Towards informatics-driven design of nuclear waste forms

Abstract

Informatics-driven approaches, such as machine learning and sequential experimental design, have shown the potential to drastically impact next-generation materials discovery and design. In this perspective, we present a few guiding principles for applying informatics-based methods towards the design of novel nuclear waste forms. We advocate for adopting a system design approach, and describe the effective usage of data-driven methods in every stage of such a design process. We demonstrate how this approach can optimally leverage physics-based simulations, machine learning surrogates, and experimental synthesis and characterization, within a feedback-driven closed-loop sequential learning framework. We discuss the importance of incorporating domain knowledge into the representation of materials, the construction and curation of datasets, the development of predictive property models, and the design and execution of experiments. We illustrate the application of this approach by successfully designing and validating Na- and Nd-containing phosphate-based ceramic waste forms. Finally, we discuss open challenges in such informatics-driven workflows and present an outlook for their widespread application for the cleanup of nuclear wastes.

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1 Introduction to nuclear waste forms

The United States manages large volumes of nuclear wastes generated from nuclear weapons production during World War II and the Cold War. These wastes are primarily stored at US Department of Energy sites such as Hanford (near Richland, Washington), Savannah River Site (near Aiken, South Carolina), Oak Ridge National Laboratory (ORNL; in Oak Ridge, Tennessee), Los Alamos National Laboratory (in Los Alamos, New Mexico), and the Idaho Nuclear Technical and Engineering Center (near Idaho Falls, Idaho). These wastes range from solids to sludges to liquids and contain, in broad terms, most of the elements from the periodic table. In general, the most hazardous of these wastes are high-level tank wastes at Hanford and Savannah River which will be vitrified into borosilicate glass-based waste forms and buried in disposal facilities.

Recent increased interest in advanced nuclear power technologies has spurred interest in advanced nuclear fuel cycles.¹ Some of those fuel cycles include the generation of unique waste streams such as molten halide salts containing fission and activation products that will need to be treated for ultimate disposal.^2–6 Modern approaches that short circuit the traditional inefficient Edisonian trial-and-error style of waste form development and significantly accelerate the design of novel waste forms are in great need.

In general, options for immobilizing entire salt wastes, i.e., without any salt fraction partitioning, are limited. For such full-salt immobilization, the few demonstrated waste form options include glass-bonded sodalite and tellurite glasses,^2,7 all of which exhibit low salt loading. Alternatively, rather than immobilizing the full-salt waste, the salt can be partitioned into different constituents to make waste form fabrication and/or partition recycling easier. One method for doing this is removing the halide fraction and converting the resulting salt cations (e.g., M^m+) to other chemistries (e.g., M₂O, M₃PO₄) through a process called dehalogenation.

Dehalogenation can be accomplished through a variety of methods, including the conversion of the halide salt cation in air to oxides,^8–10 conversion of halide salt cations to phosphates through reactions with NH₄H₂PO₄,^11–13, (NH₄)₂HPO₄,¹² or H₃PO₄,^14,15 or converting fluoride-salt wastes to a different fluoride-containing compound that is environmentally stable (water insoluble), such as CaF₂, for disposal.¹⁶ Any halide-containing byproducts from the dehalogenation process (e.g., H³⁷Cl, NH³⁷₄Cl) can be captured and recycled to produce new actinide halides or directly disposed (e.g., F). Since the one of the key limiters of waste loading capacity of a waste form is the halide fraction, the dehalogenated salt products (e.g., oxides, phosphates) can be immobilized at much higher salt cation loadings than the full-salt wastes prior to dehalogenation.

Different salt-based nuclear waste streams, such as alkali-based (e.g., Li/K/Na, Li/Be) and/or alkaline earth-based ones, can thus undergo a dehalogenation process with the waste cations being converted to the same anion-type compounds (e.g., oxides, phosphates). These compounds then be mixed with glass-forming compounds to create a final waste form that meets all of the necessary product criteria based on disposal requirements. Some of the primary properties of interest for each waste composition are waste loading (salt cation loading in wt%), chemical durability (e.g., aqueous solubility, leach rates), processability/manufacturability (e.g., higher melt viscosity, lower melt temperatures), mechanical durability (e.g., compressive strength), and radiation/thermal stability under expected radiation fields in the final product.

Historically, glass composition-property models have been developed to predict the properties required for efficient processing and acceptable product qualities of (mainly borosilicate) waste forms. These models have largely been empirical fits of glass composition-property data using single metal oxide concentrations as features. The model development process typically involved the production and development of painstaking, time-consuming, and empirically-created databases. There is an opportunity to leverage the large amounts of data present in existing chemical, physical, and thermodynamic databases to build machine learning (ML) models to predict the waste form properties of interest for an entirely new class of waste forms, e.g., dehalogenated phosphate ceramics or glasses. Thus, the empirical data collection process could be side-stepped entirely or be limited to incremental efforts in the novel chemistries and/or processes where there exist no prior data for the ML models to train on. Such an approach has the potential to bring about a paradigm shift in the design of different types of novel waste forms, including relatively simple single-phase systems (e.g., glasses, single-phase ceramics) or more complex multiphase systems (e.g., glass–ceramic hybrids, multiphase ceramics, and cermets).

2 AI/ML for nuclear waste immobilization: prior art

Approaches utilizing artificial intelligence (AI) or machine learning (ML) techniques have enabled significant advances in many materials science problems, ranging from the prediction of complex materials properties (e.g., superconducting critical temperatures of complex oxides,^17,18 the casting size of metallic glass alloys¹⁹) to the development of self-driving labs.^20,21 However, there remain certain families of materials and research questions that have not benefited as extensively from AI/ML due to their extreme complexity, of which designing new nuclear waste form materials is a prime example. In the case of nuclear waste form design, difficulties in data-driven approaches primarily arise from (1) a vast and complex design space (in terms of chemistry, phase, microstructure, processing) that spans glasses, ceramics, and glass–ceramic composites, and (2) a lack of sizeable legacy datasets for several properties a waste form must possess to be considered viable for disposal (e.g., chemical durability, radiation stability), especially for classes of materials that have not been previously explored. Further, due to the cost of performing experiments with nuclear materials and the complexity of waste form compositions, experimental data acquisition is a major bottleneck.^22–24

Nonetheless, there is a plethora of prior work on using AI/ML approaches to model waste forms,^25,26 and we highlight a few recent reports below, including both ceramic and glass based waste forms. In terms of radiation effects, Pilania et al.²⁷ used ML to explore the physical factors underlying amorphization resistance in pyrochlores (A₂B₂O₇), which have been extensively investigated for use in nuclear waste forms^28–37 and have been incorporated into some variants of the SYNROC waste form.³⁸ Here, an ML model was trained to predict the critical amorphization temperature, T_c, from simple structural features and DFT energetics. Another critical property for nuclear waste forms is chemical durability, which is significantly reduced by nepheline (NaAlSiO₄) precipitation during vitrification of certain waste glasses. To address this challenge, Sargin et al.³⁹ and Lu et al.⁴⁰ built ML models to predict nepheline crystallization behavior from glass composition, comparing several different algorithms and achieving a reasonable classification accuracy. However, due to the data acquisition challenge for nuclear waste forms, we note that the dataset sizes for the aforementioned models is relatively low for ML (on the order of 100–1000 data points), which may limit generalizability and extrapolability.

Similarly, several previous efforts focused on developing ML models for predicting properties relevant for glass waste forms, including glass density,⁴¹ viscosity,⁴² durability,^43–45 glass transition temperature,⁴⁶ thermal expansion,⁴⁷ mechanochemical wear,⁴⁸ and Young's modulus.⁴⁹ While many of these models were trained to predict a single property on a specific family of glasses (e.g., silicates), a recently developed multi-task deep neural network model, GlassNet, has been trained predict 85 various glass properties on the entire SciGlass database⁵⁰ with reasonable accuracy on most of the modeled properties.⁵¹

In addition to the direct modeling of waste form properties, the development of neural network interatomic potentials (NNIPs) trained on first-principles density functional theory (DFT) energetics of waste form-relevant systems has made significant strides. For example, NNIPs have been applied to the modeling of molten salts, enabling the accurate prediction of structural and dynamical properties at normal operating conditions, high-temperature–pressure conditions, and in the crystalline solid phase.^52,53 Additionally, Byggmästar et al. developed a Gaussian Approximation Potential (GAP) to study radiation damage in W,⁵⁴ and Ghosh et al. developed an NNIP to study Cs incorporation into hollandite (A₂B₈O₁₆).⁵⁵

While significant advances have been made in applying AI/ML methods toward modeling nuclear waste forms, approaches that utilize such methods for the design of entirely novel waste forms are not well established. Some recent works have explored the use of AI/ML to design optimal waste glass compositions,^56,57 but a unified framework for an informatics-driven design, especially in previously-unexplored chemistries encompassing ceramics, glasses, and other types of waste forms is lacking.

In the following sections, we present a generally-applicable, closed-loop, iterative design framework that effectively combines AI/ML approaches with physics-based simulations and experiments to design novel waste forms. As a topical use-case, we present the design of new phosphate-based ceramic waste forms as part of an ongoing project under the Advanced Research Projects Agency-Energy (ARPA-E) ONWARDS program, with the following initial target properties: (1) >20% waste cation mass loading, (2) <400 cm³ waste form volume/100 g salt, and (3) >30 MPa compressive strength. For each task in the presented iterative design framework, we discuss available methods and tools, and best practices to use them, with illustrative examples from the phosphate-based waste form design problem. We also present a single end-to-end pass (i.e., the first iteration) through the design process for the same problem, resulting in successful synthesis of phosphate waste forms that satisfy all the target criteria listed above.

3 A system design view of nuclear waste forms

A system design approach primarily uses the well-established processing–structure–property–performance (PSPP) paradigm to understand the underlying materials problem.^58,59 Namely, the ways in which a material is synthesized and processed (including the various steps and their respective parameters and conditions) determines its structure, both on a microscopic level (i.e., the phases that are formed, their atomistic structure, etc.) as well as on a macroscopic level (i.e., the grain structure, level of porosity and other defects, etc.). The overall structure of a material, in turn, determines its properties, which, in turn, determine its performance in the target application. This system design approach can be encapsulated in a so-called “system design chart” where individual components of the PSPP systems are connected by causal links or edges in the forward direction (i.e., from processing to structure to properties to performance), which often capture domain knowledge for the problem. Note that several processing steps can affect multiple structural features of a material, and several structural features can contribute to the properties of the overall material, and so on. Drawing up a system design chart for the materials problem at hand should therefore be the first step in any non-trivial materials design endeavor.

An example system design chart for the recently-proposed phosphate glass waste forms¹² is shown in Fig. 1. Here, a typical processing step can involve dehalogenation of the salt waste via reagents such as NH₄H₂PO₄, followed by melting with glass-forming chemicals (GFCs) such as Fe₂O₃, and eventual quenching and/or slow cooling of the melt. The waste form resulting from such a process can be crystalline, glassy, or a mixture of both, i.e., various crystalline phases embedded in a glassy matrix. In addition, depending on other processing steps such as sintering or hot isostatic pressing (HIP), the waste form may have porosities ranging from 2% to 30%. The properties of interest of the resulting waste forms can then be obtained by direct experimental measurement during one of the processing steps (e.g., melting temperature and viscosity during the melting step) or once the processing is complete (e.g., waste salt cation loading, volume, and compressive strength of the final solid waste form). Finally, within limitations of available experimental facilities, the waste form can be tested to determine its actual performance in near-operation conditions (e.g., chemical durability in aqueous conditions).

Fig. 1 A sample system design chart for nuclear waste form design.

4 A system design-informed sequential learning framework

Sequential learning (or active learning) (SL) is an iterative approach for the design of novel materials via data-driven models. A typical SL workflow consists of the following iterative loop: (1) machine learning (ML) models are trained on existing materials property data, (2) these ML models are then used to make property predictions in an as-yet unexplored design space, (3) the ML predictions, and uncertainties in such predictions, are used to identify the most-promising candidate material(s) to evaluate next, (4) the objective evaluation (via simulations and/or experiments) of the selected candidate(s) is performed, and (5) the loop iterates to augment the training data with results from the objective evaluations, retrain and/or refine the ML models, make new candidate predictions, and so on. SL workflows have been previously shown to significantly accelerate the discovery of new materials, ranging from small molecules to catalysts to semiconductors and others,^60–64 as well as to rapidly optimize manufacturing and processing parameters.^65,66

The construction of such an SL workflow for the materials design problem at hand can greatly benefit from being informed by the underlying system design chart. An example of a system design-informed SL workflow for phosphate waste forms is shown in Fig. 2. An effective translation from the system design chart to a closed-loop SL workflow requires: (1) an initial data generation or collection effort upfront, ideally pertaining to the design spaces of interest, (2) various aspects of processing, structure, and properties to be adequately captured in the representation or schema used to store materials data, (3) physics-based simulations or ML models that can predict both (a) the mixture of phases and their structure in the final resulting waste forms based on their initial compositions and relevant processing conditions, and (b) the target waste form properties of interest, with robust uncertainty estimates, and (5) a scheme to filter and rank candidates in the target design space that takes into account model predictions and uncertainties, as well as any other application-based constraints not previously accounted for. In each of the above steps, incorporating domain knowledge and expertise can have a significant impact on the effectiveness of the SL workflow; this is further discussed in the context of each individual SL task below.

Fig. 2 A sample sequential learning driven workflow for the design of novel waste forms, informed by the underlying system design chart.

4.1 Dataset construction

The first step in an SL workflow is dataset construction. This can involve (1) data collection from various sources such as existing materials databases, published materials property datasets, and individual or small sets of data scattered across literature, (2) data generation using relatively inexpensive physics based simulations (see Section 4.2) and/or a small set of baseline experimental measurements, and (3) data engineering of the collected and/or generated data. The latter step encompasses data curation (e.g., outlier removal, imputation of missing values), data fusion (e.g., combining data for a given set of materials from different experimental measurements or simulations), and other related post-processing.

What datasets should one construct for their effective use in waste form design? The target datasets should be informed by the underlying system design chart. That is, datasets should be constructed for the target materials properties that correspond to the waste form performance criteria. This identification of materials properties data that correlates with waste form performance is non-trivial and requires significant domain knowledge integration for extracting maximum utility. We provide a sample list of openly-available datasets of materials properties and related performance criteria in Table 1, some of which were used for phosphate waste form design in this work. Note that while a 1 : 1 correspondence between materials property and performance metric is ideal (e.g., leach rate [property] ⇒ chemical durability [performance]), ML models can leverage indirectly-related data (e.g., cohesive energy [property] ⇔ melt temperature [property] ⇒ processability [performance]) via approaches such as transfer learning^71–76 (see Section 4.3).

Table 1 A few sample open materials property datasets used to optimize the corresponding waste form performance; here, used to design phosphate-based waste forms

Performance metric	Material property	Dataset source	Dataset size
Mass loading	Formation energy	Materials project^67	∼140 000
	Thermodynamic stability	Materials project^67	∼140 000
Mechanical durability	Bulk modulus	Materials project^67	∼5640
Thermal stability	Melting temperature	Literature^68	∼250
	Liquidus temperature	SciGlass^50	∼45 280
Processability	Glass transition temperature	SciGlass^50	∼91 650
	Viscosity	SciGlass^50	∼6290
Chemical durability	Cohesive energy	Materials project^67	∼140 000
	Solubility	IUPAC-NIST^69	∼670
	Leach rate	ALTGLASS^70	∼2400

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Lastly, while the dataset construction task is often overlooked in favor of exploring more sophisticated ML algorithms, improving data quality and quantity is perhaps the most critical enabler of the eventual success of an informatics-driven approach to materials design. A recent “renaissance” towards data-centric AI (in contrast to model-centric AI) across several other fields is ongoing.^77–79

4.2 Physics-based simulations

Physics-based simulations, such as those based on first-principles density functional theory (DFT), molecular dynamics (MD), calculation of phase diagrams (CALPHAD), and other approaches can be used to perform “computational experiments”, at various time and length scales, that can both augment datasets of materials properties as well as provide faster feedback within an SL design loop. We discuss some of the more widely-used physics-based modeling approaches below, with a focus on how they can be integrated within an SL workflow.

4.2.1 Ab initio atomistic simulations. Ab initio density functional theory (DFT) and related methods can be used to calculate a variety of properties of a material based only on its crystal structure, including but not limited to formation energy, bulk density, elastic moduli, electronic band structure and gap, and others. Due to some underlying approximations in the approach (e.g., the exchange–correlation functional in DFT), the accuracy of a DFT calculation can vary significantly based on the target material/property and its applicability is often limited to materials with unit cells consisting of not more than hundreds of atoms. Nonetheless, these calculations are an invaluable tool to generate initial training data for ML, validate ML-predicted properties of a waste form candidate, improve the description of a waste form composition, etc. within an SL workflow. As an example, below we highlight how a DFT-calculated property (compound formation energy) can be used to estimate the ground state mixture of phases from the overall composition of a waste melt, enabling ML property predictions for those phases.

An analysis of the convex hull of compound formation energies as a function of phase composition can be used to identify the thermodynamically stable phases in a given chemical space.^80–84 Combined with linear programming approaches, a convex hull of a chemical space can rapidly predict compounds that are stable/synthesizable as well as the ground state mixture of phases that can be expected to form under thermodynamic equilibrium conditions at any given composition in that chemical space (see Fig. 3a).⁸⁷

Fig. 3 (a) The convex hull of formation energy in a model A–B system. Blue circles indicate stable phases (“on the hull”), and red diamonds (〈u〉) indicate phases that are metastable or unstable (“above the hull”). Under perfect thermodynamic equilibrium conditions, the unstable/metastable phases will decompose into a mixture of stable phases. For example, AB is predicted to decompose into a mixture of A₃B + AB₃, and its distance to the convex hull (“E_HD”) is an indicator of the thermodynamic drive for the decomposition (or relatedly, likelihood of its experimental synthesis^85,86). (b and c) Slices of the convex hull computed using DFT-calculated formation energies in the Na–Fe–P–O chemical system, with sodium and iron phosphate (3+ and 2+, respectively) end members.

The grand potential ϕ of a collection of phases in a chemical space is given by

where x_i is the relative amount of phase i, μ_j is the chemical potential of element j, and C_i,j is the fraction of element j in phase i. The ground state mixture of phases (i.e., the x_i) can be determined by minimizing ϕ with respect to and , provided the free energies G_i and the list of stable phases in the chemical system (the C_i,j terms) are defined. Typically, tools such as high-throughput DFT are used to calculate the energies of the various phases in the chemical space of interest in a self-consistent fashion, and the phase free energies are approximated by their respective 0 K formation energies. Although using 0 K energies from a computational tool such as DFT ignores finite-temperature effects, kinetic effects, and other factors, and can result in differences between predictions and experimental observations (e.g., see two slices of a DFT-computed Na–Fe–P–O convex hull in Fig. 3b and c, where two experimentally-reported phases, Na₃Fe₂(PO₄)₃ and Na₆Fe₃(PO₄)₄, are predicted to be unstable), this approach is still invaluable for estimating ground state mixture of phases with reasonable accuracy,^85,86 particularly for systems that are not experimentally explored or fully described within the CALPHAD approach (see Section 4.2.3).

4.2.2 Semi-empirical atomistic simulations. Interatomic potential based atomistic simulations, e.g., classical molecular dynamics (MD) and Monte Carlo (MC) methods, are computationally much more efficient and can deal with much larger systems (up to millions or billions of atoms) and longer time scales (micro to milliseconds) than ab initio methods. Such semi-empirical atomistic simulations have been successfully used to study multicomponent glass structures, radiation effects in metals, ceramics and glasses, and to study the effect of microstructure on mechanical properties.^88–90

The fidelity of an MD simulation strongly depends on the type and quality of the underlying interatomic potential (IAP), making the development of an IAP and its parameterization a significant portion of the overall simulation effort. For example, the development of IAPs for the modeling of borosilicate glasses, a widely-used nuclear waste glass, has been a challenge due to the strong dependence of boron coordination on thermal history and composition, which in turn affects properties such as mechanical strength and corrosion rate. However, recent efforts in IAP development have led to successful simulations of borosilicate glasses across a wide range of compositions where boron coordination is in good agreement with experimental observations.^91–93 Also, the development of efficient “reactive” potentials allow for simulations of reactions of aqueous solution with glasses.^94–98

Further, these large-scale atomistic simulations enable so-called quantitative structure–property relationship (QSPR) analysis, linking properties of a material to its atomistic structural features. QSPR analysis has been applied to a wide range of properties such as glass transition temperature, dissolution rate, Young's modulus, and hardness of glass materials, correlating trends in these properties to descriptors derived from structural features such as bond angles, bond energies, coordination numbers, network connectivity (e.g., see Fig. 4a for a visualization of a iron phosphate glass, showing various Q_n units, where n is the number of bridging oxygen atoms connected to network formers Q, Fe₂O₃ and P₂O₅).¹⁰¹ A simple statistical model (e.g., linear or multilinear regression) can then be used to identify relationships between such descriptors and the target property of interest (e.g., see Fig. 4b for an example of modeling Young's modulus as a linear function of a glass network strength descriptor, F_net^99,100).

Fig. 4 (a) Visualization of part of the atomic structure of a sodium iron phosphate glass with composition 5% Na₂O–35% Fe₂O₃–60% P₂O₅, showing the various Q_n units in the glass network (where n is the number of bridging oxygen atoms (red spheres) connected to network formers, Fe₂O₃ and P₂O₅), and (b) linear correlation between Young's modulus and QSPR-based network strength descriptor, F_net, calculated using bond strength of diatomic cations (“BS-dc”) and a multiplicative factor (“m_x”).^99,100 The blue circles are MD-calculated Young's moduli values for various Na–Fe–P–O glass compositions (the error bars indicate variance in the calculated moduli across different structures at the same composition), and the solid black line shows a linear fit with an r² score of 0.95.

4.2.3 Thermodynamic modeling. The Calculation of Phase Diagrams (CALPHAD) approach is a commonly used thermodynamic formalism, along with related numerical methods, to assess phase equilibria in materials under various conditions. Within the CALPHAD approach, thermodynamic properties (e.g., Gibbs energy, specific heat) of phases in a system are described using mathematical models with adjustable parameters, which are then optimized by fitting to all available thermochemical information about phases/sub-systems to arrive at a consistent description of the multicomponent system of interest.

The CALPHAD method can be used to study the typically multicomponent, multiphase nuclear waste materials, e.g., calculate phase diagrams of the waste form systems (e.g., the Fe₂O₃–P₂O₅ iron phosphate system, shown in Fig. 5), equilibrium and non-equilibrium phase evolution during solidification of a molten salt mixture, estimating transformation temperatures between waste-relevant phases, waste salt or waste element solubility limits, and so on.

Fig. 5 The Fe₂O₃–P₂O₅ phase diagram (a) computed using the CALPHAD approach (the liquidus curve is highlighted in solid green), and (b) calculated using ab initio DFT (blue circles indicate “on the convex hull” stable phases and red diamonds indicate “above the convex hull” 0 K metastable phases).

While CALPHAD-based approaches can be used to study the behavior of waste-relevant systems under non-ideal conditions (e.g., as a function of temperature, pressure, chemical potential, pH), they are often limited by the availability of fully-assessed thermodynamic databases for the systems of interest. While ab initio methods can be used to calculate the properties (e.g., formation energy) of novel phases from scratch, such data may suffer from limitations related to “ideal conditions”, as discussed in Section 4.2.1. For example, we compare the 0 K phase diagram calculated using DFT with that from CALPHAD for the Fe₂O₃–P₂O₅ system in Fig. 5. Note that two of the experimentally observed phases, Fe₃PO₇ and Fe₄(P₂O₇)₃, described correctly by CALPHAD are predicted to be metastable at 0 K by DFT. Further, for systems with missing or partial CALPHAD assessments, there are no currently available tools that enable a seamless integration of such ab initio data with existing thermodynamic databases to extend their capabilities to the partially assessed/unassessed system (e.g., the Na₂O–Fe₂O₃–P₂O₅ system, discussed in Section 4.5).

4.3 Machine learning models

The various waste form-relevant property datasets described in Section 4.1 can be used as input to train ML models that can predict those properties of novel waste forms. While sizeable research efforts have gone into ML algorithms and models for predicting materials properties, there are a few considerations to keep in mind specifically towards their effective applicability in an SL workflow for waste forms. First, the crystal structure of a novel waste form is often unknown or yet to be characterized. Even though ML models that are built using crystal structure-based descriptors as input show higher accuracy in predicting materials properties,¹⁰² this information is often a priori unavailable. Thus, ML models that can predict waste form properties purely based on the melt composition alone tend to be of higher value than those that require crystal structure as input. Several approaches have been developed to generate physical descriptors based on chemical composition of a material (“featurization”), e.g., Magpie,¹⁰³ Deml,¹⁰⁴ Matscholar,¹⁰⁵ MEGNet,¹⁰⁶ including off-the-shelf software tools with implementations of such approaches, e.g., matminer,¹⁰⁷ that can be used to featurize waste form compositions.

Second, the type of AI/ML model to build, e.g., based on ensemble approaches such as random forests (RF) and gradient-boosted trees, or deep neural net (NN) approaches such as ElemNet,¹⁰⁸ Roost,¹⁰⁹ CrabNet,¹¹⁰ depends on the target property and the size of the available data. Tree-based models have been shown to generally outperform deep learning models on tabular data, especially for small-to-medium sized datasets typical of materials properties (10²–10³ examples).¹¹¹ While NN-based models are useful for certain larger property datasets (e.g., CrabNet for predicting the formation energy, or GlassNet to predict the glass transition temperature from an input composition), we urge researchers to test “simpler” ML models using off-the-shelf tools such as scikit-learn¹¹² and lolo,¹¹³ as these can often be more interpretable and provide superior extrapolation performance.¹¹⁴

Third, we underline the importance of assessing the performance of ML models using evaluation methods and metrics that are directly relevant to the task at hand. Typically, ML model accuracies are reported using metrics such as r² values or mean absolute errors (MAEs) evaluated in random cross-validation or a random train/validation/test split of the available data, which can be informative of within-distribution generalization but not of out-of-distribution performance. For example, an ML model for predicting melt temperatures trained on chemical composition based Magpie features¹⁰³ and a dataset of congruent melting temperatures of stoichiometric compounds has a reasonable accuracy (MAE of ∼200 K, consistent with prior work⁶⁸) in random cross-validation (Fig. 6a). However, the model fails to capture the liquidus temperature trends in a target design space of the Fe₂O₃–P₂O₅ system, especially the two deep eutectic regions around 40% and 60% P₂O₅ content, where the prediction errors are higher than 400–450 K (see Fig. 6b). In short, using a test strategy (e.g., leave-one-cluster-out [LOCO] cross-validation for estimating extrapolative performance¹⁸) that is representative of the actual environment that the ML model will be eventually used in, is critical for a true assessment of its performance. Further, an accurate evaluation of a model is crucial to improving it, e.g., overcoming data scarcity with strategies such as transfer learning^71–76 or prioritizing targeted data collection (see Fig. 6b, which shows how adding even a small fraction [10–20%] of data from the target design space of Fe₂O₃–P₂O₅ enables the ML model to learn the liquidus curve more effectively than from congruent melting temperatures alone).

Fig. 6 (a) An actual-vs.-predicted parity plot for an ML model trained in predict congruent melting temperatures of stoichiometric compounds, evaluated using 5-fold random cross-validation. (b) Performance of the melting temperature model in predicting liquidus temperatures of a waste form-relevant target system (here, Fe₂O₃–P₂O₅). The dashed black lines represent “ground-truth” liquidus temperatures from CALPHAD, and the blue circles are liquidus values predicted by the melting temperature (T_m) model. The yellow-to-red solid lines show predictions from an ML model that is trained on a dataset with increasingly higher amounts of the target liquidus data (T_L) in the training set, tested on the held-out liquidus data. These results show that the performance of the model, especially in the two deep eutectic-related compositions of 40% and 60% P₂O₅ can be improved significantly augmenting the training dataset with even a small sampling (10–20%) of the target design space.

Lastly, while most of the focus of ML-based modeling of materials properties has been towards developing more accurate models, the efficacy of an ML model within an SL workflow goes beyond simply the model accuracy.¹¹⁵ In other words, it is indeed possible to have performant SL workflows with lower-accuracy ML models. Further, robust uncertainty estimates along with ML model predictions are crucial not only to calibrate user confidence in the model predictions but also for ranking and selecting candidates within an SL workflow (see Section 4.4).

4.4 Candidate ranking and selection

Once ML models for target properties have been trained, the next task in the SL workflow is to rank and select candidates in a target design space for objective evaluation. The target design space is defined by the target application, and can be identified by the underlying system design chart. As discussed earlier (see Section 3), for phosphate-based nuclear waste forms, the design space is often bound by the composition of the liquid melt, which in turn is defined by the waste stream, the reagents used for dehalogenation, glass-forming chemicals (GFCs), and other additives added during the processing step. From a well-defined design space, candidates can be generated via simple enumeration (e.g., a uniform grid of compositions within a Gibbs triangle for a ternary design space). This enumerated list of candidates can be further filtered down using domain knowledge and/or outputs of simulations as an intermediate step.

As illustrative examples of phosphate-based waste form design, we generate candidates in the ternary Na₂O–Fe₂O₃–P₂O₅ and quaternary Na₂O–Nd₂O₃–Fe₂O₃–P₂O₅ spaces (where Na/Nd are the waste cations, Fe₂O₃ acts as the GFC), using a uniform grid spacing of 2.5% along each composition axis. We then apply a combination of domain knowledge-informed rules as well as results from DFT convex hull analysis to filter out candidate compositions with the following characteristics (and are therefore unviable): (1) no salt cations in the composition, (2) unreacted elements, unreacted waste salt, or no phosphates in the ground state phase mixture predicted by the convex hull analysis, and (3) ground state phase mixtures that contain binary alkali compounds (e.g., Na₃PO₄) that are known to be water soluble. The resulting filtered list of viable candidate compositions is much smaller (see Fig. 7).

Fig. 7 Candidate generation, ranking, and selection in two sample waste form design spaces: (a) Na₂O–Fe₂O₃–P₂O₅ and (b) Na₂O–Nd₂O₃–Fe₂O₃–P₂O₅ (with a constant 5 mol% Nd₂O₃). Composition axes are in mol%. In both design spaces, a significant fraction of the candidate compositions are filtered out: with no salt waste or phosphate phases (open circles), with unreacted waste (dark grey filled circles), with binary alkali phases (light grey filled hexagons). Viable candidate compositions are shown as hexagons colored according to (a) salt cation loading in wt%, and (b) a scaled score that quantifies the likelihood of improvement simultaneously over all target metrics over reasonable baseline values; in the latter panel, candidate compositions with salt cation loading >20 wt% are shown as hexagons outlined in bold. The candidates selected for experimental synthesis and validation from the two design spaces, (a) A2 and (b) E1, E2, are annotated.

All the relevant target properties for each viable candidate can then be predicted using the previously trained ML models. The scoring of candidates, and the identification of candidates to prioritize for experimental validation and testing (the “acquisition function”), can be performed one of several ways depending on the design problem. For the design of waste forms in particular, it is desirable for the acquisition function to be: (1) multiobjective, to simultaneously optimize several target properties such as mass loading, waste form volume, compressive strength, and so on; (2) enable end users and domain experts to weight the different targets or success criteria differently. For example, Vienna and Kim¹¹⁶ used a penalty-based approach to multiattribute optimization of borosilicate glasses:

where P_i is penalty associated with the ith property, w_i is the ith property weighting parameter, T_i is the ith property target value, Y_i is the ith property predicted value, and L_i,L and L_i,U are the lower and upper limit for property i. For each formulation, an optimization is performed by minimizing the sum of penalties for multiple properties; (3) able to leverage ML model predictions as well as prediction uncertainties to score candidates. Acquisition functions traditionally used for Bayesian optimization are well-suited for this purpose, such as probability of improvement over a specified baseline, , where x is the target variable to be maximized (e.g., bulk modulus), x₀ is the current baseline performance, and μ(x) and σ²(x) are the predicted value and uncertainty from the relevant ML model. For multiple objectives, a scalarizing function that combines a_PI for all the targets can be defined (e.g., simply a product of a_PI values for uncorrelated targets).

We demonstrate two different strategies for candidate selection in the two example design spaces. For the Na₂O–Fe₂O₃–P₂O₅ design space, we show a single-objective optimization targeting salt cation loading in wt%, which can be estimated from DFT-based convex hull analysis without the need for ML surrogates. We choose a candidate composition from among the ones showing the highest loading (∼18%; higher than the state-of-the-art iron phosphate glasses^6,12,117) for validation (e.g., “A2” in Fig. 7a). For the Na₂O–Nd₂O₃–Fe₂O₃–P₂O₅ design space, we show a multi-objective optimization with the following three objectives as well as a hard constraint of salt cation loading >20%: (1) minimizing melt temperatures (<1173 K, the lowest liquidus temperature in the Fe₂O₃–P₂O₅ binary; see Fig. 5a), (2) minimizing the waste form volume to immobilize 100 g of salt (<100 cm³, lower than state-of-the-art phosphate glass waste forms^6,12,117), and (3) maximizing bulk modulus (>48 GPa, the highest moduli reported for Fe₂O₃–P₂O₅ glasses^118,119). We use the product of the individual a_PI scores as a cumulative PI score [scaled to lie within 0–1 using simple min–max scaling; i.e., a_scaled = (a − a_min)/(a_max − a_min)] as the aggregate metric, representing the likelihood of improvement over all three targets simultaneously (note that the term “likelihood” is used loosely, and is not exactly equivalent to the statistical concept of likelihood), to rank candidates and choose the most promising ones for experimental synthesis and validation (e.g., “E1” and “E2” in Fig. 7b). Note that the final bulk modulus objective used here is different from the predefined project objective (>30 MPa compressive strength). This change is motivated by the lack of compressive strength data available for use in building ML models (for a discussion of this change and general guidelines on how to approach such data-scarce objectives, see Section 6).

4.5 Experimental validation

The final task in a SL workflow is the validation of selected candidates, and closing the loop by providing feedback for the refinement of next round of predictions and candidate selection tasks. Ideally, the synthesis of the target materials and any post-processing steps should mimic the final production process as closely as possible, although this is not always practical (e.g., an industrial process that is hard to replicate in a laboratory). Similarly, the characterization of the synthesized materials and measurement of properties should enable direct feedback into the earlier prediction and selection tasks in the SL workflow; note that this is not always possible, e.g., properties that are hard to measure experimentally, mismatch in target performance metrics and properties for which there exists data to train ML models.

Here, we synthesize the three selected candidate waste forms (A2, E1, and E2; e.g., see Fig. 8a for an image of a quenched sample) and perform post-synthesis heat treatment to simulate a canister pour-and-cool for final disposal (see Section 7.3 for details of the heat treatment profiles used). We then characterize the heat-treated samples using X-ray diffraction (e.g., see Fig. 8b for Rietveld refinement for the as-quenched A2 candidate)—determining the phases in each sample and quantifying their distribution (Table 2). We follow up with measurements of candidate properties such as density (i.e., to validate storage volume estimation) and compressive strength.

Fig. 8 A2 candidate waste form: (a) picture of the as-quenched sample, (b) Rietveld refinement of the diffraction pattern of the as-quenched sample.

Table 2 Phase distribution (in wt%) for samples A2, E1, and E2 observed in experiment using XRD analysis post heat treatment (SC2 profile for samples A2 and E1, and SC1 profile for sample E2; see Section 7.3), and the corresponding thermodynamic ground state phase distribution predicted by DFT and CALPHAD (sample A2 only). R_wp is the weighted profile residual for the XRD refinements

Phase	ICSD#	Sample A2 [SC2]			Sample E1 [SC2]		Sample E2 [SC1]
		Expt.	DFT	CALPHAD	Expt.	DFT	Expt.	DFT
NaFe3(PO4)3	61 696						33.06
NaFePO4	193 244						30.97
NaFeP2O7	237 850				15.96
Na3Fe3(PO4)4	95 532		36.70					36.92
Na2Fe3(PO4)3	200 238	24.41			11.10
Na3Fe2(PO4)3	66 405	23.52			8.10		6.55
Na3Fe(PO4)2	85 558		60.92			79.79		43.98
NaNd(PO3)4	401				9.80
NdPO4	79 750				10.06	20.21	11.54	19.10
NaPO3	174 201			11.0
Na3PO4	33 718	12.45
Na4P2O7	10 370			31.0
Na5P3O10	25 837			13.0
FePO4	79 906	17.51		27.0
Fe2O3	22 505	9.82	2.38	18.0	8.08		13.37
Amorphous		12.30			36.90		4.51
SUM		100.00	100.00	100.00	100.00	100.00	100.00	100.00
R wp		6.38			9.29		7.78

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5 An end-to-end workflow run

To illustrate the end-to-end workflow of an informatics-driven framework, we consolidate and present one full SL iteration for one of the candidate waste forms discussed previously, E2, from the Na₂O–Nd₂O₃–Fe₂O₃–P₂O₅ chemical space (with Na₂O and Nd₂O₃ from the waste stream, Fe₂O₃ added as a GFC, and P₂O₅ from the dehalogenation process).

We first generate an enumerated design space of potential candidates in the quaternary space, as a uniform grid with 2.5 mol% spacing along each composition axis, for a total of 12 341 candidate compositions. For each candidate composition, we then apply DFT convex hull analysis to predict the ground state phase mixture, and apply domain knowledge filters to exclude unviable compositions, as described in Section 4.4. The E2 candidate (composition 37.5% Na₂O + 5% Nd₂O₃ + 20% Fe₂O₃ + 37.5% P₂O₅ in mol%) passes all the domain knowledge filters, and a mixture of {50% Na₃Fe(PO₄)₂ + 21.43% Na₃Fe₃(PO₄)₄ + 28.57% NdPO₄} (mol%) is predicted to be the thermodynamic ground state at its overall composition. We then estimate (a) the salt cation loading (25.29 wt% of Na, Nd combined), (b) the waste form storage volume per 100 g of nominal waste salt (an equivalent amount of NaCl + NdCl₃) as the mole fraction weighted DFT-calculated volume of the ground state phase mixture (54.36 cm³) (c) target properties (e.g., melt temperature, bulk modulus), using pretrained ML models. Note that the ML models used here are trained to predict the properties of the individual phases in the mixture, and the overall properties (and uncertainties) of the mixture are approximated as the mole fraction-weighted mean. For the E2 candidate, e.g., the predicted bulk modulus is 75.6 ± 20.1, the predicted melt temperature is 1480.7 ± 229.3. We calculate the probability of improvement for each property (e.g., ) and aggregate the individual scores into a cumulative likelihood of improvement metric scaled to lie between 0 and 1 using a simple min–max scaling (see Fig. 7b).

We then perform experimental synthesis and heat treatment for the target E2 composition, and characterize the phases present using X-ray diffraction (see Table 2). We measure properties such as density (3.164 g cm⁻³, and corresponding waste form volume) and compressive strength (38.3–40.2 MPa; see Fig. 9), with all properties surpassing the target performance criteria listed at the end of Section 2. The characterization data and measured properties are used to refine the outputs from the physics-based modeling tasks, as well as to augment existing training data and retrain ML models for the next SL iteration; these efforts will be reported in the future.

Fig. 9 Compressive strength of the E2 waste form (tested on two different samples).

6 Discussion and outlook

While such as informatics-driven workflows are powerful tools for accelerating materials design in general, and waste form design in particular, some challenges remain and we discuss a few salient ones below.

The prediction of the exact phases formed during synthesis (and their respective distributions) given the overall composition and a set of experimental conditions and post-processing steps remains an open question. Analyses based on DFT convex hull and related thermodynamic stability measures, while still extremely useful, suffer from limitations related to DFT accuracy and ideal conditions (zero temperature and, often, ambient pressure). As seen from Table 2, the mismatch between computational predictions and experimental realization is large. Some phases that have been reported experimentally are predicted to be 0 K unstable by DFT (e.g., Na₂Fe₃(PO₄)₃ found in both A2 and E1 candidates is predicted to be 60 meV per atom above the convex hull of formation energy). Some other phases are excluded from the DFT analyses due the nature of the problem set up; i.e., with the overall melt modeled as a mixture of the corresponding oxides, certain compounds with mixed elemental oxidation states (e.g., NaFe₃(PO₄)₃ found in the E2 candidate, with two nominal Fe³⁺ and one nominal Fe²⁺) or oxidation states different from the one in the melt ingredient (e.g., NaFePO₄ with Fe²⁺, in contrast to the GFC additive Fe₂O₃ in the melt) are naturally excluded. The latter problem can be mitigated by performing DFT convex hull analyses with varying elemental chemical potential (e.g., μ_O) but increases the computational expense and complexity of the task. While CALPHAD approaches can consider effects of variables such as temperature and chemical potential is calculating phase diagrams, they are severely limited by the lack of assessed databases, especially for compositions relevant to phosphate-based waste forms (e.g., current CALPHAD databases in FactSage do not include thermodynamic data for any quaternary phases in the Na–Fe–P–O chemical space [resulting in the mismatch for sample A2 in Table 2], and assessed databases for the Na–Nd–Fe–P–O chemical space do not exist). Further, while recent work has shown the possibility of predicting crystallization pathways from amorphous matter using a combination of ab initio methods and deep learning,¹²⁰ further efforts are needed to develop a holistic framework that fully bridges the gap between computational design and experimental realization of materials.

Another general challenge is the lack of large, well-curated datasets to train ML models to predict several waste form-relevant properties of interest. For example, properties such as compressive strength are not widely reported for a large set of materials and are not trivial to calculate using ab initio techniques. Similarly, chemical durability measurements of ceramic waste forms are scattered across the literature and no centralized curated dataset exists (in contrast to traditional borosilicate glass-based waste forms, e.g.). While some of these challenges may be mitigated by using domain knowledge integration (DKI; see below) or using techniques such as transfer learning,^71–76 we urge for investments into large-scale data collection and curation of waste form properties that can benefit the community as a whole.

For effective use of informatics-based approaches for the design of waste forms, especially in the context of the above-mentioned challenges, it is crucial to leverage DKI from experts in various parts of the SL workflow. Such DKI can take many forms: (1) identifying data-abundant waste form properties that are correlated with data-scarce performance metrics (e.g., cohesive energy to inform chemical durability, viscosity and liquidus temperature to inform processability, bulk modulus to compressive strength [as used in this work]), (2) defining and constraining the design spaces of interest (e.g., identifying the correct target alkali and rare-earth waste cations, and any potential pool of additives that can be used in waste form fabrication), (3) filtering undesirable candidates (e.g., excluding candidate waste form compositions that are expected to form water-soluble alkali phosphates), and so on.

In particular, while a one-to-one correspondence between target waste form properties and available datasets is ideal, such a fit can be hard to find in practice. Some approaches to tackle this challenge are to (1) perform an initial set of “data-generation” experiments, typically exploratory in nature, (2) perform physics-based simulations that can predict the target property or a close surrogate (e.g., if they are cheaper than experiments), or (3) leverage domain knowledge to identify datasets of a different property that is expected to be correlated with the target property (e.g., if neither experiments nor physics-based simulations are viable). The compressive strength of a phosphate waste form (and the related target performance of >30 MPa) is a property in the latter category. There exists no large curated dataset of compressive strength of ceramics, and generating such a database via experiments or physics-based simulations is prohibitively time-consuming and expensive. So, here we choose to rely on a relatively large dataset of DFT-calculated bulk moduli, a property we expect to be correlated with compressive strength (e.g., a relation between the two quantities has been reported in composite materials such as concrete;¹²¹ similarly, a correlation between elastic and plastic responses during deformation in aluminate spinels, oxynitrides, and nitrides has been reported,¹²² but a clear quantitative relationship in ceramics is unknown). For this choice of a property that is different (but expected to be correlated) to the target property, we still need to choose a suitable baseline value to improve upon in the sequential learning workflow, and we thus choose the highest moduli reported for Fe₂O₃–P₂O₅ glasses, 48 GPa. While we need more data to make a quantitative claim about the relationship between bulk modulus and compressive strength for phosphate ceramics, results presented here demonstrate that the strategy of optimizing for closely-related data-abundant property in place of a data-scarce target property is a viable one in a real-world materials design scenario.

We emphasize that the system design-informed, sequential learning-driven framework presented in this work is a general materials design framework that can be, and has been, successfully applied to designing materials for a wide variety of applications.^{19,44,60–63,65,74} In the context of nuclear waste disposal, the particular end-to-end workflow we demonstrate here can most readily be applied to wastes which require immobilization in glass, ceramic, or ceramic–glass composite waste forms, including LAW, HLW, and high-sodium wastes (e.g., borosilicate glass is already being used to immobilize HLW at West Valley in New York¹²³). For other classes of materials, while the overall design framework presented here can still be applied, some of the specific tools used in this work (e.g., atomistic physics-based simulations, phase diagram calculations) may not be applicable, and alternate experimental and simulation techniques need to be identified. For instance, LAW at Savannah River Site is being immobilized in Saltstone (salt cake mixed with concrete and fly ash), geopolymers (amorphous, aluminosilicate-based inorganic polymers) have been investigated as binders for granular mineral wastes produced by fluidized bed steam reforming for LAW at Hanford, and hydroceramics have been shown to be effective for immobilizing sodium-bearing LAW at INL.¹²³ Such systems cannot be modeled effectively using current ab initio modeling approaches, but the design framework is still applicable.

Overall, the proposed approach of using informatics-based approaches for waste form design has the potential to drastically reshape the way that these types of efforts are conducted across the world. The largest impact of this method, in contrast to the Edisonian trial and error approach, is the potential to cut the time required to find optimal solutions by an order of magnitude (or more). Tapping into crystallographic databases, thermodynamic property databases, phase diagrams, as well as other material property databases, this informatics approach can be used to design waste forms containing anywhere from high amorphous (glassy) phase fractions to high crystalline fractions as well as mixtures thereof. Finally, we note that the opportunities afforded by such approaches extend far past borosilicate and phosphate waste forms, and these techniques can be used for optimize the processing history and conditions to effectively fabricate the next generation of waste forms as well as aiding eventual scale-up efforts for promising waste forms once so identified. Some challenges related to bridging the gap between predictions and laboratory realization of waste forms exist, but current approaches already present avenues for significant acceleration of the design of novel waste forms.

7 Methods

7.1 Thermodynamic analysis

All DFT-based convex hull analysis was performed using data openly available from the Materials Project⁶⁷ and the pymatgen¹²⁴ software package.

All CALPHAD calculations were performed using the commercial FactSage v8.1 software¹²⁵ and associated databases.

7.2 Machine learning

All machine learning models reported in this work were built using the open-source lolo¹¹³ random forests library, using 128 tree estimators and all other default hyperparameters, trained on some of the open datasets listed in Table 1 (i.e., those for melting temperature and bulk modulus). The models used Magpie features generated using only the material compositions, as implemented in the matminer software package.¹⁰⁷ The additional Fe₂O₃–P₂O₅ liquidus data used in Fig. 6 was calculated using FactSage, and the data is made available.

7.3 Experimental synthesis and characterization

The raw reagents (i.e., Na₂CO₃ [Aldrich, 99.9% trace metals], Nd₂O₃, Fe₂O₃ [Baker, 100.4%], and NH₄H₂PO₄ [Sigma Aldrich, ≥98.5%]) were batched using an analytical balance (Mettler Toledo ME204). The reagents were then loaded into 250 mL alumina crucibles (ACC3742, McDanel Advanced Ceramic Technologies) with a Pt/10% Rh lid and melted in a high-temperature furnace (Deltech Furnaces, Inc.) for 2 hours delay at 1250 °C, with a ramp heating rate of 5 °C min⁻¹, followed by quenching on an Inconel plate. The quenched materials were ground to a fine particle size in a tungsten carbide milling chamber and run through slow cooling process with an aliquot of the ground material within a 10 mL alumina crucible (ACM3760, McDanel Advanced Ceramic Technologies).

Two custom heat treatment profiles were employed: “SC2” for samples A2 and E1, and “SC1” for sample E2. SC2 included a ramp rate of 5 °C min⁻¹ from room temperature to 1250 °C, a dwell for 1 hour at 1250 °C, a −5 °C min⁻¹ cooling to 500 °C, a reheat at 5 °C min⁻¹ to 600 °C, a dwell for 12 hours, a ramp cool at −0.03 °C min⁻¹ to 465 °C over the course of 75 hours, and ended with a −1 °C min⁻¹ ramp cooling rate. SC1 included a ramp rate of 5 °C min⁻¹ from room temperature to 1250 °C at 5 °C minute⁻¹, held for 1 hour, ramp cooled at −25 °C minute⁻¹ to 1000 °C, and then ramp cooled to room temperature at −0.1 °C minute⁻¹. These custom heat treatments were designed to homogenize the melt at 1250 °C, rapidly cool to above the glass transition temperature (T_g), reheat and then cool slowly to initiate crystallization.

Once the samples were cooled, vertical slides were prepared using glycol suspensions and a slow speed Buehler diamond saw. The samples were removed from their alumina crucibles, ground in a tungsten carbide milling chamber to a fine particle size and analyzed using X-ray diffraction. Then, a known amount (5 mass%) CeO₂ of a NIST Standard Reference Material (SRM-674b) was added and ground for 30 additional seconds. Samples doped with CeO₂ were run with a Bruker D8 Advance diffractometer in a scan range of 5–90° 2θ, with a 0.01486° 2θ step angle, with 1 second dwells per step. The diffraction patterns were analyzed using Bruker Topas (version 5) software with PDF5+ International Centre for Diffraction Data (ICDD) and Inorganic Crystal Structure Database (ICSD).

Compressive strength tests were run in duplicate on an Instron 5582 (ID 5582R1924) using a fixed rate of 0.1 mm minute⁻¹ in accordance with ASTM C1358-18. Samples were prepared in a 2 : 1 geometry of height : width using a series of procedures from a diamond wire saw, a slow-speed diamond blade saw, and polishing processes. The preferred sample geometry is a cylinder but making cylinders from these samples proved difficult due to small sample size. Thus, rectangular prisms were made that were on the order of ∼6 mm tall by ∼3 mm wide. Samples were loaded until failure and the compressive strength was reported as the maximum uniaxial compressive stress reached when the material failed.

Data availability

Associated research data has been deposited into a repository on figshare.¹²⁶ The following datasets are made available: (1–3) Formation energy, cohesive energy, and bulk modulus data retrieved from the Materials Project database in September 2022. (4) Melting temperatures of solids extracted from the literature.⁶⁸ (5) Liquidus temperatures of waste form-relevant binary–binary systems, calculated as part of this work using FactSage v8.1. (6) Aqueous solid solubility extracted from IUPAC-NIST Solubility Database, Version 1.1.⁶⁹ (7) The Accelerated Leach Testing of GLASS (ALTGLASS) dataset of short- and long-term product consistency tests (PCT, ASTM C1285 A and B) on waste glasses.⁷⁰ (8) Candidate waste form compositions generated and evaluated in this work, including ML predictions, DFT-based convex hull analysis and ground state phase predictions. In addition, sample Python scripts to train machine learning models of melting/liquidus temperatures, to perform convex hull analysis and predict ground state phases, as well as to calculate probability of improvement scores with appropriate scaling are also made available.

Author contributions

Conceptualization: VIH, JES, BJR, JDV, JD; data curation: VIH, MP, SIA, XL, JO, RJS, EKN, BJR, JDV, TM, JK, TN; formal analysis: VIH, MP, JO, RJS, EKN, BJR, JDV, TM, JK, TN; funding acquisition: VIH, JES, BJR, JDV, JD; investigation: VIH, MP, JO, RJS, EKN, BJR, JD, TM, JK, TN; methodology: VIH, SIA, JES, MP, XL, RJS, EKN, BJR, JDV, TM, JK, TN; project administration: JES, JDV, JD; resources: BJR, JDV, JD; software: VIH, SIA; supervision: JES, BJR, JDV, JD; validation: VIH, BJR, TM; visualization: VIH, MP, BJR, JD, TM, JK, TN; writing – original draft: VIH, SIA, JES, MP, XL, BJR, JDV, JD, TM; writing – review and editing: all authors.

Conflicts of interest

There are no conflicts of interest to declare.

Acknowledgements

The information, data, or work presented herein was funded in part by the Advanced Research Projects Agency-Energy (ARPA-E), U.S. Department of Energy, under Award Number DE-AR0001613. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. Pacific Northwest National Laboratory (PNNL) is operated by Battelle Memorial Institute for the DOE under contract DE-AC05-76RL01830. The authors thank Anthony Guzman (PNNL), Irving L. Brown (PNNL), and Timothy J. Roosendaal (PNNL) for help with preparing samples for compressive strength, which proved to be quite challenging, and Jarrod V. Crum (PNNL) for reviewing the manuscript.

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