Valence Electron Concentration Research Articles

Discovering superhard high‐entropy diboride ceramics via a hybrid data‐driven and knowledge‐enabled model

AbstractMaterials descriptors with multivariate, multiphase, and multiscale of a complex system have been treated as the remarkable materials genome, addressing the composition–processing–structure–property–performance (CPSPP) relationships during the development of advanced materials. With the aid of high‐performance computations, big data, and artificial intelligence technologies, it is still a challenge to derive an explainable machine learning (ML) model to reveal the underlying CPSPP relationship, especially, under the extreme conditions. This work supports a smart strategy to derive an explainable model of composition–property–performance relationships via a hybrid data‐driven and knowledge‐enabled model, and designing superhard high‐entropy diboride ceramics (HEBs) with a cost‐effective approach. Five dominate features and optimal model were screened out from 149 features and nine algorithms by ML and validated in first‐principles calculations. From Shapley additive explanations (SHAP) and electronic bottom layer, the predicted hardness increases with the improved mean electronegativity and electron work function (EWF) and decreases with growing average d valence electrons of composition. The 14 undeveloped potential superhard HEBs candidates via ML are further validated by first‐principles calculations. Moreover, this EWF‐ML model not only has better capability to distinguish the differences of solutes in same group of periodic table but is also a more effective method for material design than that of valence electron concentration.

Theoretical analysis and design of Ti-based shape memory alloys correlating composition and electronic properties to transformation temperatures for high temperature applications

The influence of electronic parameters, such as the atomic number of the alloy (Z), valence electron concentration (cv), number of valence electrons per atom (ev/a ratio), and pseudopotential radius (P), on the phase transformation temperatures (Ms, As) of binary, ternary and quaternary shape memory alloys (SMAs) based on Ti is investigated. These temperatures (Ms, As) are correlated with the composition and ev/a of the alloys using multiple linear regression. The dependence of thermal hysteresis on the atomic radius of the alloys is also studied. The alloys investigated comprise of both low (ev/a < 5) and medium (5 < ev/a < 7.5) valence electrons groups. The phase transition temperatures are raised with an increase in ev/a ratio, Z and P for medium ev/a group alloys, while for low ev/a group alloys, the trend is quite the opposite. As cv increases, the transformation temperatures (TTs) decrease for the medium ev/a group, while they increase with cv for the low ev/a group. The two different trends observed for low and medium ev/a groups are explained in the light of elastic modulus of the alloys. The effect of Z, cv, electron orbital occupancy and alloying elements on the elastic modulus, and in turn on the temperatures of transition of the alloys, is also discussed.

Metallurgy, superconductivity, and hardness of a new high-entropy alloy superconductor Ti-Hf-Nb-Ta-Re

We explored quinary body-centered cubic (bcc) high-entropy alloy (HEA) superconductors with valence electron concentrations (VECs) ranging from 4.6 to 5.0, a domain that has received limited attention in prior research. Our search has led to the discovery of new bcc Ti-Hf-Nb-Ta-Re superconducting alloys, which exhibit an interesting phenomenon of phase segregation into two bcc phases with slightly different chemical compositions, as the VEC increases. The enthalpy of the formation of each binary compound explains the phase segregation. All the alloys investigated were categorized as type-II superconductors, with superconducting critical temperatures (Tc) ranging from 3.25 K to 4.38 K. We measured the Vickers microhardness, which positively correlated with the Debye temperature, and compared it with the hardness values of other bcc HEA superconductors. Our results indicate that Tc systematically decreases with an increase in hardness beyond a threshold of approximately 350 HV. Additionally, we plotted Tc vs. VEC for representative quinary bcc HEAs. The plot revealed the asymmetric VEC dependence. The correlation between the hardness and Tc, as well as the asymmetric dependence of Tc on VEC can be attributed to the simultaneous effects of the electronic density of states at the Fermi level and electron-phonon coupling under the uncertainty principle, especially in the higher VEC region.

Design of a nickel–cobalt based eutectic high entropy alloy (NiCo)1.7AlCrFe with hierarchical microstructural length scales

ABSTRACT A combined nickel–cobalt-based eutectic high entropy alloy (Ni-Co-based HEA) has been designed based on the valence electron concentration (VEC) value. The microstructure of the developed alloy consists of a face-centered cubic (FCC) and ordered cubic (B2) structure, which was validated using CALPHAD. The microstructure is hierarchical with length scales in micrometers (lamellar eutectic morphology) and nanometers (B2 length scale). B2 phase precipitation was observed in the supersaturated FCC primary phase. The orientation relationship (OR) between the phases exhibited a strong Kurdjimov-Sachs (KS) type OR, typical of FCC/BCC interface structures. This EHEA displayed good yield strength at room temperature and strain rate insensitivity in the 10−3–10−1 strain rate range.

Prediction of glass-forming ability in ternary alloys based on machine learning method

In this paper, six machine learning (ML) models including k-nearest neighbour, adaptive boosting, gradient boosting regression tree, extreme gradient boosting tree, support vector regression (SVR) and random forest were built in the frame of ternary alloys to predict the critical size of glass formation (Dmax) based on 840 alloy data. Composition, atomic size difference, enthalpy of mixing, electronegativity difference, and valence electron concentration were selected as input features. The features were selected by Pearson correlation analysis and target value of Dmax was transformed into the logarithmic form. The determination coefficient and root mean square error were used to evaluate the prediction performance of models. The result showed that the SVR model has a good generalization ability. Additionally, the prediction ability of SVR was verified in Zr−Al−Co and La−Al−Co alloy systems. It is thus suggested that ML has great potential in predicting and pinpointing the amorphous composition with good glass-forming ability.

Valence electron concentration- and N vacancy-induced elasticity in cubic early transition metal nitrides

Motivated by frequently reported deviations from stoichiometry in cubic transition metal nitride (TMNx) thin films, the effect of N-vacancy concentration on the elastic properties of cubic TiNx, ZrNx, VNx, NbNx, and MoNx (0.72≤x≤1.00) is systematically studied by density functional theory (DFT) calculations. The predictions are validated experimentally for VNx (0.77≤x≤0.97). The DFT results indicate that the elastic behavior of the TMNx depends on both the N-vacancy concentration and the valence electron concentration (VEC) of the transition metal: While TiNx and ZrNx exhibit vacancy-induced reductions in elastic modulus, VNx and NbNx show an increase. These trends can be rationalized by considering vacancy-induced changes in elastic anisotropy and bonding. While introduction of N-vacancies in TiNx results in a significant reduction of elastic modulus along all directions and a lower average bond strength of Ti–N, the vacancy-induced reduction in [001] direction of VNx is overcompensated by the higher stiffness along [011] and [111] directions, resulting in a higher average bond strength of V–N. To validate the predicted vacancy-induced changes in elasticity experimentally, close-to-single-crystal VNx (0.77≤x≤0.97) are grown on MgO(001) substrates. As the N-content is reduced, the relaxed lattice parameter a0, as probed by X-ray diffraction, decreases from 4.128 Å to 4.096 Å. This reduction in lattice parameter is accompanied by an anomalous 11% increase in elastic modulus, as determined by nanoindentation. As the experimental data agree with the predictions, the elasticity enhancement in VNx upon N-vacancy formation can be understood based on the concomitant changes in elastic anisotropy and bonding.

On europium digallides

AbstractThe europium gallide Eu1‐xGa2+3x (x is between 0.06 and 0.07) has been synthesized and its crystal structure has been refined from single crystal X‐ray diffraction data. Eu1‐xGa2+3x crystallizes with hexagonal structure related to the AlB2‐type (space group P6/mmm, a=4.354 Å, c=4.485 Å). The structure refinement finished with R(F)=0.015 for 1603 reflections revealing a defect occupation of the Eu position (93 %) and additional Ga position, yielding the non‐integer element ratio. From magnetic susceptibility measurements, EuGa2 and Eu1‐xGa2+3x phases show the 4f 7 (Eu+2) oxidation state. The analysis of effective atomic charges within the Quantum Theory of Atoms in Molecules reveals the formation of four‐bonded Ga species with low electron demand (in comparison to the three‐bonded in AlB2‐type and four‐bonded in the KHg2‐type structures) and shows, that Eu1‐xGa2+3x represents a new way of stabilizing intermetallic structures formed at valence electron concentration, which does not allow to apply Zintl‐like electron counting.

Alloying Effect on Elastic and Mechanical Properties of Refractory Medium-Entropy Alloys from First-Principles Calculations

There is still a challenge to rapidly and accurately acquire the composition dependence of mechanical properties of multi-principal element alloys due to their huge composition space and complex electronic structures. Aiming to straightforwardly tune the mechanical properties through alloying, we systematically calculate the elastic and mechanical properties for four selected body-centered-cubic (bcc) Ti–Nb–M (M = Zr, Mo, Sn, Ta) refractory medium-entropy alloys (RMEAs) using a first-principles method. It is shown that the elastic stability of the bcc RMEAs can be significantly (weakly) improved with the increase in Nb, Mo, and Ta (Sn) content, but it is insensitive to Zr. The elastic stability, elastic modulus, hardness, and strength generally enhance, whereas Pugh’s ratio and Zener anisotropy (AZ) decrease with the increase in valence electron concentration. Additionally, there is a correlation between the magnitude of the AZ and the shape of the single-crystal Young’s modulus (E[hkl]). The largest (smallest) E[hkl] appears in the <111> (<100>) direction. Particularly, the most isotropic Ti60Nb20Mo20 has the highest hardness and strength among all the Ti–Nb–M alloys. Moreover, the change of elastic properties induced by alloying is related to the change of density of states. This work sheds deep light on the design of high-performance Ti-based multi-principal element alloys.

Evolution of the Phase and Hydrogen Storage Properties of New-Type VTiCrFeMn High-Entropy Alloy Prepared via Mechanical Alloying

High-entropy alloys (HEAs) have ample space for composition design, so finding an ideal hydrogen storage material is a great challenge. This work used the atomic radius difference, valence electron concentration, entropy, enthalpy, and new design parameters to develop a new-type alloy with a BCC structure. The V35Ti25Cr20Fe10Mn10 alloy was selected by semi-empirical formula calculation, subsequently, the alloy was successfully synthesized by mechanical alloying, and hydrogen storage properties were evaluated. The activated sample shows an excellent kinetic performance of hydrogen absorption, but its maximum hydrogen storage capacity is only about 0.34 wt.%. The results of the PCI test show that the hydrogen absorption way may be composed of the H-solid solution in the alloy and the formation of a few hydrides, which determines the hydrogen storage capacity. Significantly, the design method reported in this paper can meet the composition design of most HEAs, which has a broad application prospect.

Nanoparticle-strengthened Ni2CoCrNb0.2 medium-entropy alloy with an ultrastrong cryogenic yield strength fabricated by additive manufacturing

To improve the yield strength of metallic materials at low temperatures, a strategy of combining the calculation of phase diagrams (CALPHAD) technique with the overall valence electron concentration (OVEC) principle is applied, and a Ni2CoCrNb0.2 medium-entropy alloy (MEA) with D022 superlattice (noted as the γ′′ phase) is designed. Bulk MEA samples without defects were successfully fabricated using laser additive manufacturing (AM), followed by solution treatment at 1200 °C for 1 h and then aging at 650 °C for 120 h. The nanoscale γ′′ phase precipitated. The tensile results indicated that the MEA had superior yield strengths of ∼1180 MPa and ∼1320 MPa and tensile strengths of ∼1335 MPa and ∼1552 MPa at 293 K and 77 K, respectively. The yield strength obtained was superior to that of currently reported medium/high-entropy alloys and typical advanced cryogenic steel. The mechanical properties of the Ni2CoCrNb0.2 MEA demonstrated a strong temperature dependence, and the increased yield strength was mainly attributed to the increase in lattice friction stress at low temperatures. This research provides a new strategy for producing materials with ultrastrong cryogenic yield strengths by AM.

Dynamic compression behavior of TiZrNbV refractory high-entropy alloys upon ultrahigh strain rate loading

In this study, the dynamic compressive response behavior of a body-centered cubic (BCC) single-phase TiZrNbV refractory high-entropy alloy (RHEA) was investigated under impact at speeds of 313–1584 m s–1 using two-stage, gas-gun-driven, high-speed plate-impact experiments; recovery sample analysis; and theoretical calculations. The strain rate and pressure were approximately 107 s−1 and 5.07–29.37 GPa, respectively. The results showed that the TiZrNbV RHEA had a Hugoniot elastic limit of 4.12–5.86 GPa and a spall strength of 1.84–2.03 GPa. The initial yield strength of the alloy showed a strong strain-rate dependence and could be described by the modified Zerilli–Armstrong model, while the phonon-damping effect was the main reason for its high strain-rate sensitivity. Microstructural analysis showed that the dynamic deformation of the TiZrNbV RHEA was controlled by the dislocation slip, dislocation proliferation, intersection of the deformation bands, and grain refinement. The analysis also showed that the intergranular, transgranular, and mixed-type cracks dominated the spall failure of the material. The dynamic Hall–Petch effect and pinning from the lattice distortion led to high dynamic yield strength. The critical strain rate for the phonon drag effect was positively related to the relative atomic mass and local strain field of the metals. Within the experimental loading range, the RHEA showed good structural stability, and simultaneously, the theoretical calculation method for the equation of state based on a cold-energy mixture could accurately predict its shock-response behavior. The valence-electron concentration (VEC) had a direct effect on the shock-compression properties of the HEAs; higher VEC implied more difficulty in compressing the HEAs. The findings of this study provide insights into understanding the mechanical response characteristics of RHEAs under extreme conditions such as high-speed impact and ultrahigh strain-rate loading.

Searching for medium entropy alloys with desired mechanical property by adaptive design combined with experiments

To overcome the problems caused by the “trial and error” method in the vast element composition space and to rapidly design and discover new medium entropy alloys (MEAs) with required mechanical properties, an efficient data-driven machine learning (ML) strategy based on random forest regression (RFR) algorithm was proposed, utilizing existing data. The alloys were represented by chemical composition and additional features composed of material descriptors. Through correlation analysis, recursive elimination, and exhaustive screening, three key additional features from 20 material descriptors were identified, including six power of work function (W6), valence electron concentration (VEC), and mean Yong's modulus (E). Furthermore, an Efficient Global Optimization (EGO) algorithm was adopted to search for the optimal alloys to guide synthesis. After evaluating >448,000 kinds of alloy systems, the most strongly recommended alloy Al52Co24Cr18Ni4Mn2 with the highest predicted hardness value of 802.5 HV was determined. Interestingly, experimental validation of the synthesized alloy revealed mainly BCC and ordered B2 structure, with a hardness of 815.4 HV, which is approximately 5.2% higher than the highest value in the original dataset. Given the independence of the problem, the proposed approach could be extended to other potential multi-component regression problems.

Microstructure, hardness and wear performance of quaternary CrNbTiZr refractory medium-entropy alloy coating fabricated on a commercial Zr alloy by pulsed laser cladding

To attain enhanced surface hardness and wear resistance, a quaternary CrNbTiZr refractory medium-entropy alloy (RMEA) coating was fabricated on a commercial Zr alloy (Zr-702) by a pulsed laser cladding technique. Phase constitution, microstructural feature, hardness and wear performance of the RMEA coating were systematically investigated. The specific influence of Zr diluted from the substrate on the coating was explored. The results reveal that the cladding zone is mainly featured by a BCC solid-solution phase and net-like Cr2Zr intermetallics. Such microstructural features agree with those predicted based on atomic radius difference, Allen electronegativity and valence electron concentration. Allotropic solid phase transformation occurs in the heat-affected zone, resulting in a mixed microstructure of blocky grains (α-Zr) and martensitic laths. The hardness of the RMEA coating (637 ± 46 HV) is ∼3.3 times that of the Zr-702 substrate, which results from joint contribution from solid-solution, low-angle grain boundary and second-phase strengthening. The RMEA coating exhibits a wear rate ∼56% lower than the substrate, demonstrating significantly improved wear resistance. Both oxidation and adhesive wear are determined to be dominant wear mechanisms of the RMEA coating, accounting for its enhanced wear performance.

Revisiting the phase stability rules in the design of high-entropy alloys: A case study of quaternary alloys produced by mechanical alloying

High-entropy alloys (HEAs) are potential candidates for many structural applications because of their outstanding mechanical properties. Achieving solid solution (SS) in HEAs is desirable since intermetallic compounds (IMs) can, in most cases, be deleterious. While empirical phase stability rules have been widely used to predict SS or IM phases in HEAs, there are substantial reported cases of their breakdown. To assess the effectiveness of the empirical rules—atomic size difference (δ), mixing enthalpy (ΔHmix), mixing entropy (ΔSmix), entropy to enthalpy ratio (Ω), electronegativity difference (△χ), and valence electron concentration (VEC), a systematic study that isolates the effect of processing pathway was conducted. As a starting point and as a benchmark, an existing equiatomic AlCoCrFe BCC HEA was correctly predicted (following the classical empirical rules) and developed. This was followed by the SS FCC prediction and development of five novel semi-equiatomic AlTiCuZn-based quaternary HEAs (Al0.57TiCuZn, AlTi0.45CuZn, Al0.45TiCuZn, AlTiCu1.76Zn, and AlTiCu2Zn) by mechanical alloying. Among the AlTiCuZn-based HEAs, only AlTi0.45CuZn showed SS, but with FCC + HCP phase contradicting the traditional VEC rule that predicts FCC. Using AlTi0.45CuZn as a guide, conservative SS formation criteria for AlTiCuZn-based HEAs were determined—Ω≥ 1.6, δ≤ 5%, and ΔHmix≥ -8 kJ/mol, while retaining conventional ΔSmix and △χ range. Also, FCC + HCP phase formation is stable at 7.5 ≤VEC≤ 8.4. The revised rules were verified by correctly predicting the phases in newly-fabricated non-equiatomic HEAs—AlTi0.37CuZn0.97 and AlTi0.56Cu1.24Zn1.2. This study shows that the range of empirical phase stability model values is not fixed, but alloy-dependent due to the differences in bonding nature of HEA-constituting elements.

Theoretical study of thermodynamic and magnetic properties of transition metal carbide and nitride MAX phases

We systematically perform density-functional theory (DFT) calculations for all possible ${\mathrm{M}}_{n+1}{\mathrm{AX}}_{n}$ (MAX) phases with transition metal M=Sc to Au (excluding Tc), A in group IIIA-IVA, X=C, N, and $n=1,2,3$, a total of about 1200 systems. The thermodynamic stability is determined by comparing the formation enthalpy (at 0 K) against all possible combinations of unary, binary, and ternary boundary phases (available from online DFT databases). Thereby, we identify 124 so far unknown phases (in terms of both experimental synthesis and other theoretical predictions), of which 54 are carbides and 70 are nitrides. Among all stable MAX phases, we identify nine with magnetic properties. In addition to already known and synthesized magnetic phases $({\mathrm{Cr}}_{2}\mathrm{AlC},$ ${\mathrm{Cr}}_{2}\mathrm{GeC},$ ${\mathrm{Cr}}_{2}\mathrm{GaN},$ and ${\mathrm{Mn}}_{2}\mathrm{GaC})$, we predict five more MAX phases with magnetic ordering $[{\mathrm{Mn}}_{2}\mathrm{A}(=\mathrm{Ge}, \mathrm{Sn})\mathrm{C}, {\mathrm{Cr}}_{3}\mathrm{A}(=\mathrm{Ga}, \mathrm{Ge}){\mathrm{N}}_{2}, \mathrm{and} {\mathrm{Cr}}_{4}\mathrm{A}(=\mathrm{Ge}){\mathrm{N}}_{3}]$. Evaluating previously suggested descriptors for the stability of MAX phases [valence electron concentrations (VECs), differences in atomic radius difference $\mathrm{\ensuremath{\Delta}}{R}_{\mathrm{at}}$, and differences in electronegativities $\mathrm{\ensuremath{\Delta}}\ensuremath{\chi}$], we find that $\mathrm{\ensuremath{\Delta}}{R}_{\mathrm{at}}$ does not correlate with stability and stable phases are characterized by $\text{VEC}&lt;5.5$, $\mathrm{\ensuremath{\Delta}}\ensuremath{\chi}&gt;1.5$. The reverse is, however, not true; for example, a MAX phase with $\text{VEC}&lt;5.5$ and $\mathrm{\ensuremath{\Delta}}\ensuremath{\chi}&gt;1.5$ is not necessarily stable.

Thermodynamic evaluation of the phase stability in mechanically alloyed AlCuxNiCoTi high-entropy alloys

This study investigates the phase stability of AlCuxNiCoTi high-entropy alloys with varying amounts of copper. The regular solution and Kohler’s models are used to thermodynamically evaluate the alloys and determine a symmetrical enthalpy across their compositions. The study calculates the Gibbs energy of mixing for a quinary alloy system using the extended Miedema theory and considering the change in enthalpy of mixing of the solid-solution and amorphous phases of 10 binary alloys. By comparing the experimental and predicted results, this study suggests that thermodynamic properties can help predict stable phases in multicomponent alloys comprising equiatomic and near-equiatomic elements. Furthermore, the structural instability of the solid-solution state resulting from differences in valence electron concentrations of the constituent elements in binary mixtures is discussed, highlighting the influence of alloying elements on the phase stability of multicomponent alloys.

Development of Materials for Solid State Hydrogen Storage

Series of high entropy alloys designed on Hume-Rothery criterion was prepared and the probability of the empirical approach to hydrogen storage materials preparation was investigated. Calculated HEA’s with equimolar compositions were selected from the list of alloys with limited VEC (valence electron concentration), ΔS and ΔH. The phase composition of prepared materials was compared with the prediction model and material characteristics such as chemical composition, and phase composition were studied. In this article material characterization of predicted high-entropy alloys with various prediction parameters values will be presented in terms of empirical prediction methods for solid-state hydrogen storage materials.

Estimation of Shear Modulus and Hardness of High-Entropy Alloys Made from Early Transition Metals Based on Bonding Parameters

The relationship between the tendencies towards rigidity (measured by shear modulus, G) and hardness (measured by Vickers hardness, HV) of early transition metal (ETM)-based refractory high-entropy alloys (RHEA) and bond parameters (i.e., valence electron concentration (VEC), enthalpy of mixing (ΔHmix)) was investigated. These bond parameters, VEC and ΔHmix, are available from composition and tabulated data, respectively. Based on our own data (9 samples) and those available from the literatures (47 + 27 samples), it seems that for ETM-based RHEAs the G and HV characteristics have a close correlation with the bonding parameters. The room temperature value of G and HV increases with the VEC and with the negative value of ΔHmix. Corresponding equations were deduced for the first time through multiple linear regression analysis, in order to help design the mechanical properties of ETM refractory high-entropy alloys.

Synthesis and evaluation of rocksalt structure (Mo, W) carbides via vanadium additions

The mechanical properties of rocksalt (B1) binary, ternary, and high entropy transition metal ceramics exhibit strong correlation with their valence electron concentration (VEC). With respect to maximizing ductility and hardness, it has been proposed that materials with VEC values between 9 and 10 will generally possess superior performance. While many of the transition metal cations inherently crystallize in the B1 structure, the VEC=10 Group-VIB elements don't form room temperature stable B1 carbides. This work demonstrates the ability to stabilize bulk, spark plasma sintered samples of the Group-VIB containing carbides with VEC values in this range. The stabilization of molybdenum and tungsten carbide as room-temperature B1 structures via the addition of different amounts of vanadium and nanoindentation measurements correlated with VEC are the primary goals of this work. The minimum atom percent vanadium to form a single-phase B1 carbide is thoroughly explored and verified via a combination of XRD, EDS, and EBSD.

Predicting single-phase solid solutions in as-sputtered high entropy alloys: High-throughput screening with machine-learning model

Searching for single-phase solid solutions (SPSSs) in high-entropy alloys (HEAs) is a prerequisite for the intentional design and manipulation of microstructures of alloys in vast composition space. However, to date, reported SPSS HEAs are still rare due to the lack of reliable guiding principles for the synthesis of new SPSS HEAs. Here, we demonstrate an ensemble machine-learning method capable of discovering SPSS HEAs by directly predicting quinary phase diagrams based only on atomic composition. A total of 2198 experimental structure data are extracted from as-sputtered quinary HEAs in the literature and used to train a random forest classifier (termed AS-RF) utilizing bagging, achieving a prediction accuracy of 94.6% compared with experimental results. The AS-RF model is then utilized to predict 224 quinary phase diagrams including ∼32,000 SPSS HEAs in Cr-Co-Fe-Ni-Mn-Cu-Al composition space. The extrapolation capability of the AS-RF model is then validated by performing first-principle calculations using density functional theory as a benchmark for the predicted phase transition of newly predicted HEAs. Finally, interpretation of the AS-RF model weighting of the input parameters also sheds light on the driving forces behind HEA formation in sputtered systems with the main contributors being: valance electron concentration, work function, atomic radius difference and elementary symmetries.