1. Introduction
Metallic alloys with superior mechanical properties at elevated temperatures remain in high demand for engineering applications, including gas turbines, nuclear reactors, and aerospace propulsion systems. Due to the intrinsic limitations caused by the melting point of materials, the temperature-bearing capacity of traditional nickel (Ni)-based superalloys is approaching the service limit. Inspired by the need to develop high-temperature (HT) structural materials, refractory high-entropy alloys (RHEAs) were introduced in 2010
[1] and have attracted attention due to their ability to retain high strength at and above 1000 °C
[2]. RHEAs are composed of several alloying components (typically four or more), with concentrations ranging from 5 to 35 atomic percent (at%). With the inclusion of refractory elements with high melting points from group 4 (Ti, Zr, and Hf), group 5 (V, Nb, and Ta), and group 6 (Cr, Mo, and W), several RHEAs have already demonstrated HT strength with that of superalloys in performance
[3]. In addition, the structural stability originating from the high entropy effect makes RHEAs highly desirable for use at elevated temperatures. However, most RHEAs suffer from a lack of ductility, with a compressive fracture strain < 10% at room temperature (RT), and thus are not machinable. In fact, RT brittleness and machinability are significant bottlenecks in the development of RHEAs, limiting their applications as structural materials
[4]. Therefore, achieving a simultaneous improvement in HT strength and RT ductility is a primary technical challenge that must be achieved in order for RHEAs to replace Ni-based superalloys
[5].
The addition of refractory elements to RHEAs typically improves their HT strength but degrades their RT ductility. Several efforts have been made to develop RHEAs with two target properties that trade off with each other. NbMoTaW(V) alloy offers good HT strength but poor RT ductility
[6], while TaNbHfZrTi alloy offers excellent RT ductility but weak HT strength
[7]. Han et al.
[8],
[9] added titanium (Ti) to synthesize TiNbMoTaW(V) with improved strength and ductility compared with the two base alloys by controlling the Ti content. Juan et al.
[10] designed RHEAs by adding molybdenum (Mo) to TaNbHfZrTi to improve its yield stress at 1000 °C. Similar work on the property optimization of RHEAs through composition modification has been undertaken for Al
xTiZrHfNbTa
[11], HfMo
xNbTaTiZr
[12], Al
xNbTaTiV
[13], V
xNbMoTa
[14], and HfNb
xTa
0.2TiZr
[15]. Given that the improvement of one property always comes at the expense of the other, the design of RHEAs by tuning the composition of certain elements is mostly dependent on experience and intuition. In turn, the complex composition of RHEAs and their enormous search spaces severely restrict the further development of promising alloy systems.
Theoretical models and empirical rules have been utilized to predict the strength and ductility of RHEAs via high-throughput composition screening and rapid design. For the design of RHEAs with HT strength, Maresca and Curtin
[16] studied the mechanistic origin of high strength in RHEAs with a single-phase body-centered cubic (BCC) structure and developed a solid solution strengthening theory for strength prediction. They then used this theory to perform a computationally guided search across 10
7 BCC high-entropy alloys (HEAs) to identify alloy compositions with potentially high HT strength
[17]. Their work presented over 10
6 possible ultra-strong RHEAs for future exploration, which remains a vast space to search. Moreover, RT ductility was not considered, nor was the experimental synthesis of the alloys.
For ductility design in RHEAs, Sheikh et al.
[18] proposed a valence electron concentration (VEC) rule suggesting that single-phase BCC RHEAs composed of elements from groups 4, 5, and 6 should have high toughness when VEC ≤ 4.4 but exhibit RT brittleness when VEC ≥ 4.6. Based on an analysis of the RT properties of reported RHEAs, Senkov et al.
[19] further noted that there is a dependence between the RT ductility and VEC of RHEAs, but there is no clear boundary of VEC values separating toughness from brittleness. At present, there is a lack of adequate physics-based models or criteria to guide the rational design of RHEAs, especially regarding their multi-objective properties. Furthermore, given a palette containing more than ten principal elements (an unexplored space with billions of alloy compositions), the rapid design of RHEAs with improved performance is difficult using traditional methods.
Recently, there has been considerable interest in the use of machine learning (ML) methods to address complex problems in materials science
[20],
[21],
[22]. In particular, ML techniques have been widely employed to accelerate composition design
[23],
[24],
[25]. ML-based approaches have also been applied to study materials with multiple objectives, including copper alloys optimized for tensile strength and electric conductivity
[26], ferroelectric materials designed for good structural stability and desired Curie temperatures
[27], and superalloys designed for three or more performance requirements
[28],
[29]. Using ML to assist in the design of materials with multi-objective optimization (MOO) on demand, Gopakumar et al.
[30] proposed an approach to guide composition selection in order to improve the search efficiency of target materials with desired properties in a vast space. Guo et al.
[31] presented a method that integrated molecular dynamics simulations, ML models, and genetic algorithms to design the stiffness and critical resolved shear stress of the CoNiCrFeMn HEA system. Khatamsaz et al.
[32] used Bayesian optimization to explore the MoNbTiVW alloy system with density-functional-theory-derived ductility indicators.
Identifying RHEAs with optimal strength and ductility—properties that typically compete with each other—is essentially a MOO problem. To solve it, three issues need to be addressed:
(1) Rapid prediction or evaluation of the target properties to be optimized. Given the lack of empirical rules and valid physics-based models, the prediction of the strength and ductility of RHEAs can be implemented by ML models, which can be efficient, fast, and reliable in predicting the dependence of properties on alloy composition.
(2) Efficient and optimized searching within the vast space of compositions. Given the large unexplored space of RHEAs, effectively searching for compositions with desired properties is essential, as an exhaustive search is intractable due to limited computational resources. Metaheuristic algorithms, such as the genetic algorithm, simulated annealing, and cuckoo algorithm, are ideal for searching in a complex landscape.
(3) Consideration of the balance between antagonistic objectives. In MOO, the corresponding Pareto plot is in two dimensions, where the axes are the properties defining a characteristic locus on which materials lie so that one of the objectives can only be improved in value by degrading the other. The locus or boundary points define a Pareto front (PF) representing the optimal trade-off between multiple objectives—such as the strength and ductility of RHEAs. All the candidate samples on the PF are equally favorable and are expected to provide flexibility for the decision-maker. Thus, selecting the candidate alloys for experimental synthesis and characterization to obtain optimized multiple properties is an important feature to be incorporated.
Our objective is to accelerate the search for RHEAs while simultaneously improving their HT strength and RT ductility—that is, their yield strength at 1000 °C () and fracture strain at RT (). More broadly, we seek to provide an efficient ML-based approach to address common problems in complex materials design: limited data information, a vast composition space, and the need to satisfy multiple property demands. Thus, we employ a MOO framework that combines ML, a genetic search, cluster analysis, and experimental feedback to address these problems in the design of RHEAs and to target the composition space of RHEAs for alloys with optimal strength and ductility.
We synthesized 24 RHEAs, as the result of six iterations with four compounds synthesized and characterized in each, and identified the ZrNbMoHfTa alloy system as having potential for HT applications. More specifically, alloy Zr0.13Nb0.27Mo0.26Hf0.13Ta0.21 shows superior mechanical strength with a yield approaching 940 MPa at 1200 °C and a fracture strain of 17.2% at RT. Its high yield strength at 1200 °C exceeds those of all reported RHEAs, irrespective of RT ductility, while 1200 °C also breaks the service temperature limit for Ni-based superalloys. The prominent heat resistance and good structural stability of this alloy augur well for its structural application at elevated temperatures, while its RT ductility improves the alloy’s workability. This work serves as a basis for the optimization of multiple properties of RHEAs; it can be further applied to accelerate the compositional design of other alloys or materials systems.
2. An ML-based framework for MOO design
Fig. 1(a) schematically shows our MOO strategy for the optimal design of RHEAs, which has a tight correspondence with experiments. We utilize a MOO strategy that was previously employed in the engineering design of a diesel engine combustion chamber
[33],
[34], albeit with the addition of feedback loops involving materials synthesis and characterization. The ML models are independently built and evaluated based on the collected data samples of the two target properties. For a small-data-trained ML model, it becomes essential to consider the uncertainty of the target prediction, especially in a large search space
[20],
[35]. The expected improvement (EI) indicator balances exploitation (i.e., choosing a material with a high predicted target) with exploration (i.e., using the predicted uncertainties to study regions of the search space where the model is less accurate). This process has been demonstrated to be efficient for single-objective optimization in materials design by rationally selecting candidate compositions for experimental feedback
[23],
[24].
Fig. 1(b) compares the material selection using the EI indicator with uncertainty considerations and direct ML prediction. Material candidate #2 is finally selected as a potential optimal composition based on its higher EI utility value, rather than candidate #3, which has the highest ML-predicted property. Thus, we use the EI indicator for the two target properties for the Pareto plot instead of the direct ML-predicted properties, as shown in
Fig. 1(c) (see supplementary data in Appendix A for details). A genetic search has been previously used in the composition design of HEAs
[31],
[36] and Ni-based superalloys
[37]. Here, the calculated EI values based on the ML predictions serve as input to a standard, efficient genetic algorithm—that is, the non-dominated sorting genetic algorithm (NSGA)-II
[38], which performs a heuristic search to yield the PF and fronts of dominated solutions at the end of each genetic iteration after selection, crossover, and mutation. More specifically, after population initialization, we utilize bootstrap sampling to build additional datasets based on the training data. After training the models with the bootstrap samples, the mean of the properties and the associated uncertainties can be obtained to calculate EI values for each alloy. One PF of EI results after several generations, including selection, crossing, and mutations. Finally, the converged optimal PF is obtained after assembling the results from 100 generations and 100 randomly chosen initial populations. (Additional details are provided in the supplementary data.) Subsequently, to guide the synthesis of unexplored compounds, we perform a cluster analysis on the PF to choose alloy candidates from the cluster centers using the
K-means method
[33],
[34],
[39], as shown in
Fig. 1(c). This step allows for an iterative improvement of the ML model by incorporating the measured results into the training dataset. (Details on the clustering analysis for the experimental selection are also included in the supplementary data.) Here, the goal is to identify alloys with an optimal combination of HT strength and RT ductility beyond the two typical RHEA systems TaNbHfZrTi and NbMoTaW(V), as shown in
Fig. 1(c).
3. RHEA design
3.1. Construction of ML models
Data on alloy samples comprising elements from refractory metals belonging to period 4 (Ti, V, and Cr), period 5 (Zr, Nb, and Mo), period 6 (Hf, Ta, and W), and aluminum (Al) were collected from the literature. All alloys were fabricated by means of arc melting to minimize property variation due to material processing. Therefore, the data entries for the initial set consisted of the reported composition (ci) and mechanical properties (y). RHEAs with interstitial element additions (e.g., oxygen, nitrogen, and carbon) were not considered. The collected as-cast alloys consisted of single-phase or multi-phase structures. Accordingly, two independent datasets of and consisting of 54 and 145 alloy samples were respectively assembled (see supplementary data).
Based on the two datasets, we trained ML models to predict the target properties of alloys by establishing relationships between the composition and properties. The molar compositions of the ten elements involved in the collected alloys were directly used as the input features. The two target properties of
and
were the ML model output. Nine models commonly used in regression were considered, which predicted the values for the
and
of RHEAs. The indicators used for the performance of the model included the root mean square error (RMSE), mean absolute error (MAE), and Pearson correlation coefficient,
r2 (see supplementary data for details of model selection). Based on the result plotted in
Fig. 2, the support vector regression model with radial based kernel (svr.r) and random forest (rf) models accurately predicted the yield strength. The correlation coefficient
r2 between the experimental value and the predicted value exceeded 0.85, with the svr.r model showing a lower prediction error (RMSE). Therefore, we selected svr.r model as the final model to evaluate HT yield strength in the subsequent genetic search. For the prediction of the RT fracture strain, the kernel ridge regression (krr), svr.r, and Gaussian process (gp) models yielded good accuracy, with a 0.8 prediction correlation coefficient. The RMSE of the three models was close, and the svr.r model showed the lowest prediction deviation (MAE). We thus selected svr.r as the base model to predict the RT fracture strain during the alloy search. As the diagonal plots in
Figs. 2(c) and
(d) show, there were relatively few outliers with a larger fracture strain, indicating that the model fit the training data well and generalized reasonably well on the test data.
3.2. Experimental procedure
The alloys were prepared by arc-melting mixtures of high-purity elements (> 99.5 weight percent (wt%)) in an argon atmosphere. Each ingot was remelted at least five times to improve its chemical homogeneity. An Instron 9657 universal testing machine was employed to measure the RT mechanical properties at a strain rate of 10–3 s–1 using cylindrical samples 3 mm in diameter and 6 mm in height. HT compression tests were conducted with a thermal-mechanical simulator (Gleeble 3800) using samples with the dimensions of 6 mm in diameter and 9 mm in height. The alloy samples were heated at a rate of 10 °C∙s–1 by electrical resistance heating, soaked at the test temperature for 5 min, and then compressed at a strain rate of 10–3 s–1. The measurements were repeated 2–3 times for each prepared sample. The alloy phase structure was determined by X-ray diffraction (XRD; D8 ADVANCE X) with Cu Kα radiation at a 6 (° )∙s−1 scanning rate.
3.3. Compositional search and experimental feedback
We defined the search space as follows: All combinations of four, five, or six of the ten elements (Al, Ti, V, Cr, Zr, Nb, Mo, Hf, Ta, and W) that appeared in the collected data were considered. The composition of each element varies in the range of 5–35 wt%, with a step size of 1 at%. Thus, the search space comprised approximately 2 billion potential compositions with unknown properties. In each genetic search, a random population containing 500 individuals was first generated from the total search space based on the genetic operations of selection, crossover, and mutation. Furthermore, to eliminate the influence of the random population initialization on the result of such a heuristic search, we performed 100 genetic optimization processes at each iteration to obtain 100 different PFs of EI values. We sorted all the alloys in the 100 PFs to determine the final PF that included many potentially optimized alloy compositions. Experimental selection was implemented by subsequent cluster analysis. The sorting and predicted properties of the alloys changed with iterative feedback from the experiments (for more detail, see Fig. S1 in Appendix A). The candidate alloys were selected from the ZrNbMoHfTa, AlTiZrCrNbMo, TiZrNbMoHfTa, AlTiZrNbHfTa, AlTiZrCrNbTa, TiZrNbHfTa, TiVZrNb, and VCrNbMoHf systems. By analyzing the experimental results of the synthesized alloys in the first five iterations, we found that alloys with desirable properties had a higher frequency of occurrence if they belonged to the ZrNbMoHfTa alloy system. Therefore, for the sixth iteration, instead of searching for alloys in the predefined space mentioned above, we only searched the composition space for the ZrNbMoHfTa system; that is, the composition range of each element was restricted to 5–35 wt% to predict the HT yield strength and RT fracture strain of 553 401 alloys.
The
K-means algorithm was then used to cluster the candidate alloys in the optimized PF of EI values to obtain alloys for experimental synthesis and characterization. Thus, we could iteratively improve the target properties by incorporating the measured results into the training dataset. During selection using
K-means, the number of clusters of samples to be specified was required, as these cluster centers provide candidates for experiments. We used the “elbow method” to determine the number of clusters by calculating the within-group sum of squares (WGSS) from the cluster centers, as the number of clusters varied. The clustering analysis for the experimental selection is included in the supplementary data (Fig. S2 in Appendix A). As the WGSS decreased very slowly after
K = 4, we considered four clusters to be a good fit to the data shown in
Fig. 3. In fact, there are multiple metrics and methods to help determine the best number of clusters, and the result is not always determined. Thus, we tried to give the result of the number of clusters determined by the two other toolboxes, including “NbClust” and “mclust,” which are widely used (shown in the supplementary data). The data showed that a cluster number of 4 was basically robust and appropriate in the present case. Accordingly, we synthesized four alloys at each iteration and performed experimental feedback, finally obtaining 24 alloys after six iterations. The clustering result and the selected alloys with their measured properties are shown in Fig. S3 and Table S2 in Appendix A.
4. Results and discussion
4.1. Improvement in RHEA properties after MOO
The alloy properties in the training dataset (i.e., T-data) are compared with those obtained after MOO in
Fig. 4. The two target properties improve as the PF significantly moves forward after six experimental iterations. As shown in
Fig. 4, there were no optimized alloys with HT strengths greater than 1200 MPa. In fact, the elements Al, Cr, and Nb usually appear in these alloys, and the formation of intermetallic phases may improve their HT strength, although it results in brittle materials at RT
[5]. An example was the alloy Al
0.025Ti
0.2V
0.075Zr
0.1Cr
0.2Nb
0.2Mo
0.2 (with 1207 MPa of
; and 4.3% of
) in the T-data. The trained ML models predicted the effect of these elements or their combinations on the alloy properties. Thus, the selected alloys derived from this alloy system often showed poor RT ductility, such as alloy E8 at iteration 2 and alloys E14 and E15 at iteration 4. After six iterations, there were 12 newly synthesized alloys that dominated at least one alloy sample in the PF of the T-data, with E24, E19, E17, and E21 forming a new Pareto optimal set (
Fig. 4). The stress–strain curves of these alloys obtained during compression at RT and at 1000 °C are shown in Fig. S4 in Appendix A. The alloy compositions and their measured properties are listed in
Table 1. From alloy E24 to alloy E21, the HT yield strength increases continuously as the RT fracture strain decreases upon the addition of Mo and a certain reduction in Hf and Zr content. Further analysis of the effects of the elements on the RHEA properties is provided in Section 4.4.
To illustrate the results of the MOO strategy, we compared the two target properties of the alloys in the original PF and the new PF. The details are shown in
Table 1, where P1–P7 represent the alloys (yield strength from low to high) in the PF of the T-data. A significant improvement in the two target properties occurs after optimization. Considering alloys with high ductility (> 50%), the
of the E24 alloy is nearly 2.5 times that of P1 (i.e., the typical TaNbHfZrTi alloy whose
is only 295 MPa). Similarly, with a high yield strength at 1000 °C (> 1000 MPa), the
of E21 is nearly three times that of P6. The alloys P2, P3, P4, and P5 are also dominated by more than one optimized material. Compared with P2, the yield strength of E24 increased by 41.7%, while its fracture strain simultaneously increased by more than 54.3%. The alloys E19 and E17 also showed improvement in their HT strength and RT ductility. Compared with the typical alloys NbMoTaW (548 MPa of
and 2.6% of
) and NbMoTaWV (842 MPa of
and 1.7% of
), most of the newly designed RHEAs showed significant improvement in both HT yield strength and RT ductility.
4.2. Structure and softening resistance of the optimized alloys
We investigated the phases of the optimized RHEAs before and after compression deformation at 1000 °C in order to further explore their structural stability for potential engineering applications at elevated temperatures. According to the XRD results shown in
Figs. 5(a) and
(b), the phases of the as-cast alloys E24, E19, and E17 are composed of a disordered BCC solid solution, while alloy E21 exhibits a BCC structure with a minor Laves phase. The XRD patterns suggest that the phase structure is the same before and after HT deformation, indicating the favorable structural stability of the optimized RHEAs. The same characteristics apply to other alloys with improvements, as shown in Fig. S5 in Appendix A. Other than the BCC phase, Laves peaks appear for the four alloys E13, E21, E22, and E23. The presence of Laves phases always decreases the RT ductility but increases the HT strength of RHEAs
[2],
[5]. Thus, the RT fracture strain of these alloys is below 20%, while the alloys with a single BCC phase exhibit better RT ductility, as shown in Table S2. We noted that the fracture strain of alloys E4 and E24 was over 50% under compression deformation; such RT ductility is comparable to those of the alloys in the typical TaNbHfZrTi system, which often show weak HT yield strength, as in the equimolar TaNbHfZrTi with a
of 295 MPa, whereas E4 and E24 showed higher HT strength, with values of 550 and 714 MPa, respectively.
Fig. 5(c) compares the yield stress at higher temperature deformation of alloy E21 with those from Ref.
[40] and shows significant improvement in softening resistance. In contrast to the typical alloys NbMoTaW and NbMoTaWV, the yield strength at 1200 °C (
) for E21 increased by 85.8% from 506 to 940 MPa and by 27.9% from 735 to 940 MPa, respectively. This strength retention suggests that 1200 °C may not approach the break temperature (the temperature at which alloys rapidly lose strength
[41]) of alloy E21, and the simultaneous lower density and better RT ductility also make E21 promising for extremely high temperatures. In fact, alloy E17, which has a single-phase structure, also shows good temperature resistance, and its yield strength at 1200 °C approaches 669 MPa, which probably originates from its significant solid solution strengthening. Furthermore, we compare in
Fig. 5(d) the specific yield strength at 1200 °C (
/
ρ, where
ρ is the density) of the newly synthesized alloys with the reported RHEAs, as the density constraint is relevant to HT structural application. Alloy E21 exhibits an HT specific yield strength superior to that of existing RHEAs, whether for single-phase or multi-phase structures, and even irrespective of the synthesis process. In Fig. S7 in Appendix A, we also give the specific yield strength at 1000 °C (
/
ρ) for all the prepared alloys compared with the T-data. The designed alloys with equivalent
/
ρ levels show better RT ductility. This improvement in multiple properties suggests that these RHEAs are potential candidates for the replacement of traditional superalloys for HT applications, especially the ZrNbMoHfTa system.
4.3. Strength-ductility analysis of the designed alloys
Metallic alloys generally exhibit a considerable decrease in strength at temperatures above approximately 0.6
Tm (
Tm is the melting points). This favorable temperature resistance of RHEAs results from the addition of refractory elements with extremely high melting points. Senkov et al.
[40] explored the relationship between melting points and the yield strength of RHEAs and suggested the design of RHEAs with high
Tm, with both single-phase and multi-phase structures. Here, we estimated the melting points of the designed RHEAs using the rule of mixtures (
Tm = ∑
ci ×
Tmi, where
ci is the concentration of component
i, and
Tmi is the melting point of component
i) and investigated the dependence of
on
Tm. The result is depicted in
Fig. 6(a). For alloys with a higher
Tm of above 2150 °C, there is a clear tendency for the strength at 1000 °C to increase with increasing
Tm, consistent with the previous analysis
[40]. However, this relationship is not applicable to Al-containing RHEAs with low melting points, such as E2, E8, E9, E14, E15, and E16, which are marked in a red circle in
Fig. 6(a). Deviation also occurs for RHEAs with RT brittleness, such as E6. The addition of Al to RHEAs lowers the alloy density and improves the oxidation resistance. The nature of the bonding between Al and the refractory transition metals is significantly different from the bonding of refractory transition metals with each other. Al tends to have large negative enthalpies of mixing with refractory metals, and the resulting bonds are stiff with strong angular characteristics, which can result in different mechanical behaviors for Al-containing RHEAs
[42].
Solid solution strengthening is considered to be responsible for the high strength of BCC RHEAs
[16]. The plot of
Fig. 6(b) for
versus the atomic radius difference (δ
r), representing solid solution strengthening, shows that BCC RHEAs with high δ
r values tend to have higher strength. In fact, thermal activation
[16] and edge dislocation motion
[19] should also be considered for a more reliable prediction of the HT strength of RHEAs, which may explain the deviation in
Fig. 6(b). Sheikh et al.
[18] noted that decreasing the VEC can be favorable for RHEAs’ RT ductility, which is consistent with our synthesized Al-free RHEAs. However, the plot of RT ductility versus VEC (
Fig. 6(c)) shows that the VEC rule is invalid for the reported and designed Al-containing RHEAs. Recently, Yang et al.
[43] revisited the empirical VEC concept with a high-throughput calculation of phase diagrams (CALPHAD) approach in a typical AlCoCrFeNi system. Senkov et al.
[19] investigated the correlations to improve the RT ductility of RHEAs, and a revised VEC criterion suggested that commercially relevant levels of ductility are correlated with a compressive yield strength of ≤ 1500 MPa. However, these suggested criteria are mainly based on statistical analysis of limited data. Hence, our experimental results demonstrate that, even though a physics-based model and the VEC concept can be used as qualitative indicators of strength and ductility for RHEAs designed with a single-phase structure, additional work is still required for a rapid compositional search for RHEAs. The present ML-based approach is a step in this endeavor.
4.4. Effect of elements on RHEA properties
Based on the potential for engineering applications at elevated temperatures, the pseudo-ternary composition diagram for the distribution of HT yield strength and RT fracture strain for the ZrNbMoHfTa system is plotted in
Fig. 7. Our goal was to investigate the effect of changes in elements on alloy properties. As shown in
Fig. 7(a), increasing Mo significantly contributes to the strength of RHEAs at elevated temperatures, in line with previous studies on metallic alloys
[44] and HEAs
[45],
[46]. In compositions lower in Nb and Ta and higher in Hf and Zr, however, the influence of Mo on HT strength is limited. At a given Mo composition, increasing Nb and Ta can improve the HT strength, and this effect is significant when the Mo content is high. The Mo composition of the RHEAs with the highest
was 25 at% rather than at the peak content (35 at%), meaning that the HT strength is not determined by a single component; rather, there is a synergistic effect of elements on the properties.
As shown in
Fig. 7(b), the composition range of ZrNbMoHfTa with good RT ductility is given by a Mo content of no more than 20 at%, Hf and Zr between 35 and 55 at%, and Nb and Ta in the range of 35–55 at%. It has been shown that decreasing the VEC by means of alloying improves the RT ductility
[18],
[19], even though there is no quantitative relationship. It can be seen that the addition of Mo will deteriorate the RT ductility, which may be due to the maximum VEC of Mo for the five elements. In addition, when the Mo content is high (∼30 at%), increasing the composition of Nb and Ta (20–50 at%) and decreasing that of Hf and Zr (20–50 at%) will significantly reduce the RT ductility, in line with the VEC rule. For low Mo content (< 20 at%), changing the Hf, Zr and Nb, Ta compositions has no clear influence on the toughness of the alloys. We also plotted the distribution of properties dependent on composition in Fig. S8 in Appendix A by multiplying the
from
Fig. 7(a) and the
from
Fig. 7(b) to provide an intuitive index for the MOO design of RHEAs.
In
Fig. 7(c), we compare the composition evolution of the alloys on the PFs before and after MOO to provide a more general reference for RHEAs. For alloys with
over 50%, substituting Ti with Mo can improve the HT strength without ductility loss, such as the evolution of alloys from P1 (ZrNbTiHfTa) to E4 (Zr
0.1Nb
0.21Ti
0.18Hf
0.21Ta
0.2Mo
0.1), E5 (Zr
0.23Nb
0.23Mo
0.11Hf
0.23Ta
0.2), and E24 (Zr
0.24Nb
0.29Mo
0.09Hf
0.2Ta
0.18). The addition of Al, V, or Ti has no effect on strength, as illustrated by E1 (Ti
0.26V
0.23Zr
0.25Nb
0.26) and E16 (Al
0.06Zr
0.21Nb
0.18Ti
0.16Hf
0.26Ta
0.13). Whether before or after optimization, alloys with Al, Cr, and/or Nb always show poor RT ductility, regardless of their strength, as shown by the alloys P7 (Al
0.025Ti
0.2V
0.075Zr
0.1Cr
0.2Nb
0.2Mo
0.2), E2 (Al
0.23Ti
0.18Zr
0.12Cr
0.13Nb
0.19Mo
0.15), E8 (Al
0.15Ti
0.14Zr
0.22Cr
0.12Nb
0.22Ta
0.15), E14 (Al
0.17Ti
0.19Zr
0.11Cr
0.13Nb
0.2Mo
0.2), and E15 (Al
0.14Ti
0.2Zr
0.07Cr
0.19Nb
0.2Mo
0.2). This is likely due to the formation of intermetallic compounds that embrittle the alloys. Alloys toward the middle of the PF largely belong to the ZrNbMoHfTa system; from high
to low
, the trend in Mo composition is consistent with the aforementioned analysis. This outcome suggests that MOO essentially iteratively modifies the alloy components, including the elements and their compositions, assisted by the ML model.
5. Summary and perspective
This work demonstrated an acceleration of the compositional design of RHEAs with HT strength (for applicability) and favorable RT ductility (for machinability) over a vast compositional space using an ML-based approach. With this approach, we synthesized and characterized 24 predicted compositions. The two target properties improved as the PF moved forward significantly after six experimental iterations. Four compositions showed outstanding combinations of HT yield strength and RT fracture strain spanning the ranges of 714–1061 MPa and 17.2%–50.0%, respectively. We identified the ZrNbMoHfTa alloy system as being promising for engineering applications at elevated temperatures. In particular, we synthesized the alloy Zr0.13Nb0.27Mo0.26Hf0.13Ta0.21, which exhibited superior mechanical strength, a yield approaching 940 MPa at 1200 °C, and a fracture strain of 17.2% at RT. The heat resistance and favorable structural stability of this alloy suggest considerable potential in structural applications under extreme temperatures.
We have demonstrated an efficient ML-based approach to solve common problems in complex materials design: the need to optimize with limited data information, explore a vast composition space, and achieve multiple design objectives. The proposed design framework can be applied to other alloy or materials systems, such as amorphous alloys, superalloys, and shape memory alloys, as well as functional materials such as ferroelectrics. In RHEA design, in addition to superior HT strength and adequate RT ductility, other properties such as density, oxidation resistance, and HT creep behavior require consideration. Although the oxidation resistance of RHEAs is at least an order of magnitude higher than that of conventional refractory alloys, it remains far inferior to that of superalloys
[47]. Indeed, it is possible to tailor the constituent elements in order to enhance the oxidation resistance of RHEAs at elevated temperatures by adding Al, Cr, and Si to the alloy. Hence, to address realistic engineering applications, it is necessary to perform MOO with three or more properties or objectives. In the present case, we used a cluster analysis to determine the alloy candidates for experiments in order to update the PF through feedback. With more objectives, we will increase the number of experimental iterations to update the PF of the properties for a given material system. Therefore, efficient selection strategies (in our case, cluster analysis) are essential to reduce the overall budget of experimental and computational costs.
Another aspect of the model relates to the feature/descriptor choice, as applied to the materials systems. Here, we only used composition to define the search space. Often, features based on the composition and basic physical or chemical properties of elements can improve the prediction accuracy at the expense of the dimensionality of the feature space. However, it is possible that the features after genetic operations (e.g., crossing and mutation) may no longer correspond to specific alloy compositions. In that case, it may be necessary to effectively reduce the space with the help of domain knowledge.
For ML-based material design with high-dimensional small data samples and a large composition space, it is necessary to take into account the uncertainties caused by changes in input features. Multiple data sources and missing data will also lead to uncertainties that need to be addressed. We used the EI as a utility indicator to guide alloy selection for experimental iteration in the present case. Other utilities, such as the knowledge gradient and upper confidence bound, or PF indicators, such as hypervolume-based indices and the R2 index, need to be investigated to improve the efficiency of MOO. Uncertainties due to data noise are another issue in materials design that must be accounted for. We could draw on the experience of image data augmentation approaches to introduce noise into experimental data in order to generate further pseudo samples.
Acknowledgments
The authors gratefully acknowledge the financial support of the National Key Research and Development Program of China (2021YFB3802100), the National Natural Science Foundation of China (52203293), and the Innovation Centre of Nuclear Materials Fund (ICNM-2022-ZH-02).
Compliance with ethics guidelines
Cheng Wen, Yan Zhang, Changxin Wang, Haiyou Huang, Yuan Wu, Turab Lookman, and Yanjing Su declare that they have no conflict of interest or financial conflicts to disclose.
Appendix A. Supplementary material
Supplementary data to this article can be found online at
https://doi.org/10.1016/j.eng.2023.11.026.
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