Disordered enthalpy-entropy descriptor for high-entropy ceramics discovery

High-entropy ceramics (HECs) are a burgeoning class of materials with diverse applications including thermal barrier protection, wear-resistant coatings, thermoelectrics, batteries, and catalysis. Their desirable mechanical properties, thermal and chemical stability, and high-temperature plasmonic resonance make them attractive for extreme environments. However, the discovery of single-phase HECs has primarily been driven by experimental trial and error, with the exception of high-entropy carbides, which have been explored computationally using the entropy-forming-ability (EFA) descriptor. This slow pace of discovery highlights the need for more effective theoretical tools. This research addresses this challenge by defining a new concept, "functional synthesizability," and developing a novel descriptor, the disordered enthalpy-entropy descriptor (DEED), to quantify it. Unlike previous approaches that focus solely on intrinsic material properties, DEED considers the chosen synthesis process—in this case, hot-pressed sintering—as a crucial factor in determining synthesizability. This process-centric view allows for a more practical and effective approach to materials discovery, moving beyond the limitations of intrinsic synthesizability assessments.

Existing literature on high-entropy ceramics emphasizes their potential but acknowledges the challenges in their synthesis and discovery. Several review articles highlight the diverse applications and unique properties of HECs, underscoring their importance in various technological fields. Computational methods, particularly density functional theory (DFT), have been used to study the properties of HECs, but previous descriptors, such as EFA, have shown limitations, especially in systems with non-homogeneous enthalpy landscapes. The lack of comprehensive ab initio enthalpy data for various HECs has also hindered computational discovery efforts. Studies on the synthesizability of materials have often lacked a clear consensus, leading to inconsistencies in predicting which materials can be successfully synthesized under specific conditions. This paper addresses the need for a more robust and predictive descriptor.

The researchers defined functional synthesizability as a function of the chosen manufacturing process, in this case hot-pressed sintering. The DEED descriptor was developed to quantify functional synthesizability by balancing the entropic gain in generating disorder against the enthalpy costs. The entropic gain is estimated through the statistical ensembles of configurational formation enthalpies, reflecting the degeneracy of states. The enthalpy cost is determined by the ensemble of distances of these formation enthalpies from the convex hull, representing the Gibbs free energy of stable configurations. DEED uses the first and second moments of the thermodynamic density of states (Ω(E)), associating enthalpy loss to the former and entropy gain to the inverse of the latter. The inverse of DEED is correlated with the miscibility gap critical temperature, providing a connection to an observable quantity and allowing for the prediction of functional synthesizability based on the relationship between synthesis and miscibility gap temperatures. To make calculations feasible for a large number of chemical compounds, the authors developed a convolutional algorithm, convolutional partial occupation (cPOCC), drastically reducing computational requirements. This algorithm partitions the calculations into smaller, manageable subsystems. For validation, 952 disordered ceramics (carbides, carbonitrides, and borides) were characterized using the aflow++ ab initio framework. Nine new carbonitrides and eight new borides were synthesized experimentally based on DEED predictions, using hot-pressed sintering after a 24-hour dry mixing with extra carbon to promote oxide reduction. X-ray diffraction (XRD) and scanning electron microscopy (SEM) were used to characterize the synthesized materials, confirming DEED's predictive capabilities. The compensation temperature (Θ), which is inversely proportional to DEED, serves as a fingerprint of the order-disorder transition and provides another metric for assessing the ease of synthesis.

The DEED descriptor successfully predicts the functional synthesizability of high-entropy ceramics, outperforming previous methods like EFA and VEC, especially for carbonitrides and borides. The study successfully synthesized 17 new single-phase high-entropy carbonitrides and borides, validating the DEED predictions. Specifically, the newly synthesized single-phase carbonitrides include (HfNbTiVZr)CN, (HfNbTaTiV)CN, (NbTaTiVZr)CN, (HfTaTiVZr)CN, and (MoNbTaTiZr)CN. The newly synthesized single-phase borides include (HfMoNbTaZr)B₂, (HfNbTaTiV)B₂, (CrMoTiVW)B₂, and (CrHfNbTiZr)B₂. DEED also showed its ability to predict the formation of microstructures, such as pearlite-like lamellae in multiphase carbonitride systems, indicating potential for controlling microstructure through compositional design. The convolutional algorithm (cPOCC) proved effective in accelerating calculations, enabling the analysis of a large dataset of ceramic compositions. The compensation temperature (Θ), derived from DEED, showed a correlation with the order-disorder transition temperature, providing further validation of the descriptor's physical meaning. The results demonstrate that DEED is robust and applicable across various chemical systems and crystal structures, offering a new framework for the accelerated discovery of novel high-entropy ceramics.

The success of DEED in predicting the synthesizability of both carbides and carbonitrides, and borides, demonstrates its broad applicability. The superior performance compared to EFA and VEC highlights the importance of considering both entropic and enthalpic factors in a balanced way. The successful experimental synthesis of the predicted materials strongly validates the theoretical framework. The ability to predict not only the formation of single-phase materials but also multiphase systems with specific microstructures opens new avenues for tailored materials design. The development of the cPOCC algorithm significantly reduces computational cost, making DEED a practical tool for high-throughput materials discovery. The correlation between the compensation temperature (Θ) and the order-disorder transition temperature further reinforces the physical relevance of DEED. The presented findings significantly advance the field of high-entropy ceramics by providing a robust and efficient tool for accelerating the discovery of novel materials with desired properties.

The study successfully introduces DEED, a robust thermodynamic descriptor that accurately predicts the functional synthesizability of high-entropy ceramics. The descriptor's ability to capture the balance between enthalpy and entropy costs proves superior to existing methods. The development and successful implementation of the cPOCC algorithm significantly enhances computational efficiency, expanding the scope of DEED's application. Future research could extend DEED's application to other types of disordered ceramics, such as high-entropy oxides, and explore the possibilities of integrating machine learning techniques to further refine the predictions and accelerate the discovery process.

While DEED demonstrates excellent predictive power, some limitations exist. The accuracy of DEED relies on the accuracy of the underlying DFT calculations and the approximations inherent in the cPOCC algorithm. The definition of functional synthesizability is specific to the hot-pressed sintering process used in this study, and the applicability to other synthesis methods requires further investigation. The current study focuses on transition metal carbides, carbonitrides, and borides; extension to other chemical systems requires additional validation. Finally, experimental conditions like sintering temperature and time can influence results, indicating that DEED should be used in conjunction with careful experimental design.