES) analyses, combined with DFT calculations, elucidate that increased configurational entropy optimizes the formation energy of oxygen vacancies on HEO surfaces. This facilitates lattice oxygen participation in the OER via the formation of stable oxygen vacancies and intermediates such as O₂²⁻, as confirmed by in situ Raman spectroscopy (Figures 13K and 13I). Additionally, the sluggish diffusion effect inherent to high-entropy systems ensures exceptional stability under harsh acidic conditions, with FeCoNiIrRu HEA nanoparticles (NPs) maintaining robust performance (overpotential: 286 mV at 20 mA cm⁻²) and negligible metal leaching during prolonged operation (Figures 13E–13H).¹¹⁸ These results underscore the synergy between entropy-driven structural stability and optimized electronic states in achieving high-performance OER catalysts.

Figure 13.

Electrocatalytic oxygen evolution performance and reaction mechanisms of high-entropy oxides: effects of configurational entropy and active site engineering

(A–D) OER polarization curves, Tafel plots, EIS spectra, and TOF values of HEOs with varying configurational entropy.¹¹⁷ Copyright 2022, Wiley-VCH.

(E and F) OER polarization curves and overpotentials at 20 mA cm⁻² for FeCoNiIrRu/CNF and related materials.

(G and H) Mass activities and Tafel plots of FeCoNiIrRu/CNF and related materials at 1.5 V vs. RHE.

(I and J) Free energy diagrams for OER on FeCoNiIrRu surfaces at different sites.

(K and L) In situ Raman spectra of FeCoNiIrRu at various potentials.¹¹⁸ Copyright 2022 Elsevier B.V.

DFT calculations and operando spectroscopic evidence converge on a “dual-site” pathway for water dissociation on FeCoNiMnRu HEA surfaces (Figures 14A–14F). Co atoms exhibit the lowest dissociation barrier (0.34 eV) for H-OH bond cleavage, substantially lower than those on Fe (0.70 eV), Ni (0.63 eV), and Ru (0.67 eV) sites,^71,121 establishing Co as the primary site for water activation. The resulting H∗ intermediates then migrate to Ru sites, where a near-thermoneutral hydrogen adsorption free energy (ΔG_H∗ = −0.07 eV) promotes facile desorption, in contrast to the overly strong binding on Co (ΔG_H∗ = −0.43 eV). Operando Raman spectroscopy corroborates this cooperative sequence. At 60 mV bias, distinct vibrational signatures of surface-bound hydroxyl species emerge at Fe-O (215/290 cm⁻¹), Co-O (585/704 cm⁻¹), and Ni-O (446/530 cm⁻¹).^122,123 Concurrently, Ru-H stretching modes at 2069/2092 cm⁻¹ provide direct evidence for preferential hydrogen adsorption on Ru (Figure 6E). As the potential increases to 180 mV, the intensities of all characteristic bands scale proportionally, indicating a positive correlation between intermediate coverage and catalytic activity and further validating the Co-Ru synergistic mechanism.¹²⁴ in situ Raman (Figures 14G–14I) shows that as the number of elements increases, the evolution of OER intermediates gradually transitions from a single Ni-O vibration (Ni, NiFe, AEM path) to Ni-OO-Co μ -peroxide species (NiFeCoCr and NiFeCoCrW₀.₂, LOM path). NiFeCoCrW₀.₂ exhibited the strongest 1069 cm⁻¹ Ni-O-Co signal in the full potential range, directly confirming the Ni-O-CO dual-site-dominated LOM mechanism, which corresponds to a significant increase in activity and ph-dependence.

Figure 14.

Atomic-scale mechanism and operando spectroscopic insights into water dissociation and hydrogen evolution on high-entropy alloy catalysts

(A) Atomic configurations of the catalytic sites on FeCoNiMnRu HEA captured at four successive stages of H₂O dissociation.

(B) Reaction-energy profile for water dissociation across distinct catalytic sites of the FeCoNiMnRu HEA surface.

(C) Gibbs free-energy (ΔG_H∗) diagrams for hydrogen adsorption on the same catalytic sites.

(D–F) Operando electrochemical-Raman spectra of FeCoNiMnRu/CNFs acquired during HER in 1.0 M KOH.¹¹⁹ Copyright 2025, Springer Nature.

(G) Schematic of the in situ Raman cell configuration.

(H–L) Voltage-dependent Raman spectra (1.1–1.6 V vs. RHE, 1 M KOH) for Ni (H), NiFe (I), NiFeCo (J), NiFeCoCr (K), and NiFeCoCrW₀.₂ (L).¹²⁰ Copyright 2022, Springer Nature.

Based on the breakthrough in the research of the dual-site cooperative mechanism of high-entropy alloys, the future in situ characterization techniques will make a leapfrog development toward high spatiotemporal resolution multimodal integration and artificial intelligence-driven deep analysis.

• (1)Nanometer-millisecond level dynamic tracking: Develop in situ electrochemical TERS technology with sub-5 nm resolution, combined with femtosecond-level ultrafast XAS/RAU application, to real-time image the spatial distribution of specific intermediate species (such as Co-O∗/Ru-H∗) on the surface of HEA, and capture the generation kinetics of transient oxygen species (∗OOH);
• (2)Extreme operating condition analysis: Build an in situ XPS platform resistant to strong acids/alkalis and a high-temperature/high-pressure (>80°C, 5 bar) spectroscopic cell, to quantitatively monitor the evolution of multiple element valence states and the formation mechanism of Cr/Mo passivation layer under industrial current density (>500 mA/cm²);
• (3)Intelligent spectral disentanglement revolution: Through machine learning to decouple overlapping signals in complex spectra (such as Fe-O/Ni-O vibrations), integrate multimodal data streams to establish dynamic structure-activity relationship models, guiding the design of acid-resistant non-precious metal HEA catalysts;
• (4)Cross-scale correlation at the device level: Utilize embedded micro-sensors and cryo-electron microscopy technology to reveal the failure mechanisms from atomic distortion (lattice strain) to mesoscopic structure (reconstruction layer shedding), ultimately achieving a closed-loop research paradigm of rational design—condition verification—failure traceability.

Current challenges and future directions

Although high-entropy alloys (HEAs) have shown great application potential in multiple fields, there are still some key technical bottlenecks in their research and development process. Among them, the trade-off between computational accuracy and efficiency, as well as the insufficiency of dynamic interface modeling, are two main challenges.

Technical bottleneck

From a theoretical-computation perspective, HEA catalysts outperform conventional alloys in three overarching dimensions: “complexity under control, statistical averaging, and clear structure-property mapping.” Specifically, the advantages can be articulated as follows.

• (1)Statistical active-site descriptors: The near-equimolar, multi-principal-element surface of HEAs generates an astronomical number of local configurations that cannot be enumerated individually. Theoretical treatments therefore employ joint distributions of generalized coordination numbers and element-pair probabilities to collapse the configurational space into a statistically tractable ensemble, whereas traditional alloys—with only a few ordered motifs—lack such a statistical framework.
• (2)Continuously tunable electronic-structure window: DFT reveals that the d-band centers of HEAs shift linearly with elemental composition and ratio, enabling efficient prediction of adsorption energies via virtual-crystal approximations or special quasi-random structures. Conventional alloys, limited to narrow compositional ranges, exhibit discrete d-band jumps that preclude continuous optimization.
• (3)Quantitative models of synergistic effects: Charge transfer and strain coupling between neighboring dissimilar metals in HEAs can be unified within effective-medium theory, elucidating multi-element synergy in lowering reaction barriers. Traditional alloys confine such synergy to bimetallic interfaces and cannot capture higher-order cooperativity.
• (4)High-throughput stability screening: Phase-stability criteria based on enthalpy-entropy competition allow rapid computational screening for HEA compositions that resist segregation and oxidation. Traditional alloys require individual calculation of ordered-phase formation energies, with computational cost scaling exponentially with the number of components.
• (5)Machine-learning-DFT closed-loop acceleration: The vast compositional space of HEAs is ideally suited to machine-learning regression of adsorption energies and volcano plots; only a small set of representative DFT labels is needed. Traditional alloys, with limited configurational diversity, gain comparatively little from such ML workflows.

The complex chemical composition and structural characteristics of high-entropy alloys make their calculation and simulation extremely complex. The core predicament currently faced by the application of DFT in high-entropy alloys lies in that as the number of principal components increases, the chemical disorder leads to an exponential expansion of the configuration space, and conventional supercell sampling can no longer be exhaustive. At the same time, to ensure accuracy, a large system and high K-point density must be adopted, resulting in a sharp increase in computational load O(N³), while simplified SQS or cluster expansion sacrifices reliability. More importantly, the existing linear scaling or QM/ML hybridization algorithms have not been deeply optimized for multi-principal component systems, resulting in low parallel efficiency, large memory bottlenecks, and the lack of a unified open-source platform and automated workflow. This limits both “accuracy and efficiency”, seriously hindering the high-throughput screening and design of large-scale HEA. High-precision calculation methods, such as the HSE06 functional, although they can provide more accurate electronic structure and energy calculation results, their calculation costs and time consumption also increase significantly. This high computational cost limits its application in large-scale and high-throughput screening, especially when exploring multi-component systems. Therefore, how to improve the calculation efficiency while ensuring the calculation accuracy is one of the urgent problems to be solved at present. Researchers are exploring methods that combine machine learning with high-throughput computing in the hope of completing the screening and optimization of complex systems within a reasonable time.

HEAs often involve complex interface and surface phenomena in practical applications, such as the solid-liquid-gas three-phase interface. However, at present, the modeling of such multi-physics field coupled dynamic interfaces is still in the early stage. The existing simulation methods have limitations in describing the atomic arrangement, electronic structure, and dynamic behavior at the interface, and it is difficult to accurately predict the chemical reactions and physical processes at the interface. This deficiency in modeling ability not only limits the understanding of the interfacial properties of HEAs, but also affects their performance optimization in applications such as catalysis, corrosion, and wear. Therefore, developing a multi-physics coupling model that can accurately describe the behavior of dynamic interfaces is an important direction for promoting the application research of high-entropy alloys.

Maximal breach path

To address the challenges in the research and development of HEAs, a comprehensive strategy integrating multiscale modeling, AI-driven automation, and experimental-computational-data triadic synergy has been proposed. Multiscale modeling establishes a cross-scale framework by combining QM, MD, and continuum models, enabling seamless coupling from atomic to device scales. At the atomic level, quantum mechanical methods such as DFT are employed to calculate interatomic interactions, electronic structures, and defect states, providing insights into fundamental physicochemical properties—for instance, predicting stability, and reactivity through band structure and density of states analysis. MD simulations then leverage these atomic-scale parameters to investigate mesoscale phenomena, including microstructure evolution, diffusion dynamics, and mechanical responses over extended spatiotemporal scales. Continuum models further bridge these results to macroscale properties, such as thermal, electrical, and mechanical performance, achieving hierarchical performance prediction. Experimental validation using advanced characterization techniques (e.g., TEM, XRD, and APT) ensures model accuracy and guides iterative optimization.

AI-driven automation revolutionizes the entire HEA development workflow. A unified platform architecture integrates GANs for composition design, GNNs for synthesis pathway prediction, and reinforcement learning for performance optimization. These models are trained on extensive datasets encompassing experimental results, computational outputs, and historical literature, enabling intelligent iteration. For example, GANs generate novel compositions by learning composition-property relationships, while GNNs predict optimal synthesis conditions by correlating process parameters with target properties. Data mining techniques extract implicit knowledge, such as elemental synergy and process-property correlations, to refine AI models and enhance predictive capabilities.

Experimental-computational-data triadic synergy is achieved through open-source database expansion, standardized data sharing, and data-driven optimization. Enhanced databases, such as materials project with dedicated HEA modules, facilitate global data accessibility and interoperability. High-throughput experiments rapidly synthesize and characterize diverse HEA samples, generating rich datasets that validate computational predictions and inform model refinement. Concurrently, computational simulations guide experimental priorities—for instance, identifying catalyst candidates with predicted high activity for targeted validation. Data-driven models, built via machine learning and statistical analysis, uncover hidden trends and accelerate performance optimization through closed-loop feedback between experiments and simulations.

This integrated strategy—combining multiscale modeling precision, AI-driven automation efficiency, and triadic synergy robustness—transcends traditional trial-and-error limitations. It not only accelerates the discovery and application of HEAs in structural and functional materials but also establishes a paradigm for understanding complex structure-property relationships in multicomponent systems, paving the way for next-generation high-performance materials.

Techno-economic

HEAs demonstrate transformative potential in techno-economic applications by leveraging multi-element synergy to achieve a triple breakthrough in performance enhancement, cost reduction, and resource sustainability.

Noble metal reduction and substitution

Ir/Pt-containing HEAs minimize noble metal content to 5–20 at.% (e.g., PtRuRhCoNi). The HEA@Ir-MEO electrolyzer achieves a levelized hydrogen cost of $0.88/kg H₂ (energy consumption: 3.98 kWh/Nm³), reducing expenses by >30% compared to conventional systems. Noble metal-free HEAs (e.g., FeCoNi-based) deliver acidic OER overpotentials (286 mV) comparable to IrO₂ while exhibiting 5× enhanced stability (metal leaching <1%).

Life cycle cost optimization

High-entropy effects suppress elemental dissolution (e.g., <5% activity decay in FeCoNiIrRu after 500 h operation). Combined with poison-resistant self-cleaning surfaces, this reduces maintenance and regeneration costs. Multi-metal recovery technologies (hydrometallurgy coupled with ML optimization) achieve >95% metal reclamation from spent catalysts, decreasing raw material dependency.

Resource and energy efficiency

Abundant transition metals (Fe/Co/Ni/Cu) replace scarce elements, while laser-induced synthesis reduces energy consumption by 40% versus smelting. Dual-site cooperative mechanisms (e.g., Co-mediated water dissociation coupled with Ru-driven H⁺ desorption) lower the water dissociation barrier to 0.34 eV, boosting energy conversion efficiency by 15%. The NiFeCoCrW₀.₂ HEO further enables >80% NO₂⁻ selectivity in NRR, streamlining downstream separation processes.

Future advances require low-temperature electrochemical deposition and closed-loop recycling systems to further minimize manufacturing costs, accelerating HEA deployment in green hydrogen production and CO₂ conversion.

Conclusion

HEAs have attracted extensive attention due to their significant potential in structural and functional applications. This review systematically summarizes the latest research progress of HEAs in the field of electrocatalysis, covering core aspects, such as design principles, synthesis methods, performance optimization, and theoretical computational analysis. DFT calculations reveal the microscopic mechanism by which metal sites on the surface of HEAs optimize reaction pathways through multi-element synergy, providing theoretical support for catalyst design. At the same time, machine learning techniques have provided breakthrough tools for the rapid screening and performance prediction of HEAs—graph neural networks based on geometric descriptors (such as ACE-GCN, AGAT) achieve high-precision prediction of adsorption energy (MAE <0.08 eV), and through the attention mechanism, they analyze the regulatory rules of local chemical environment on catalytic activity. Although significant progress has been made, the research on HEAs still faces challenges such as the balance of computational accuracy and efficiency, and the insufficiency of dynamic interface modeling. In the future, a “experimental-computational-data” tripartite synergy is needed to deepen basic research and promote the practical application process.

Acknowledgments

This work was supported by the Key Scientific and Technological Program of Gansu Province (22ZD6NA046, 24ZD13NA019).

Declaration of interests

The authors declare that they have no conflict of interest.