High Entropy Alloy Composition Design for Mechanical Properties

Abstract

Multi-component high-entropy alloys (HEAs) are a novel class of materials exhibiting outstanding material properties that often surpassing their traditional counterparts. Despite their ubiquity, the underlying microstructure-property relationships in HEAs remain elusive. This chapter addresses this gap by exploring the application of cutting-edge machine learning tools to establish robust connections between HEAs’ chemical composition, microstructure, and mechanical response. The survey begins by discussing the current state of micro-structural characterization techniques in HEAs, giving insights into their complex underlying microstructure. The development of ML force fields for HEAs based on ab initio datasets is then highlighted, addressing challenges posed by the expansive composition space associated with HEAs. The chapter further outlines machine learning-assisted composition search strategies for HEAs with specific functional properties, offering a systematic and efficient approach to explore material properties. Overall, the present overview demonstrates the potential of machine learning in unraveling the intricate nature of HEAs and accelerating their tailored design for diverse applications.

1. Introduction

Multi-component high-entropy alloys (HEAs) represent an emerging class of materials that are composed of at least four or more elements in roughly equal proportions. As opposed to traditional alloys, typically consist of one or two primary elements with smaller amounts of additional elements, HEAs exhibit exceptional thermo-mechanical properties that often outperform the former alloys [1, 2, 3]. HEAs’ transport properties, for instance, have long been characterized by comparatively slow kinetics, as opposed to pure metals and conventional alloys, hence the term sluggish diffusion [4, 5]. Together with high entropy of mixing, severe lattice distortion, and also, the cocktail effect, these so-called “core effects” are commonly identified as the principal sources of HEAs’ superior performance (e.g. single-phase thermodynamic stability [6], creep resistance [7], and high-temperature strength [8]).

The atomistic origin of these remarkable properties presumably originate from the underlying chemical/micro-structural complexities attributed to HEAs, yet the nature of these inherent microstructure-property relationships remain elusive to this date. Empirical frameworks conventionally describe such correlations by a fairly small set of “descriptors” largely ignoring underlying scale hierarchies and intricate microstructure. Multi-scale simulation frameworks have fairly limited applicability/predictability due to the modeling gap in transferring physics-based information across different length/time-scales. Experimental investigations can only explore a small portion of the immense combinatorial space spanned by different elemental compositions in HEAs.

The above limitations demand applications of state-of-the-art machine learning (ML) tools that can help establish robust relationships between HEAs’ composition and their mechanical response. This chapter aims to conduct a thorough survey of the current state-of-the-art in data mining and pattern detection, feature extraction and analysis, physics extraction and interpretation of ML predictions relevant to HEAs’ composition design and properties. A special emphasis will be given to applications of machine learning relevant to characterization of micro-structural defects, development of machine-learned interatomic potentials via ab initio calculations, and ml-assisted composition search strategies for targeted functional properties.

The organization of this chapter is as follows. Section 2 discusses the current state of micro-structural characterization for HEAs by employing advanced imaging techniques such as digital image correlation, X-ray diffraction, and high-resolution microscopy. Material scientists utilize these techniques, along with sophisticated image processing tools and machine learning-inspired descriptors, to extract meaningful insights into structural patterns, physics, and property correlations from complex micro-structural data in HEAs. Section 3 highlights the development process of machine learning force fields for HEAs based on ab initio datasets. We present relevant challenges arising from the expansive composition space associated with HEAs and certain mitigation strategies. In Section 4, we review machine learning-assisted composition search strategies for HEAs with specific functional properties, addressing challenges in navigating their vast compositional space. Our focus will be on the efficacy and robustness of the machine learning-supported HEA design, as opposed to traditional methods like density functional theory and CALPHAD, offering a systematic and efficient approach to explore material properties and guiding experimental validation.

2. Data mining, pattern detection, and physics extraction in HEAs

Current state of the art in the high-throughput design of HEAs involves extensive applications of powerful characterization techniques which allow for gaining deep insights into inherent structural patterns, underlying physics, and structure-property correlations across a broad range of length scales [9]. Material scientists make routine use of advanced image-based techniques, such as digital image correlation (DIC), X-ray diffraction (XRD) scattering, and high-resolution scanning/transmission microscopy (STEM), capturing two-dimensional micro-structural information at micro and nano scales down to atomic resolutions. These data sets, along with simulation-generated data, often embed complex hierarchical patterns and necessitate the employment of advanced image processing tools, i.e. for tasks related to the analysis of complex atomic displacement patterns [10, 11], direct recognition of plasticity through local strain evolution [12, 13, 14, 15] as well as identification/classification of local chemical environments and structural ordering [16, 17, 18, 19, 20]. Extracting meaningful insights from micro-structural data through data analytic and feature-extraction algorithms is vitally important for HEAs. The present section highlights certain effective tools, some of them inspired by machine learning, that can help address the physics extraction aspect of HEAs.

2.1 General structural descriptors

As previously outlined, the aforementioned characterization techniques offer hierarchical insights into the underlying materials’ substructure and, therefore, the notion of relevant structural descriptors is highly scale-dependent [22]. Atomically-resolved images, obtained from high-resolution TEM typically with Å-level resolution, allow for extracting individual atomic positions ${\overrightarrow{r}}_i$, $i=1,2,\dots$ with remarkable accuracy. Such high-precision coordinates may be subsequently utilized to construct atom-based metrics and/or spatially-averaged proxies of the underlying microstructure. In this context, one may analyze statistical order parameters including two-point pair-correlation function in real space $g\left(\overrightarrow{r}\right)$ and/or structure factor $S\left(\overrightarrow{q}\right)$ [23], exploiting the overall periodicity within the crystalline structure, and seek for underlying contrasts within the local structure. The latter relates to diffraction pattern intensities $D\left(\overrightarrow{q}\right)\propto {\left|S\left(\overrightarrow{q}\right)\right|}^2$ as the key metric to describe crystal symmetries [24]. This methodology is presented in Figure 1 illustrating a schematic two-dimensional diffraction pattern and the resulting contrast owing to the defective structure [21].

Figure 1.

Two-dimensional diffraction pattern is schematically represented in a). Panel b) illustrates a pristine simple cubic structure and the same structure with vacancies along with corresponding diffraction fingerprints. The difference between the diffraction fingerprints of the defective structure and the pristine structure is depicted in c). Figure taken from [21].

Direct recognition of these patterns via machine learning tools has proven to be useful for image-based microstructure classification tasks in materials science [25, 26, 27] as well as crystallography and phase segmentation [28], and material property predictions [29, 30, 31]. Along these lines, a machine learning-based approach has been developed by Kaufmann et al. for phase selection demonstrating a rapid and autonomous method for identifying crystal symmetry based on diffraction patterns [27, 32]. Within the context of HEAs, establishing such relationships has proven challenging due to the high configurational entropy, leading to a distribution of lattice parameters and cell compositions instead of a singular unit cell and lattice constant typically observed in traditional alloys [33]. In [34], the authors proposed a topological approach analyzing experimental atom probe tomography (APT) data and atomic neighborhoods in Al1.3CoCrCuFeNi and Al0.3CoCrFeNi HEAs. Despite the presence of severe lattice distortion, their methodology achieved a high accuracy in classifying the lattice structure of atomic neighborhoods including BCC and FCC phases.

2.2 Local descriptors for chemical short-range ordering

To investigate the local chemical environment (including short-range chemical ordering), the Warren–Cowley SRO parameters ${\alpha }_{\mathit{ab}}\left(\overrightarrow{r}\right)$ [35] provide robust chemical/structural metrics probing concentration variations of type-$b$ atoms within distance $\overrightarrow{r}$ from a center type-$a$ element. This metric is applicable to experimentally derived datasets (i.e. like atom probe tomography) [36, 37, 38] and/or simulation-based datasets [19, 39, 40, 41], containing both atomic identities and three-dimensional coordinates. In this framework, any coherent compositional deviations from (statistically) random distributions of atoms within the solution matrix, the so-called random solid-solution limit, can be regarded as spatial footprints of chemical ordering [16, 37, 42, 43]. Theses order parameters relate to the spatial density correlations with the associated two-point correlation function defined as $c_{\mathit{ab}}\left(\overrightarrow{r}-{\overrightarrow{r}}^{\prime }\right)=⟨{\rho }_a\left(\overrightarrow{r}\right){\rho }_b\left({\overrightarrow{r}}^{\prime }\right)⟩$. Here the angle brackets $⟨.⟩$ denote ensemble averaging and ${\rho }_a\left(\overrightarrow{r}\right)$ refers to the local concentration of type-$a$ element. Figure 2 presents direct evidence of angstrom-level chemical short-range order in an annealed NiCoV medium-entropy alloy and associated correlation analysis featuring meaningful (anti-)correlation patterns in space.

Figure 2.

Chemical mapping in (a) illustrating the local distribution of elements in annealed face-centered-cubic NiCoV concentrated solution. Panel (b) provides magnified views of V, V-Co, and V-Ni elements. The corresponding pair correlation coefficients, $c_{\mathit{ab}}\left(r\right)$, are presented in (c) before and after tensile tests, offering insights into SRO distributions in space. Figure reproduced from [16].

The SRO patterns primarily arises from processing parameters (e.g., special heat treatment, alloy composition), yet mechanical deformation on pre-annealed alloys was further reported to facilitate the SRO formation [44]. Notably, Naghdi et al. [45] employed molecular dynamics simulations to demonstrate that nano-indentation, specifically dwell nano-indentation protocols, can manipulate chemical short-range order in equi-atomic NiCoCr alloys. Their research revealed the formation of density-wave stripe patterns under the indenter tip, mainly influenced by local stress concentrations, providing explicit validation pathways for the manipulation of SROs in multicomponent alloys. In [40, 46], combined lattice distortion and SRO effects were reported to enhance the critical resolved shear stress associated with simulated NiCoV and NiCoCr concentrated solid solution alloys.

In studying NiCoCr-based complex alloys, stacking fault energy, hardness, and fracture toughness, as bulk properties, were recently shown to strongly correlate with the degree of Ni-rich SROs and corresponding structural features [39, 41, 42, 47]. On the other hand, local misfit properties and associated atomic-scale fluctuations have been long known to serve as favored trapping sites for dislocation migration with appreciable contributions to plastic yielding properties [48, 49, 50]. Current efforts for direct measurements of local lattice misfit volumes and deviatoric strains in CSAs have been mostly limited to atomistic simulations. In this context, direct experimental characterization of local distortion properties and SROs and their collective effects on the dislocation mobility in HEAs is still largely unexplored.

2.3 Insights from microscopic deformation in polycrystalline alloys

Electron backscatter diffraction (EBSD) imaging provides detailed local crystallography information at each specific location on the studied surface and enables a thorough examination of micro texture and sample morphology down to sub-micron levels [51, 52]. These maps delineate distinct crystallographic signatures, aiding in the phase identification [32], characterization of local strain and deformation patterns [53] as well as twinning [54], (geometrically necessary) dislocations [55], and other crystalline defects within grains substructure. EBSD datasets typically contain pixel-based orientation information with a relatively high precision (order 0.1°) and fine spatial resolution (order $100$ nm) which can be conveniently processed by various statistical tools and machine learning algorithms (Figure 3).

Figure 3.

EBSD imaging serves as a foundation for machine learning applications. Panel (a) presents a snapshot of the grain structure for Mg alloy. In (b), the sample was characterized by the dislocation density (${\rho }_{\mathrm{GND}}$) both before and after deformation. The panels in (c) and (d) serve as representatives for the machine learning dataset, encompassing the two classes associated with grains before and after deformation. Figure taken from [56].

In conjunction with nano/micro-mechanical deformation tests, such datasets can give further insights into the intricate links between complex surface characteristics and mechanical behavior at microscopic levels [57, 58]. In this framework, Salmenjoki et al. [56] examined the evolution of dislocation density in polycrystalline magnesium-zinc (Mg-Zn) alloys by comparing EBSD images before and after deformation. The study aimed to predict the dislocation density levels upon deformation solely based on associated changes in grain microstructure. Based on the grain-wise information prior to loading, machine learning Graph Neural Network (GNN) models were trained representing the grain microstructure to predict geometrically necessary dislocation density within the deformed alloy. Utilizing a GNN model, a similar approach was employed in [59] investing predictability of nanoscale hardness in polycrystalline metals based on micro-structural features such as grain orientations and neighboring grain properties.

2.4 Machine learning descriptors for diverse microstructures

Machine learning-based descriptors play a pivotal role in the context of computational chemistry and materials science, enabling an efficient (yet accurate) representation of complex molecular and material microstructure [61, 62]. Among the notable descriptors, the Behler-Parinello method [63] and relevant atom-centered symmetry functions (ACSFs) stand out for their capability to capture the inherent symmetry and non-local features in atomic environments. The radial ACSF represents a two-body function that can be constructed using a Gaussian-smoothed atomic density function ${\rho }_i\left(r\right)={\sum }_{j}exp\left\{-\left(r_{\mathit{ij}}-r\right)/2{\sigma }^2\right\}$ centered at atom $i$. Here $r_{\mathit{ij}}$ denotes the pair-wise distance between center atom $i$ and a neighbor atom $j$ and $\sigma$ is the width of the Gaussian kernel. The angular ACSF include more tunable parameters leading to an immense flexibility of the function. This includes three-body terms corresponding to the angle between the vectors ${\overrightarrow{r}}_{\mathit{ij}}$ and ${\overrightarrow{r}}_{\mathit{ik}}$ with $j$ and $k$ being neighboring atoms of center atom $i$. The Smooth Overlap of Atomic Positions (SOAP) descriptor, on the other hand, excels in representing local atomic environments by making an expansion of the local atomic density based on a complete set of the spherical harmonics $Y_{\mathit{lm}}\left(\varphi ,\theta \right)$ and radial basis functions $g_n\left(r\right)$.

In terms of ML-based characterization of defects, a fairly universal strategy for crystalline solids at the atomic scale is to seek for statistical “outliers” within the multi-dimensional descriptor space. This approach is illustrated in Figure 4 providing a generic measure to describe the distortion score of local atomic environments [60]—a metric quantifying an outlier’s distance from the observed average. The proposed strategy enables automatic detection of defects, facilitating a hierarchical description of defects [64] and offering applications in advanced materials modeling, including relevant information selection for evaluating energy barriers [65, 66, 67], design of robust interatomic machine learning potentials [68, 69, 70], and high-throughput analysis of databases for materials science advancements.

Figure 4.

A single vacancy in a body-centered cubic metal is depicted in the $26$-dimensional descriptor space in (a) utilizing a variant of the SOAP descriptor. The spectral map illustrates contrasting bands arising from atomic environments in the vicinity of the mono-vacancy. Here the atoms are colored based on the first descriptor component indicated by the arrow. Panel (b) illustrates several defects in body-centered cubic crystal which are quantified based on the associated distortion scores. Figure taken from [60].

Within the context of HEAs, one promising avenue pertains to diffusion studies of micro-structural defects, typically exhibiting sluggish dynamics under extreme conditions (i.e. irradiation effects) [71]. A recent kinetic Monte Carlo (kMC) study probing the micro-structural kinetics of NiCoCr and NiCoCrFeMn alloys revealed a subdiffusive dynamics of atomic vacancies [72]. This subdiffusive phenomenon was interpreted as a universal signature of dynamical sluggishness in HEAs rooted in severe lattice distortion and chemical complexities. The concept of a distortion score is amenable to the automated classification of (irradiation-induced) defects in atomistic configurations. Furthermore, it facilitates the prediction of associated diffusion paths and energy barriers under thermal activation, which is otherwise a formidable task owing to inherent lattice disorder.

3. Machine-learned force fields

3.1 Overview

Machine Learning Force Fields (MLFFs) have emerged as a viable alternative to empirical potentials, such as the Embedded Atom Model (EAM) and Modified Embedded Atom Model (MEAM), demonstrating the capability to achieve quantum accuracy in predicting forces and energies during large length- and time-scale molecular dynamics (MD) simulations [73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87]. Various methods have been developed to date, utilizing diverse machine learning algorithms for MLFF development. These approaches include Gaussian Approximation Potentials (GAP) [88], Neural Network Interatomic Potentials (NNIPs) [82], Tabulated GAP (tabGAP) [89, 90], on-the-fly active learning of interatomic potentials such as FLARE [81] and equivariant graph neural network potentials (NequIP) [80].

MLFF development involves constructing a dataset derived from Ab initio Molecular Dynamics (AIMD) simulations, followed by training a machine learning (ML) algorithm to predict energies and forces for individual atoms within the training dataset (see Figure 5). The ML algorithms are trained using embedded atomic positions and their corresponding bond angles, which serve as local atomic environment descriptors. These features are labeled with associated forces and energies in the training dataset. In the literature, a diverse array of descriptors is employed for MLFF development. These encompass Behler–Parrinello vectors [93], atomic cluster expansion descriptors [94], Smooth Overlap of Atomic Positions (SOAP) [95], and graph-based descriptors [73], among others.

Figure 5.

MLFF overview. (a) Demonstrates an active learning framework for MLFF developement, where new congifurations are added to the training dataset based on intrinsic uncertainty values of standard gaussian process (SGP) model (figuare taken from [81]). (b) Illustrates a NNIP developement workflow, where descriptors are calculated for each atom and are fed to the input leyer of the model. The properties, such as energies and forces, are then predicted from the output layer (figure from [91]). (c) The GAP potential framework, which uses a kernel function to evaluate uncertainties of the model (figure from [92]).

3.2 Machine-learning force field development for multi-component alloys

Creating MLFFs for single-element materials is significantly simpler compared to multi-component materials. This is attributed to the expansive composition space in multi-component materials, which grows substantially with the introduction of new elements into the system. This complexity renders the development of MLFFs challenging and relatively unexplored within the realm of computational material science. The challenge linked to the composition space stems from creating the training dataset, typically generated with DFT, which is an expensive process. Moreover, training a MLFF with all possible configurations covering the entire composition space, even if feasible, becomes computationally demanding due to the high dimensionality of atomic environment descriptors for multi-component configurations and constraints posed by memory limits [96, 97, 98, 99]. Nevertheless, there have been efforts to develop MLFFs for materials with compositions involving up to five elements [90, 100, 101, 102, 103, 104, 105, 106, 107, 108], which will be discussed in the following section.

To tackle the dimensionality scaling challenge in MLFF development for multi-component alloys, the problem with SOAP descriptors is addressed by reducing the descriptor’s dimensionality from quadratic to linear with respect to the number of elements [99]. The proposed methods include, firstly, considering the recoverability of density expansion coefficients from the power spectrum to create a compressed version that can be restored under specific conditions. Secondly, introducing a generalized SOAP kernel that enables compression concerning both the number of chemical elements and the number of radial basis functions in the density expansion. The performance of the compressed descriptors is then evaluated across various datasets using numerical tests that assess the accuracy of fitted energy models and MLFFs.

In a subsequent study, Byggmastar et al. [89, 90] discovered that augmenting the training data significantly for a GAP force-field with typical high-dimensional SOAP descriptors will not achieve satisfactory accuracy. They discovered that employing uncomplicated low-dimensional two-body and three-body descriptors in a GAP proved more effective than a SOAP-GAP for crystalline alloys. Additionally, this approach provided a noteworthy advantage, as a pure 2 + 3-body potential could be tabulated and evaluated efficiently using cubic-spline interpolations (referred to as tabGAP), resulting in a two-order-of-magnitude speed-up. The tabGAP model yielded results with near-quantum accuracy for a five-element Mo–Nb–Ta–V–W alloy, demonstrating the effectiveness of low-dimensional descriptors in handling complex alloy systems.

Byggmaster et al. [90] developed a MLFF designed for the Mo–Nb–Ta–V–W quinary system, employing it to investigate segregation and defects in the body-centered cubic (BCC) refractory high-entropy alloy MoNbTaVW in the GAP framework. They observed clear ordering in the bulk alloy, primarily involving Mo-Ta and V-W binaries at lower temperatures. Simulations in radiated crystals revealed pronounced segregation of V, the smallest atom in the alloy, to densely packed interstitial-rich regions like radiation-induced dislocation loops. The utility of MLFFs in modeling intricate multi-component alloys is evident, offering significant benefits to the material science community.

To assess the capability of MLFFs in accurately predicting alloy phase diagrams, Rosenbrock et al. [103] formulated an MLFF for Ag-Pd, aiming to characterize the energy of alloy configurations across a broad spectrum of compositions. The MLFF developed exhibits high accuracy across diverse compositions, demonstrating comparable results to the cluster expansion method [109], which serves as a reliable benchmark for this system. In a subsequent study, Xie et al. [79] created an MLFF for SiC polymorphs using the FLARE active learning framework, specifically designed to capture phase transformations in this material. Their work demonstrated strong agreement with both ab initio calculations and experimental measurements, surpassing existing empirical models in terms of vibrational and thermal properties. The examples mentioned above highlight the applicability of MLFFs in capturing diverse crystalline phases of a material, provided that the MLFF is developed accurately.

Marchand et al. [104] developed a series of NNIPs for the Al-Cu system, attaining near-first-principle accuracy across various metallurgically significant aspects. These NNIPs accurately predict intermetallic compounds, elastic constants, solid-solution energetics, precipitate-matrix interfaces, stacking fault energies, and other properties. The machine learning approach, in comparison to state-of-the-art potentials, offers substantial quantitative benefits and exhibits promising transferability to defects and properties beyond the training structures, highlighting the crucial requirement for meticulous validation in specific metallurgical applications. In a subsequent study by Jain et al. [106], a set of NNIPs were created for Al-Mg-Si alloys, demonstrating significant applicability in the investigation of precipitate/matrix interfaces, antisite defect energies, and vacancy trapping in Mg-Si segregations. Furthermore, in another study by Marchand et al. [107], NNIPs were formulated for Al-Cu-Mg-Zn alloys. This research demonstrated that NNIPs could be effectively employed for more intricate systems, particularly in predicting plasticity-related properties at the quantum level.

Upon reviewing the literature on MLFF development for multi-component alloys, it becomes evident that despite significant progress in this field, there remains substantial work to be undertaken to enhance the effectiveness of MLFFs in alloy modeling across diverse aspects. While some strides have been made to extend the time and length scales of MLFFs, further improvements are needed, particularly for larger molecular dynamics simulations involving intricate nano-mechanical applications such as dislocation dynamics in nanoindentation and other nano-mechanical tests.

4. AI-assisted composition search and materials discovery

4.1 Machine learning and its role in accelerated search for compositions that exhibit specific functional properties in HEAs: An example for good practices

On the basis of the composition-property relationships, the expansive compositional flexibility afforded by multi-principal component alloys, such as High Entropy Alloys (HEAs), holds the potential to unlock a diverse array of material properties. An example of such potential is the so-called “cocktail” effect, which pertains to the synergistic interplay of various elements within high entropy compounds, giving rise to unexpected properties [110]. However, the advantages presented by such diversity hold a challenge in the disclosure of the optimal compositions for specific applications. In fact, navigating the vast landscape of HEAs compositions requires a departure from conventional trial-and-error approaches. The inherent inefficiency of such methods, both in terms of time and the uncertainty of practical utility, necessitates a more systematic exploration, especially considering the cost and complexity associated with experimental validations.

Density Functional Theory (DFT) [111] stands out as a highly precise means of simulating materials at the quantum level. However, its computational demands impose constraints on the scales of systems that can be effectively explored, rendering the comprehensive investigation of compositional spaces impractical. As a complementary tool, CALPHAD (Calculation of Phase Diagrams) [112] methods has proven to be precious in the context of HEAs [113, 114, 115, 116] and the prediction of their phase structures, overcoming the limitations of the Hume-Rothery rules [117]. These methods utilize a thermodynamic approach, combining computational and experimental data to predict phase diagrams and thermodynamic properties. Nevertheless, empirical methods, including CALPHAD, may encounter challenges when extrapolating to compositions significantly divergent from the original databases [71].

By leveraging data-driven methodologies, Materials Informatics [22] accelerates computational experiments, offering a promising avenue for exploring compositional and functional spaces within materials. This integration of advanced computational tools holds immense potential for revolutionizing our understanding of material properties and streamlining the experimental validation process.

As an introduction to such methods, one could start from an example of a general procedure in the ML supported design of HEAs for specific target properties. C. Wen et al. [118], defined a virtuous procedure for the composition search on HEAs based on Al-Co-Cr-Cu-Fe-Ni to maximize their hardness, which will serve for us as a template for the explanation of ML property-oriented materials design, and the inherent good practices. The scheme of such procedure is presented in Figure 6. The first point of any ML application is the building of a high-quality dataset; this includes to pay particular attention to the source of the data, to data-cleaning and balancing techniques. In the mentioned work, the authors (i) collect samples in the dataset only if fabricated with the very same technique; (ii) remove outliers and samples with significantly different predictions of the target property in different works; (iii) validate with experiments the hardness measures of a small random subset of samples among the selected ones, to ensure small differences.

Figure 6.

Figure taken from [118]. Visualization of the property-oriented design of HEAs with the support of ML models. Good practices in the development of a materials discovery pipeline include: (a) a high-quality dataset; (b 1-2) materials features supporting the supervised learning with the ML model subsequently selected in (c); (d) the creation of the virtual space of potential compositions to be explored; (e) the definition of utility function of support in the selection of potential candidates for experiments; (f) experiments.

The next step, is the addition of supporting features for each sample. In a general Materials Informatics framework, such features can be anything from chemical compositions to structural or electronic properties, and it is crucial to understand the so-called feature importance, namely how relevant a set of features is for the sake of the ML model performance, and to remove redundant features. As it can be seen from the procedural scheme, C. Wen et al. analyzed the importance of features by separately testing a ML model with and without materials knowledge, namely features beyond compositions that included phase formation (examples are atomic radii mismatch, mixing and configurational enthalpy, valence electron concentration, local electronegativity mismatch, cohesive energy) and strengthening theory and mechanical features (examples are modulus mismatch, Peierls-Nabarro factor, shear modulus difference).

Model selection is the crucial step in which the capabilities of different models to perform the regression task towards unseen materials properties is examined to promote the one with minimum prediction errors. After identifying the best ML model out of a set of candidates (i.e. linear, polynomial or support vector regression, or neural networks), it can be applied on the search space containing the large variety of samples (in C. Wen study case, unsynthesized alloys) with unknown target labels (hardness). During such explorations, C. Wen et al. have applied the precaution of maximizing, through the Efficient Global Optimization (EGO) [119] a utility function for the selection of the samples to be validated through experiments, the expected improvement function [120], defined as $\mathrm{EI}=E\left[max\left(y-{\mu }^{\prime },0\right)\right]=\sigma \left[\varphi \left(z\right)+z\mathrm{\Phi }\left(z\right)\right]$, where ${\mu }^{\prime }$ is the maximum hardness value in the training dataset, and $\varphi \left(z\right)$ and $\mathrm{\Phi }\left(z\right)$ are the standard normal density and distribution functions of $z=\left(\mu -{\mu }^{\prime }\right)/\sigma$, with $\mu$ and $\sigma$ the hardness and uncertainty predictions, respectively. The hardness experimental measures on the samples of confidence are then added to the training data, recalling what is known to be as Active Learning.

With such procedure, C. Wen et al., found 35 out of 42 samples of confidence with higher hardness than the maximum contained in the training dataset. Clearly, each step of the process is and contains general enough practices that can be applied to any target property during HEAs composition searches and design.

In the next section, we present and discuss examples of how Machine learning can help exploring HEAs compositions for specific target properties, in particular focusing on energy and extreme environment applications.

4.2 AI-assisted accelerated search for compositions that exhibit specific functional properties in HEAs: Energy and environmental applications

The urgent transition towards green energy solutions clearly requires sophisticated solutions in terms of the target materials properties to be adapted for specific applications (i.e. automotive, solar, catalysis), but, in the first place, in terms of the searching strategies of materials displaying them. The actual exploration process which determines potential materials candidates a is time-consuming one, ramping the scales from the quantum-level investigation to the large scale validation of their characteristics. As introduced before, Machine Learning approaches are the most advanced tool that a scientist can deploy to speed up composition searches, and in this section the use of HEAs in energy applications is presented.

In the context of hydrogen storage, for automotive applications the US Department of Energy (D.O.E) identified the very strict criteria for characterizing a suitable storage systems in terms of costs, safety, reversibility, working thermodinamic conditions and more [121]. Metal hydrides [122] have been extensively studied as a potential solution, as they are able to absorb and release a significant amount of hydrogen through a chemical reaction, offering a safer option compared to high-pressure gas or liquid hydrogen storage [123, 124, 125], because they can operate at lower pressures [126] and are less prone to leakage. However, the development of efficient metal hydrides for hydrogen storage has been a persistent challenge For example, for hydrides primarily composed of transition metals, the problem is their constrained capacity, characterized by a low H/M ratio, while the acceptable storage capabilities of metal hydrides like the AB5-type, necessitate the incorporation of rare-earth metals like lanthanum, adding complexity and cost to the system [127]. While MgH2 and certain complex hydrides come close to meeting storage targets, their practical utility is hindered by challenges such as slow kinetics, limited reversibility, and high operating temperatures. These factors collectively render them unsuitable for applications with highly dynamic loads, such as those encountered in passenger cars [128].

In this context, HEAs represent a promising solution due to the high tunability of the compositions, to the advantageous capacities of BCC structures and the lattice distorsions due to the atomic radii mismatch between the constituent species [127]. As an example, from the point of view of BCC HEAs, ${\mathrm{Ti}}_{0.325}{\mathrm{V}}_{0.275}{\mathrm{Zr}}_{0.125}{\mathrm{Nb}}_{0.275}$ and $\text{TiVCrNb}$ absorb hydrogen at room temperature, with moderate working pressure conditions, and are suitable for stationary hydrogen storage applications, while the intermetallic ${\mathrm{Mg}}_{0.10}{\mathrm{Ti}}_{0.30}{\mathrm{V}}_{0.25}{\mathrm{Zr}}_{0.10}{\mathrm{Nb}}_{0.2}$ HEA composition reported promising hydrogen storage performance as it absorbed $1.7\mathrm{H}/\mathrm{M}$ (2.7 wt% ${\mathrm{H}}_2$) at room temperature with fast kinetics [129].

A ML model for predicting thermodynamic properties of HEAs is an essential step to allow the fast screening of their potential for hydrogen storage applications. The reason lies in the thermodynamics basis of the HEA-H interaction, for a complete description of which we advice the reading of [129]. For our purposes, let us limit ourselves to mention the importance of the van’t Hoff relation

$$\mathit{ln}\left(P_{\mathit{eq}}\right)=-\frac{\mathrm{\Delta }H}{\mathit{RT}}+\frac{\mathrm{\Delta }S}{R},$$E1

which relates the equilibrium ${\mathrm{H}}_2$ pressure at a given temperature T with the enthalpy and entropy of dissociation of the hydride phase. In a remarkable contribution, Witman et al. [131] trained a ML model (Gradient Boosting Tree) on the HydPARK metal hydrides database to predict such target feature, confirming the structure-property relationships intertwining $\mathrm{\Delta }H$ and interstitial volumes, responsible for the dependence of the $P_{\mathit{eq}}\left(H_2\right)$ on the volume of the cell. The applied approach did not rely on the exact crystal structures of the systems in the dataset, making it a compositional ML model [132, 133], particularly suitable for the search of potential HEA hydrides yet to be synthesized. For this reason, in a later work, Witman et al. [132] based on the same approach the predictions of the hydride stability for a virtual space of 672 refractory HEAs (rHEAs) from the set of Al-Ti-V-Cr-Zr-Nb-Mo-Pd-Hf-Ta, after improving the model by adding reference rHEA samples in the dataset (TiVzrNbHf and TiVZrNb) with their known thermodynamic properties, as well as the binary hydride formation enthalpy per atom (given an element $i$), averaged and weighted by the composition fraction of the element $f_i$: $\overline{\mathrm{\Delta }}{H}_{\mathit{bh}}={\sum }_i{f}_i\mathrm{\Delta }{H}_{\mathit{bh},i}$. Additionally, a predictive model for the stability of single phase solid solution for a given HEA composition was developed by training a classifier on the experimental database from Miracle et al. [71], and validating its predictions on a small dataset of synthesized compounds. Combining the predictions of the augmented model for the thermodynamic stability of rHEAs and the one for the formation of single-phase SS, the authors were able to identify the samples to be reproduced, characterized by improved destabilization (higher plateau pressures). In particular, AlTiVCr was identified to have a 70x increase in the plateau pressure, and conveniently included lighter-weight, lower-cost aluminum.

Another aspect inherent to the presence of hydrogen in metals, is the hydrogen-induce deterioration of their mechanical properties, also referred to as hydrogen embrittlement (HE), for example including reduction of tensile and fatigue strength, crack initiation and propagation [134]. For these reasons, such phenomena are of remarkable interest also for the hydrogen storage community. One way of mitigating the effects of HE, is to design materials with ultra-low hydrogen diffusion coefficients, in such a way to limit hydrogen accumulation at defects. X. Zhou et al. [130], define a machine learning supported procedure to design HE-resistant high-entropy alloys (HEAs), specifically focusing on FeCoNiCrMn compositions. Combining density functional theory to obtain hydrogen solution energies in the crystal structure, their use in kinetic Monte Carlo (kMC) simulations to access diffusion coefficients, and SOAP (Smooth Overlap of Atomic Positions) [135] descriptors for the local atomic environments of hydrogen, they target to highlight the intertwined relations via ML and to consequently lower the diffusion coefficients by tuning the HEAs compositions. In particular, the whale optimization algorithm (WOA) [136] is used to optimally explore the virtual HEAs compositional space, and by training a neural network to predict the hydrogen solution energy in unseen HEA SOAP descriptors, they can then be deployed within kMC simulations to obtain the diffusion coefficients. Figure 7 reports such scheme of the procedure from their work. The model indicates that adjusting the alloy chemical composition, particularly elevating Co and Mn levels, enhances the FeCoNiCrMn alloys resilience against hydrogen embrittlement.

Figure 7.

Figure taken from [130]. Hydrogen solution energies are acquired by DFT simulations, and correlated to the SOAP descriptors of the local atomic environment. Predictions of hydrogen solution energies in unseen HEAs compositions, explored by the WOA algorithm, are then fed into kMC simulations to predict diffusion coefficients.

5. Conclusions

In this chapter, we have outlined the current state of materials informatics and its applications relevant to high-entropy alloys’ composition, microstructure, and functionality. Our special focus has been on utilizing the machine learning toolbox pertinent to high-throughput materials design, micro-structural/chemical ordering quantification/tailoring, and search strategies for novel materials, offering useful insights into inherent processing-microstructure-property relationships in HEAs. The present overview has further addressed fundamental challenges concerning the highly dimensional and scattered nature of the parameter space associated with the high-throughput design of HEAs. We have also provided guidance on relevant mitigation strategies, offering a roadmap for machine-learning practitioners exploring the complexities of materials informatics in the context of HEAs.

Acknowledgments

The authors acknowledge funding from the European Union Horizon 2020 research and innovation program under Grant Agreement No. 857470 and from the European Regional Development Fund under the program of the Foundation for Polish Science International Research Agenda PLUS, grant No. MAB PLUS/2018/8, and the initiative of the Ministry of Science and Higher Education Support for the activities of Centers of Excellence established in Poland under the Horizon 2020 program’ under agreement No. MEiN/2023/DIR/3795.