Ceramic waste forms are designed to immobilize radionuclides for permanent disposal in geological repositories. One of the principal criteria for the effective incorporation of waste elements is their compatibility with the host material. In terms of performance under environmental conditions, the resistance of the waste forms to degradation over long periods of time is a critical concern when they are exposed to natural environments.
The sustained development of nuclear energy requires the safe disposal of radionuclides produced from nuclear fission. Depending on nuclear fuel cycle options, which may involve several processes such as enrichment, off-gas capture, separation, and reprocessing, different waste streams with quite different compositions and levels of toxicity may result. Nuclear waste forms, including borosilicate glass and ceramics, are developed to immobilize these radioactive elements. However, due to factors such as the processing temperature of the glass, incorporation capacity, and long-term chemical durability in aqueous environments, borosilicate glass cannot efficiently immobilize certain radionuclides such as I-129, Cs-137, Cs-135, Tc-99, and Cl-36. If these radionuclides are released into the environment, they most likely form relatively large ionic species such as I−
, and Cl−
in most natural aqueous environmental conditions, due to their large stability field in Eh-pH phase space. Because of their low ionic potential (formal charge to radius ratio), they are less likely to form insoluble compounds through interactions with rocks and other dissolved species in the environment. Thus, they are mobile with a very limited amount of adsorption on the surfaces of rocks in the disposal environment. Furthermore, the surfaces of many silicate minerals, common in the disposal environment, are negatively charged because their pH values for the surface point of zero charge (PZC) are usually lower than the pH values of the groundwater expected to occur in the nearby field and geological formations 
. For instance, for felsic (acidic, more silica content) igneous rocks (granite and rhyolite) consisting of minerals including quartz, albite, orthoclase, muscovite, and their weathered minerals such as kaolinite, their PZCs are from 1.5 to 4.8 
. For mafic rocks such as gabbro and basalt (basic, less silica content), the PZCs of the constituent minerals such as pyroxene and forsterite become higher, at 7.5 and 9.0, respectively 
. However, these basic rocks often consist of a substantial amount of altered minerals with high surface area such as serpentine, montmorillonite, and chlorite (also common in sedimentary rock) with a PZC less than 5 
. As a result, these negatively charged surfaces cause negligible surface adsorption of the negatively charged aqueous ions, such as I−
, and TcO4−
, on the silicate materials because the species and the surfaces are both negatively charged. As such, these ionic aqueous species are highly mobile in the environment. Therefore, these abovementioned elements are considered to be the most problematic radionuclides, which have the greatest potential of long-term adverse impact on the environment. It is therefore essential that appropriate waste forms are developed to efficiently incorporate these elements with sufficiently high loading and high chemical durability in the environment.
Due to their high structural compatibility with certain nuclear waste elements, including long-lived fission products, actinides, and activation products, crystalline phases have been selected to develop ceramic waste forms that immobilize a group of radionuclides, such as synroc, or a single radionuclide, such as iodoapatite 
. Unlike glass waste forms, ceramic waste forms have existing technical challenges and are not omnipotent in the immobilization of all waste components, but they are effective in targeting certain problematic radionuclides. For instance, the iodoapatite (Pb5
I) waste form was developed to incorporate iodine-129 (~16 million years of half-life), which has very low solubility in nuclear glass, because apatite’s crystallographic channel site can accommodate large anions such as iodide 
. Apatite-structured materials (materials with apatite structure but with a composition different from natural apatite) are also capable of incorporating cesium, strontium, chlorine, rare earth elements, and some actinides 
. Similarly, sodalite-structured materials can incorporate iodide in the β-cage site of the crystal structure 
. Hollandite and pollucite waste forms have been developed to incorporate Cs+
in its large channel site and 12-coordinated crystallographic sites, respectively 
. In addition, other crystalline ceramic waste forms such as perovskite, pyrochlore, murataite, monazite, and crichtonite have been developed for the incorporation of various radionuclides, including Cs, Sr, rare earth, and actinides 
. A great advantage of these selected crystalline phases is that each of them has a flexible composition and is capable of incorporating multiple elements by chemical substitution. These synthetic ceramic phases also have natural analogues and are chemically durable in the environment 
. While their compositional flexibility is intriguing in pursuing correlation and coupling relationships between composition, structure, and properties, and provides a great opportunity to explore the compositional space to improve radionuclide immobilization, it also presents an enormous challenge for optimizing the composition of a given waste form with respect to waste element loading capacity and waste form environmental performance. This is because, as these phases have multiple substitution sites, the possible number of combinatorial substitutions at different sites and the number of materials to be tested become enormous when considering potential substituting elements from the periodic table. Therefore, approaches based on the trial-and-error method of testing the possible compositions require a substantial amount of work and decades of research and development. In this regard, methods such as machine learning and other computational methods that are capable of processing a large number of compositions for a given crystalline phase are essential in identifying and narrowing down the potential compositions that are worthy of further consideration.
In addition, ceramic waste forms need to be chemically durable in aqueous environments to ensure their long-term performance in a repository. It would be ideal if waste forms could be designed by concurrently considering both radionuclide incorporation and long-term chemical durability in aqueous environments. While it is not feasible to test the chemical durability of the waste forms under all conditions that may occur in a repository over a time period of hundreds of thousands of years, test protocols such as the Product Consistency Test 
, which was designed to evaluate and screen nuclear waste forms, may not be able to account for chemical durability under different conditions and over a long period of time, as degradation mechanisms may change with the degree of degradation and time 
. Therefore, a systematic test of the chemical durability, aiming to understand the dissolution mechanism, is necessary for the reliable prediction of long-term performance under various environmental conditions. Based on an understanding of the dissolution mechanisms, modeling methods such as those based on thermodynamic kinetic theories that relate the dissolution rate to environmental conditions are necessary for the prediction of the chemical durability over time. Only experiments or computations that are critical to parameterizing the kinetic rate equations are needed. Once the kinetic rate equations are properly parameterized, they can be used to predict dissolution rates under various environmental conditions.
2. Machine Learning for Ceramic Waste Form Design
2.1. Artificial Neural Network Simulation
Machine learning techniques have recently become widely used for material discovery 
. Among them, the artificial neural network (ANN) is a supervised machine learning technique based on statistical principles. The approach is inspired by biological neuron assemblies, their way of encoding and solving problems, and how neurons function in the human brain. The perceptron learning algorithm used in ANN accepts input, performs a computation on the input, and then produces an output 
. As shown in Figure 1
, the input data with weight and bias are first passed to the neurons in the hidden layers and processed by a training function, and the data are then passed to the output neuron. The weights and biases are continually adjusted during the ANN simulation to match the predicted results to the actually observed ones 
. To deduce the relationship between the input and output, a neural network is first trained using a given set of input-output datasets through supervised learning. After the training (supervised learning), the network needs to be validated before it can be used for prediction with an input–output characteristic approximately equal to the relationship of the training problems. Because of the modular and nonlinear activation functions, the network is in principle able to approximate any arbitrary relationship to an arbitrary degree of accuracy 
. For the application of ceramic waste forms with a given crystal structure, the composition and composition-derived properties are used as input parameters. The output parameters are selected to have prediction powers such as crystal structure features and thermodynamic properties that can be used to predict potential new compositions. This approach has been used to predict the compositions of iodoapatite and cesium hollandite and the chemical durability of pyrochlore 
and is expected, to the first order of approximation, to be able to aid the development of other nuclear waste forms for the prediction of potential new waste form compositions.
Figure 1. A schematic diagram of an artificial neural network. W and b are weights and biases for activation function and output. In this diagram, there are 6 input parameters and 1 output parameter, 1 output layer, and one hidden layer with 4 neurons used in the hidden layer.
2.2. Artificial Neural Network Simulation for Ceramic Waste Forms: Cases for Apatite-Structured and Hollandite-Structured Materials
Apatite-structured materials: ANN was used to predict new apatite-structured compositions, including iodoapatite 
. To apply ANN to predict new apatite compositions that incorporate iodide, a dataset of the compositions and fully characterized crystal structures of 86 apatite compositions was compiled and used for training and validation 
. Six parameters, i.e., the average ionic radius and electronegativity of the elements at the A, X, and Z sites of apatite structure A5
Z were used as inputs. For a large anion, iodide (I−
), with a radius of 2.2 Å, to be incorporated in apatite structure, the channel site of the apatite needs to have the right size so that there is no mismatch between iodide and the channel. It was hypothesized that apatite compositions whose channels accommodate the iodide ions (as a spherical particle) without mismatch are the most likely chemical compositions for incorporating iodide. This is a reasonable assumption due to the highly ionic nature of iodide bonding in the channel site. Therefore, channel size is likely a good indicator of possible iodine incorporation and was used as the output of the neural network. As shown in Figure 2
, the channel size was predicted from the trained network. Using 3.5% as the flexibility of the structure to accommodate iodide, the compositions for which the channel sizes are located between the purple lines are predicted to be potential apatite-structured materials incorporating iodide. The result suggests that combinations of A-site cations of Ag+
, and Cs+
, and X site cations of Mn5+
, and Re7+
, are possible apatite compositions that can incorporate iodide at the Z site. This prediction is consistent with existing data from the literature based on experimental synthesis and first-principles calculations. As shown, iodoapatite Pb5
I has been synthesized 
and the predicted channel size is within the values of possible iodoapatite compositions and is ~0.1% from the experimentally determined channel size (Figure 2
). Additionally, iodoapatite Ba5
I and Sr5
I were predicted to be potential apatite compositions, which is in agreement with first-principles calculations. Recent experiments have also confirmed some of these predictions 
. For instance, arsenate–lead iodoapatites Pb5
X (X = OH, Cl, Br, I) have been synthesized from the precipitates of solution at acidic pH and ambient temperature conditions. Their structure and compositions were well characterized and thermodynamic properties such as enthalpy were measured using melt drop solution calorimetry 
. In addition, iodoapatite (Ba5
I) has been synthesized using a high-energy ball milling machine and spark plasma sintering technique 
Figure 2. Prediction of iodoapatite compositions with iodide ions in the structural channel site. The region between the pink lines defines the average radii of possible A and X cation combinations. The channel size is calculated based on the ionic radius of iodide (2.20 Å) and coordination cations and a 3.5% prediction error. The region is also both projected onto the radius-A and radius-X planes and onto the surface of predicted channel size. The stars indicate the locations of Pb5(VO4)3I (blue), Ba5(VO4)3I (green), and Sr5(As4)3I (red) apatite compositions.
Hollandite-structured materials: A similar strategy and method were used to predict compositions of hollandite for the incorporation of cesium-137 
. Chemical substitutions in hollandite (A2
) can occur at both A and B sites, where the A site can be occupied by alkali and alkaline earth elements such as Na+
, and Ba2+
, and the B site can be occupied by various di-, tri-, and tetravalent cations such as Mg2+
, and Si4+
. Both sites can have substantial substitutions to form solid solutions. The A site in the tunnel can accommodate large cations such as cesium and barium. An interesting characteristic of the hollandite structure is that its tunnel size is largely controlled by the B site cations. A site is often partially occupied with vacancies, but both the O and B sites are usually fully occupied. The normal charge is balanced by coupled substitutions at both the A and B sites. For ANN simulations, only four parameters, i.e., the average ionic radius and electronegativity of the A and B sites of hollandite, A2
, were used as inputs (because the oxygen sites always remain fully occupied, the inputs for oxygen sites do not vary with A and B site compositional variance). The number of vacancies was not used as an input and including it did not improve the prediction. Since the objective is to find the appropriate channel size that can accommodate large cations such as Cs, the channel size is used as the output for the structural property, similar to the channel size used for iodide incorporation in iodoapatite 
. A structural stability criterion, called the tolerance factor, was also used to further narrow the ANN-predicted compositions in addition to the channel size. The tolerance factor criterion is often used in experimental studies to rule out candidate structures that are potentially less plausible 
. Figure 3
shows ANN-predicted possible compositions defined by the tolerance factor and channel size of hollandite-structured materials. The data encompass a combination of Ba and Cs with varying A-site occupancy and B-site compositions. Given the tolerance factor between 0.90 and 1.10 and channel size between 2.80 and 3.15 Å, which are reasonable ranges based on the experimental observations in the dataset, the compositions located within the rectangular box (Figure 3
) can be considered possible hollandite compositions. A wide range of previously unexplored Cs–hollandite compositions were evaluated as possible, including M4+
at the B site. These compositions are likely to be potential candidates for immobilizing Cs based on the ANN predictions of their channel size 
. In particular, a combination of some of the aforementioned variable-valence M3+,2+
cations, such as Fe3+,2+
, has also been predicted to be highly possible. These hollandite compositions can also be candidates for accommodating the chemical changes as a result of the β decay of Cs-137 due to radioparagenesis 
. Although Ti-based hollandite compositions are the most extensively investigated, encompassing most recent studies 
, several new Ti-based hollandite compositions were also predicted to have potential for the immobilization of Cs-137 
Figure 3. Prediction of Cs–hollandite compositions based on the channel size and the tolerance factor (τH) with a combination of Cs and Ba at the A site. B-site compositions comprise combinations of 3+ and 4+ cations from the entire dataset considered in this study.
3. Computational Materials Design and Performance Prediction of Ceramic Nuclear Waste Forms
Understanding the mechanisms of the incorporation of the waste elements in ceramic waste forms and their dissolution kinetics in the environment plays an important role in nuclear waste management. Such an understanding is essential for the performance evaluation of nuclear waste forms in the environment. Examples of ceramic waste forms based on apatite-structured and hollandite-structured materials demonstrate a methodology with which new iodoapatite and Cs–hollandite compositions were predicted using a combination of artificial neural network simulations and first-principles calculations. The dissolution rates were predicted under various environmental conditions using thermodynamic rate equations. By integrating the predictions of the compositions of waste forms and their performance in aqueous solutions in a holistic framework, this strategy could be used to design ceramic waste forms with optimal incorporation capacity and environmental performance by varying the chemical compositions.
Figure 4 summarizes this approach, which can provide a path for accelerated ceramic waste form development and performance prediction for ceramic nuclear waste forms. In this approach, material design and performance modeling are considered simultaneously. The design of ceramic waste forms considers the loading capacity of waste elements and focuses on problematic radionuclides. Certain crystalline ceramics can be targeted using thermodynamic principles in combination with machine learning and first-principles calculations. Performance modeling considers the dissolution kinetics associated with rate-determining critical processes using rate equations. The intrinsic properties in the rate equations are parametrized using a series of experiments or predicted based on relationships between structure, composition, and properties. Critical experimental tests are carried out to understand the dissolution reactions and benchmark intrinsic properties predicted by theories. The verification of modeling results outside the range of available data can be performed to validate the predictions of the modeling. Models and model parameters are open for revision as new data emerge. The verified predictive models will be able to predict new waste forms, their incorporation capacity, and long-term dissolution performance. The composition of a given waste form can then be optimized in terms of immobilization efficiency and chemical durability.
Figure 4. A schematic diagram for the development of advanced ceramic waste forms based on a holistic approach by simultaneously considering material design and performance prediction and model verification.
The methodology outlined above for apatite-structured and hollandite-structured materials is also suitable for other complex materials with chemical substitutions on multiple structural sites. Materials such as perovskite, pyrochlore, murataite, monazite, and crichtonite, although dissimilar to apatite-structured and hollandite-structured materials in structure and composition, also have multiple crystallographic site substitutions, flexible crystal structures, and complex chemical compositions. These materials are suitable for the immobilization of various waste streams and waste elements, including fission products, actinides, and activation products. While the structural flexibility and compositional complexity offer plenty of room for the optimization of these nuclear waste forms, the composition space becomes too populated to be handled by experiments alone. It is hoped that the approach proposed here of combining computational approaches with predictive modeling will provide a robust strategy to accelerate the development of ceramic waste forms.