Discovering the Ultralow Thermal Conductive High-Entropy Pyrochlore Oxides Through the Hybrid Knowledge-Assisted Data-Driven Machine Learning

doi:10.21203/rs.3.rs-2436395/v1

Download PDF

Article

Discovering the Ultralow Thermal Conductive High-Entropy Pyrochlore Oxides Through the Hybrid Knowledge-Assisted Data-Driven Machine Learning

https://doi.org/10.21203/rs.3.rs-2436395/v1

This work is licensed under a CC BY 4.0 License

Version 1

posted

You are reading this latest preprint version

Lattice engineering and distortion have been considered one kind of effective strategies discovering advanced materials. The instinct chemical flexibility of high-entropy pyrochlore oxides (HEPOs) motivate/accelerate to tailor the target properties through phase transformations and lattice distortion. Here, a hybrid knowledge-assisted data-driven machine learning (ML) strategy is utilized to discover the HEPOs with low thermal conductivity (𝜅) through 17 rare-earth (RE = Sc, Y, La ~ Lu) solutes optimized A-site of A₂B₂O₇. A designing routine integrating the ML and high throughput first-principles has been proposed to predict the key physical parameter (KPPs) correlated to the targeted 𝜅 of advanced HEPOs. Among the smart-designed 6188 (5RE_0.2)₂Zr₂O₇ HEPOs, the best candidates are addressed and validated by the principles of severe lattice distortion and local phase transformation, which effectively reduce 𝜅 by the strong multi-phonon scatting and weak interatomic interactions. Particularly, (Sc_0.2Y_0.2La_0.2Ce_0.2Pr_0.2)₂Zr₂O₇ with predicted κ below 1.59 Wm⁻¹ K^− 1 is selected to be verified, which match well with the experimental κ = 1.69 Wm⁻¹ K^− 1 at 300K and could be further decreased to 0.14 Wm⁻¹ K^− 1 at 1473K. Moreover, the coupling effects of lattice vibrations and charges on heat transfer are revealed by the cross-validations of various models, indicating that the weak bonds with low electronegativity and few bonding charge density and the lattice distortion (r*) identified by cation radius ratio (r_A/r_B) should be the KPPs to decrease κ efficiently. This work supports an intelligent designing strategy with limited atomic and electronic KPPs to accelerate the development of advanced multi-component HEPOs with properties/performance at multi-scales.

Physical sciences/Materials science/Theory and computation/Computational methods

Physical sciences/Materials science/Structural materials/Ceramics

High-entropy oxides

Thermal conductivity

Pyrochlore

Key physical parameter

First-Principles

High-entropy oxides (HEOs) have been considered as one kind of promising thermal barrier coating (TBC), which can meet the demands of the high-temperature application in jet and industrial gas-turbine engines ^1–3. The appealing functionality can be originated from their superior compositional flexibility, which can be described as large lattice distortion, sluggish kinetics, increased size and mass disorder ^4–7. These descriptive parameters could be treated as the genes/basic building-blocks or the so-called key physical parameters (KPPs) addressing the correlations between mechanical and thermal properties, which present a relatively straightforward physics-informed relationship with the target property/performance ^4,8,9, such as lattice distortion for solid-solution strengthen and alloy composition for thermal expansion coefficient and heat capacity ⁹. Since the A₂B₂O₇ type high-entropy pyrochlore oxides (HEPOs) have emerged as leading candidates for a new generation of TBC with significantly lower thermal conductivity and chemical stability at higher temperatures ¹⁰. It is essential to develop a heat transfer model via as few atomic and electronic KPPs as possible from the present various ones ^3,11, thus, proposing a strategy accelerating the development of advanced HEPOs with intrinsically low thermal conductivity.

Low thermal conductivity (κ) is the basic criterion for the selection of new candidate HEPOs as TBC materials, which has garnered attention in relative research. Both established electronic-structure optimizations and phonon scattering modulation strategies are available for reducing thermal conductivity by using various intrinsic and extrinsic effects ^12,13. Attributing to the potential low κ of HEPOs largely originates from the lattice distortion effect ^6,7,14, lattice engineering has been inspired and developed as a viable strategy to optimize both electronic and atomic environments by employing the local lattice distortions constituting an effective descriptor ⁴. For instance, size disorder ^7,14 has been proposed to decrease the κ by inducing severe anharmonicity, which results in enhanced multi-phonon scatting. Moreover, the glass-like phenomenon ¹⁵ and liquid-like behavior ¹⁶ could yield ultralow lattice thermal conductivity by the resonant phonon scattering and elimination of some vibrational modes, indicating that serious lattice distortion would be an effective strategy to reduce the κ ¹⁷. Furthermore, low κ has also been correlated with order-disorder phase transformation ^18,19. In A₂B₂O₇, the structure types can be transformed from pyrochlore to fluorite by controlling the relative ionic size difference of the A- and B-site cations and their ionic radius ratio (r_B/r_A), and then manipulating the lattice distortion and the random oxygen sublattices ²⁰. These unique structural features make it possible to control order-disorder phase transformation by adjusting the local r_B/r_A Doping on A and/or B sites could not only dominate the phase stability and radiation resistance ²¹ but also control the oxygen vacancy concentration and its migrations ¹⁸. Additionally, κ can be further decreased by lone-pair electrons ²² and resonant of chemical bonding ²³ with lattice softening mode. These reported electronic-structure basis modulation strategies could provide a better understanding of the physics underlying connections between weaker bonding and lower κ, and of the ability to become a key physical descriptor ¹². However, it is critical to discover and design the promising HEPOs with ultra-low κ in such a great amount of untapped composition A-site space of A₂B₂O₇ drops ^5,6,14. Therefore, employing the lattice engineering/distortion and order-disorder phase transition to modify the electronic structure and phonon transport of HEPOs could be an intelligent designing strategy for the development of advanced ultralow thermal conductive HEPOs.

It is understood that atomic vibration or the phonon plays the dominant role in optimizing the heat conduction of semiconductors and insulators. Making an accurate description of lattice thermal conductivity (κ_L) is time-consuming attributing to the normal scales linearly, even increasing exponentially with the number of atoms ²⁴. The quasi-harmonic Debye model ²⁵ is always selected as the suitable method for implementation in high throughput screening of target low 𝜅 HEPOs ^26,27, which is the most efficient approach by identifying trends and simple descriptors (Grüneisen parameters and Debye temperature) addressing the accuracy-efficiency trade-off in the prediction of thermal properties ²⁶. Meanwhile, Debye temperature can be affected dramatically by the Grüneisen parameters, which can be obtained/predicted by various definitions from both macroscopic and microscopic levels ²⁸. Moreover, the data-driven and data-intensive machine learning (ML) algorithms have become promising viable strategies, which offer novel solutions to dig out the hidden connection among existing databases and predict desired material target properties ^8,9,29,30. Recently, it has been emphasized on the importance of direct descriptor-target prediction ⁸ and multifaceted modeling approach ⁹ for materials design, such as applying the integrated ML, computational thermodynamics and lattice engineering to examine the shape-memory behavior of zirconia ceramics ⁴. The investigations of modeling κ by ML methods have concentrated on establishing composition-property regression models ^31,32 and ML interatomic potentials accelerated approach ^33,34. Domain knowledge-assisted selection of KPPs will overcome the limitations of the explainable ML models in opening the black box ²⁹ and enhance the predictive ability of addressed models/algorithms. Furthermore, optimizing domain knowledge in the feature selection process of ML regression could accelerate the identification of the advanced key design principles beyond simply discovering target materials ^4,9. Accordingly, coupling the first principles with ML predictions will not only realize multiscale cross-validation to promote the process of finding desired HEPOs but also unravel the physics of the underlying process from an atomic and electronic point of view ^29,35,36. Therefore, considerable efforts have been carried out to make ML more transparent and interpretive ³⁷, including identifying the features most driving model predictions ³⁵ or creating intelligible explanations of a model ³⁰.

In the present work, a hybrid knowledge-assisted data-driven ML strategy is utilized to discover the HEPOs with low 𝜅 through optimizing the A-site of the A₂B₂O₇ lattice with 17 rare-earth (RE) solutes. Here, RE solutes include both the heavy RE elements (HRE, Tb ~ Lu, Sc, Y) and the light lanthanide (La ~ Gd), presenting their contributions to the principle KPPs utilized to address the target properties. Since knowledge-based modeling will be the key feature connecting physical properties in multi-scales ^30,35, our proposed domain knowledge-assisted ML hybrid DFT-based thermodynamic calculations and experiments, present the phonon and electronic contribution for 𝜅.

The proposed hybrid knowledge-assisted data-driven machine learning strategy for HEPOs design combines the lattice engineering and distortion design criteria, the domain knowledge-assisted ML algorism, the high throughput first-principles calculations/validation, and the further experimental validation (Fig. 1). Firstly, the promising HEPOs components are generated under the domain knowledge assisted explainable ML algorism, consisting of five main parts, including the datasets consisted of 171 reported data in literature and summarized in Table S1, the cross-comparison 8 ML models for model selection, the feature extractions for KPPs classification guided by domain knowledge, the predictive performance with interpretable ML and the smart design HEPOs via the proposed model in terms of addressed KPPs. Secondly, the selected candidate from the ML strategy is further investigated by the first-principles study, which includes the thermodynamic combinatorial models for computational verification and various atomic and the electronic level methods supporting an insight into the physical mechanism of KPPs. Finally, the most promising composition is selected for the following experimental measurement and characterization. Therefore, based on first principles predictions and experimental verification, the identified descriptors for efficiently decreased 𝜅 will be further fed back to the domain knowledge, addressing the most precise model with the fewest KPPs. All the details will be discussed in the following.

Hybrid knowledge-assisted ML Design criteria

In the present work, the ML investigations consist of data preparation, feature preselection, ML model selection, knowledge-based key feature selection and predictions ^8,9. The main purpose/motivation is to reveal the intrinsic features closely related to the heat transport behavior of HEPOs via a domain knowledge-guided interpretive ML strategy, achieving the target performance orient design for HEPOs efficiently. Based on the established feature pool in Table 1, the feature selection process has been departed into three steps to identify the most relevant descriptors, including a Pearson correlation analysis, model selection and use of the cross-validation method. To reach acceptable accuracy, the first step involves feature filtering. As shown in Fig. 2a, 13 features were selected first and the Pearson correlation coefficient method was used to evaluate these features, which with a high correlation coefficient are considered highly correlated. In particular, those intrinsic properties related to structural information will be efficiently addressed and considered as the proper descriptor in ML. The six descriptors down selected are m_avg, m*, r_A/r_B, r*, E_neg and VEC to represent crystal (highlighted in red) and electronic (highlighted in blue) properties as input features. Several well-known ML models, including ET, RF, XGBoost, GNB, KNN, LR, DT, and SVM, were built through training using the dataset with the target property and the preselected features. The cross-validation method was utilized to estimate the test error for these models, and the best-performing model is chosen by considering its predictions. Specifically, the performance of several ML models has been evaluated by the R² and RMSE values from all the models for all possible subsets of the six retained features as shown in Fig. 2b. The XGBoost model has the lowest RMSE and higher average R² on the test observations and is used as the preferred ML model.

In order to further address the dominate KPPs, it is essential to find the optimum feature combination that yields the best prediction efficiency and remains knowledge-based selections. The SHAP values of 6 pre-selected features were calculated with the best model via XGBoost, as presented in Fig. 2c. it is worth mentioning that the XGBoost algorithm with a wrapper-based method was employed to test all possible sets and subsets of combinations of pre-selected features. The mean RMSE based on different subsets of features is plotted in Fig. 2d, presenting/indicating different combinations of feature sets have different predictabilities, while the lowest RMSE can be achieved by only one feature but three features can reach similar performance. It is understood that the consideration of domain knowledge is an integral part of feature selection, which could build the scientific connection between descriptors and heat transport mechanism by introducing the physical meaningfulness of the specific KPPs.

Combing Fig. 2c and Fig. 2d with both the SHAP analysis and physical considerations, r*, r_A/r_B, and E_neg are correctly identified as the three KPPs attain low-κ HEPOs, representing for scatting lattice distortion effect, pyrochlore to fluorite transformation and correlating to the bond strength of constituent atoms, respectively. r* and r_A/r_B are structural in nature from the perspective of the molecular level, and E_neg originates from potent insight into the electronic structure. Specifically, r* is used to represent complex structures with strong anharmonicity and has been widely applied in guiding the advanced development of high-entropy alloys ^4,9,38. It is essential to seek the large r* among HEPOs composed elements, which would be treated as one kinds of KPPs. Moreover, different values of r_A/r_B correspond to the different structures, which is important for representing the complex structure as the chemical complexity and oxygen vacancy is determined by structure type. It has been further evidenced that the conductivity of structures with imperfections can be tuned by manipulating the concentration of oxygen vacancies ³⁹. Furthermore, E_neg is a parameter to represent the ability to attract electrons inducing bond strength, scatting the electronic contribution, and contributing to the lattice distortion effect ¹³.

Figure 2e plots the predicted 𝜅 at room temperature as a function of the reported 171 values (listed in Table S1) from the ML model using the XGBoost. The ML model accurately predicted 𝜅 across a diverse range of HEPOs, along with a reference line (y = x) to facilitate visualization. It achieves R² of 0.985 implying the high performance of the XGBoost model in predicting thermal conductivity, and further demonstrating the accuracy of the knowledge-based feature selection/classification.

Smart design HEPOs via addressed KPPs

The interactions of ML with fundamental knowledge feedback a series of composite for five-component pyrochlore oxide (5RE_0.2)₂Zr₂O₇ (RE = Sc, Y and La ~ Lu) with the lower κ, considering an A-site disordered HEPOs. Within all possible subsets of combination among 17 RE elements, the visualizations of the predicted 𝜅 datasets of 6188 (${C}_{17}^{5}$) kinds of HEPOs compounds at 300 K are displayed in Fig. 2f, arranged in order of the distribution frequency. The correlations between three KPPs (r*, r_A/r_B and E_neg) and the 6459 samples (6288 points relate to the current work and the other 171 are taken from the publications) are examined in Figure S1. It can be seen that thermal conductivity presents a weak systematic dependence on E_neg but shows a linear correlation of r* and r_A/r_B. To clearly distinguish among the vast composition space, the 6188 HEPOs were divided into 8 groups (G1 to G8), and most components are distributed in G1 with a lower κ around 1.55 Wm⁻¹ K⁻¹. To further understand the underpinnings for the existence of the KPPs to heat transfer, Fig. 2g plots the distribution of κ from G1 to G8 versus r*, r_A/r_B and E_neg, respectively. These immediately relevant trends reconfirm that κ has a good correlation with the overall size disorder ^{4,9,38 14}. Moreover, lattice distortion is affected by both atomic size mismatch and electronegativity. All the compositions in G7 and G8 mainly have E_neg < 4.76 and r_A/r_B < 1.46 and higher 𝜅 (Fig. 2g). While fluorite being stable in the region r_A/r_B < 1.46, the electronegativity difference between ions could affect ionic bond strength thus κ ¹³.

To determine the most effective combination of RE elements for target performance, the distribution function of RE elements in the corresponding group is displayed in Fig. 2h. The frequency distribution of the RE elements for the component of HEPOs in the group (G1) with the lowest κ and the higher κ groups are compared in Figure S2 and Figure S3. It is found that HRE (Tb ~ Lu, Sc, Y) is mainly concentrated in G8 with higher κ. The phase structure of A₂B₂O₇ changes with RE cations radius ¹⁹ exhibiting the pyrochlore structure with light lanthanide (La ~ Gd) and the fluorite type with HRE. Accordingly, the structure of G8 is mainly fluorite-type consistent with those aforementioned findings in Fig. 2g. On the contrary, the G1 group has lower κ which is mainly composed of RE elements with high radius differences. With the decreasing size of the RE³⁺ radius in the order of La ~ Lu, the distribution of RE elements frequency in G1 presents a U-shaped relationship, indicating the importance of the lattice distortion effect.

For further validation of these promising predictions, one composition (5RE_0.2)₂Zr₂O₇ HEPOs in G1 was chosen based on Fig. 2. Since higher Y contents can offer improved insulating potential and stabilized phase ⁴⁰ and the highest frequency of Sc elements appearing in G1 of the lowest κ, Sc and Y have been selected as the essential HRE elements constructing (5RE_0.2)₂Zr₂O₇ HEPOs although Sc is expensive and Y has a lower frequency in G1. Moreover, considering the U-shaped relationship between light lanthanide and HRE and the more computationally expensive nature of HRE for 4f shells, three light lanthanides (La, Ce and Pr) were chosen to result in severe lattice distortion by inducing the large cation size difference with the selected HRE (Sc, Y). Therefore, (Sc_0.2Y_0.2La_0.2Ce_0.2Pr_0.2)₂Zr₂O₇ in G1 with the theoretical κ below 1.59 Wm^− 1 K^− 1 was selected and further verified experimentally and theoretically.

Thermodynamic combinatorial models for cross-validation

The electronic transport behavior of the composition designed with the present approach has been fully studied in this section for realizing the cross-validations. The high-throughput first-principles calculations were performed to evaluate the 𝜅_L with the thermodynamic considerations and to verify the importance of the design criteria. Figure 3 shows the combinatorial schema (details in “Methods”) of Debye temperature $\varTheta$, Grüneisen parameters γ to the 𝜅_L of RE₂Zr₂O₇ (RE = Sc, Y, La, Ce and Pr) oxides and (5RE_0.2)₂Zr₂O₇ HEPOs with temperatures ranging from 200 to 1600 K. In Fig. 3a, the ${\varTheta }_{\left(1\right)}$ curve are clearly lower than other all curves calculated by Eq. (4) and significantly different from experimental data, which is attributed to the regardless of the volumetric effects at higher temperatures for Eq. (3). When the Grüneisen parameters (approximations shown as ${\gamma }^{\left(1\right)}$ for thermal formulation, ${\gamma }^{\left(2\right)}$ for Slate, ${\gamma }^{\left(3\right)}$ for Dugdale–MacDonald, ${\gamma }^{\left(4\right)}$for Vaschenko–Zubarev and ${\gamma }^{\left(5\right)}$ for modified free-volume) are compared in Fig. 3a, the calculated Debye temperatures ${\varTheta }_{\left(m\right)}\left({\gamma }^{\left(n\right)}\right)$ using these five γ have a remarkably little variation for La₂Zr₂O₇, lying in the range 400–523K at room temperature. As all the analytical forms of $\gamma$ are based on the assumption, which are failed to describe the thermoelastic behavior of HEPOs with great accuracy, the $\gamma$ down selected is ${\gamma }^{\left(4\right)}$ for more theoretically correct ⁴¹. ${\varTheta }_{\left(2\right)}\left({\gamma }^{\left(3\right)}\right)$ will be the best descriptor for the calculation of the 𝜅_L. The comparison between ${\kappa }_{\left(1\right)}\left({\varTheta }_{\left(2\right)}\right({\gamma }^{\left(3\right)}),{\gamma }^{\left(3\right)})$, ${\kappa }_{\left(2\right)}\left({\varTheta }_{\left(2\right)}\right({\gamma }^{\left(3\right)}),{\gamma }^{\left(3\right)})$ is depicted in Fig. 3b. The ${\kappa }_{\left(2\right)}\left({\varTheta }_{\left(2\right)}\right({\gamma }^{\left(3\right)}),{\gamma }^{\left(3\right)})$ reproduces the experimental 𝜅_L values of RE₂Zr₂O₇ better than the ${\kappa }_{\left(1\right)}\left({\varTheta }_{\left(2\right)}\right({\gamma }^{\left(3\right)}),{\gamma }^{\left(3\right)})$. In addition, our approach allowed us to study not only the perfect crystal lattice, but also to predict the effect of the presence of micropores on 𝜅. The effective 𝜅 could be normalized by ${\kappa }_{\text{eff}}={\kappa }_{L}(\text{1}-\frac{4}{3})\varphi$, where the porosity ϕ sets as 5% in accordance with experiment ¹⁵. Therefore, the major contribution to the deviation between the prediction ${\kappa }_{\left(2\right)}$ with 5% porosity according to the reported data of La₂Zr₂O₇ is mainly corresponding to the relatively low 𝜅_L of zirconate intrinsically ³.

As shown in Fig. 3c, the effects of temperature and RE elements on thermal transport have been exhibited, where the 𝜅_L values are obtained by the ${\kappa }_{\left(2\right)}$ with 5% porosity. It could be seen that two regions are delineated. At stage I, 𝜅_L obeys the 1/T law by anharmonic Umklapp phonon - phonon scattering, which donates the enhanced thermal resistance and the anharmonic effect caused by the intensive phonon excitations ¹¹. Afterward, 𝜅_L decreases to a plateau-like region entitled as Stage II, presenting its independence of temperature, and accessing the minimum 𝜅_L. Moreover, La₂Zr₂O₇ has the lowest 𝜅_L values in RE₂Zr₂O₇ oxides, which is in line with that La₂Zr₂O₇ is acknowledged as the third generation of TBC materials ⁴². It is well-documented that a cocktail of the properties of complex HEPOs often exceeds those predicted by the rule of mixtures ². Consequently, (Sc_0.2Y_0.2La_0.2Ce_0.2Pr_0.2)₂Zr₂O₇ has the lowest 𝜅_L even at those investigated high temperatures, decreasing from 1.32 Wm⁻¹ K^− 1 at room temperature 300K to 0.18 Wm⁻¹ K^− 1 at high temperature 1600K.

As a test of the predictive capabilities of thermodynamic simulation in κ, experiments have been performed to determine the thermal transport performance of the synthesized composition (Sc_0.2Y_0.2La_0.2Ce_0.2Pr_0.2)₂Zr₂O₇ (more details can be found in Supplementary Information). As shown in Figure S4, the intrinsic 𝜅 of HEPO was calculated by eliminating the thermal radiation effect, ranging from 1.69 Wm⁻¹ K⁻¹ to 0.14 Wm⁻¹ K⁻¹ at 300K to 1473K, which matched well with the predicted κ below 1.59 Wm⁻¹ K⁻¹ by ML algorism and κ = 1.32 Wm⁻¹ K⁻¹ by first-principles at 300K. Meanwhile, both high-throughput first-principles calculation and experimental methods only contain intrinsic Umklapp scattering and extrinsic porosity scattering mechanisms. The difference of κ between thermodynamics calculations and experiments at room temperature and high temperature could be attributed to the larger porosity from the realistic experiment (ϕ = 6.8%).

With the guidance of our proposed model integrating various mechanisms impeding heat transfer, the physical and chemical properties of oxides are intimately linked by their local structure and disorder, identified as r* in this work. The previously reported room temperature thermal conductivities of the RE₂Zr₂O₇ oxides, HEPOs on A-site (HE₂Zr₂O₇) or B-site (RE₂HE₂O₇) is plotted with r*, as shown in Fig. 3d and the references listed in Table S1. Compared with the RE₂HE₂O₇, the lower κ even at a lower lattice distortion level is presented in the HE₂Zr₂O₇ system, indicating a greater potential to adjust components for desired properties by A-site optimization.

Atomic and electronic basis for the domain knowledge design criteria

Although the HEPOs have phonon-dominated heat transport, the electronic contribution to conduct heat is also nonnegligible. Lower 𝜅 can be obtained through highly disordered interatomic forces resulting from charge disorder among ionic bonds ⁶. Especially, the major roles of KPPs can be uncovered from insight into the atomic and electronic basis for the coupling effects of lattice vibrations and charges on heat transfer.

In the view of strong atomic bonding characterized by the high bonding charge density (∆ρ) ^36,43,44, the peaks of its line profile curve can be employed to quantitatively characterize the bonding strength of RE–O1, RE–O2 and Zr–O2 in RE₂Zr₂O₇. As displayed in Fig. 4, the yellow isosurfaces identify the atomic sites absorbing electrons (∆ρ > 0), and the blue identifies contributing electrons (∆ρ < 0). Attributing to the size mismatch and valence electron difference of the RE atoms, the highest distribution of ∆ρ or the bonding strength is between Zr and O2 atoms and the lowest region is between RE and O2 atoms, which relates to the trend of the difference in bond length and further explains the origin of the electronic structure scale well. Based on these various views of ∆ρ isosurface, the effect of the RE elements is presented in variety on the bond structures of the resultant RE₂Zr₂O₇. As highlighted by green dot circles in Figs. 4d and 4f, RE elements Ce and Pr result in the higher line profile values, due to the filling shells showing as either an extra point near each RE element.

In Fig. 4g, the ionic RE-O bonds present the accumulation of charges gradient distribution between RE and O atoms, ignoring the peak in RE-O1 caused by the 4f electrons of Pr and Ce. The local features of O1-RE show the peaks of each curve, indicating their bond strengths decrease in the order of Sc > Y > La > Ce > Pr. However, according to the previous observation perspective, the ∆ρ distribution between O2-Zr atoms is in the negative region implying a covalent-like bonding behavior ⁴³. This covalent bonded in ceramics could exhibit preeminent hardness, strength, chemical inertness and so on ⁴⁵. With the increase of electronegativity of RE atoms, the negative charge peak between the O2 atom and Zr atom becomes weaker, suggesting a decreased covalency of the O2-Zr bond. Meanwhile, weak atomic bonds usually lead to strong anharmonicity ⁴⁶. Therefore, electronegativity is an important parameter in understanding the nature of chemical bonding and manipulating thermal conductivity ¹³.

Accordingly, the nature of the HEPOs bonding can be further obtained from investigating the RE cation in a local disordered environment. As shown in Figure S5, the chemistry complexity in the HEPO creates mass fluctuation and strain field fluctuations, which can disturb the regular distribution of the bonding charge density. Because the original dodecahedron-type RE − O unit blocks are highly distorted in HEPO, additional peaks are apparent in the line profiles of RE-O1 in HEPO as a result of the crystal distortion compared with RE-O1 in the 1A system. Additionally, the line profile in HEPOs is bilateral asymmetry, which is attributed to the relatively low symmetry of the local atomic environment.

Numerous experimental and theoretical investigations have demonstrated that in addition to phonon scattering caused by typical oxygen vacancy, local ionic charge disorder and weakened interatomic interactions also contribute to the reduction of κ of HEPOs ^6,13. It is worth mentioning that the bond length dispersion caused by the different A-site RE atoms reflects the details of their strain field fluctuations affected zone, as shown in Fig. 5a. The statistics of the nearest-neighbor RE/Zr-O bond length represent the formation of the local strain and the electron redistribution through modifying the crystal morphology. For example, the lattice distortion associated with La is stronger than those of the other RE atoms in RE₂Zr₂O_7, which has the largest ironic radius. Meanwhile, the La₂Zr₂O₇ pyrochlore possesses the largest bond length and the La atom has the lowest electronegativity which weakens the interatomic interactions and results in the lowest 𝜅_L values, as shown in Fig. 3c. However, attributing to the complex surrounding environment in HEPO, the bond length distribution of La atoms is in a medium range. Accordingly, as displayed in Fig. 5a, the affected zones of Sc atoms are the biggest, reflecting the strong anharmonicity caused by the smallest cation radio exhibited by Sc³⁺. It is indicated that the highly distorted environment and electronegativity play the dominant role in optimizing the ultralow κ of HEPOs and oxides.

The difference of the predicted κ among thermodynamics calculations, ML and experiment is not only traced back to the absence of electron contribution in DFT simulation or neglection structure defect in the database but also can be attributed to the coupling effect of lattice distortion and chemical charge. The Bader analysis was utilized to study valence charge disproportionation. In Fig. 5b, an increased valence state of Ce cation is captured, which indicates that Ce exists in two oxidation states, Ce³⁺ (0.114 nm) and Ce⁴⁺(0.097 nm), thus decreasing the bond length of Ce–O. By contrast, the Bader charge of other RE elements remained unchanged with the + 3 valence state, indicating the movement of electrons to O after high entropy engineering. To illustrate this charge regulation, Fig. 5c shows the spatial redistribution of bonding charge density. The differences in spatial shapes of ∆ρ correspond to the different oxidation states ⁴⁷. The transformation of the oxidation state of Ce is induced by internal charge compensation of HEPOs, driven by severe lattice distortion ⁵. The observed difference in the local bonding charge density of the three Ce atoms originates from the contrast in the activation of oxygen anions associated with local structural distortions. Meanwhile, bond-valence sums of oxygen anions reveal that HEPO structures exhibit weakly bonded oxide ions, which are missing in the RE₂Zr₂O₇ pyrochlore systems ⁴⁸.

The unexpected charge transformation acts as a key role in the phonon scatting, and this would lead to the appearance of the discordance of predicted κ between the theoretical simulation and ML model. The substitution of Ce³⁺ for Ce⁴⁺ would drive the structural transformation from the ordered pyrochlore structure to the defective fluorite structure. Correspondingly, many oxygen vacancies will be generated to cause phonon scatting, decreasing κ significantly ¹⁹. Utilizing the strength of the detailed XRD structural-scale characterization technique, dual-phase main peaks can be observed in the enlarged image of the illustration in the XRD pattern, implying the structural transformation, as presented in Figure S6. It confirms that the charge state transformation properties in (Sc_0.2Y_0.2La_0.2Ce_0.2Pr_0.2)₂Zr₂O₇ are a direct consequence of its local structure distinctiveness. Moreover, with the smaller size of Ce⁴⁺ and the relatively weaker Ce-O bonds, the localized Ce⁴⁺ cation acts like a “rattling” atom in the cation sublattice, inducing the phonons scatting to reduce the κ ⁴⁹.

In this work, the integrated explainable ML algorism with high throughput first-principles and experimental verification is utilized to discover advanced low κ HEPOs and reveal the atomic/electronic structure-dominated thermal transport mechanism of RE₂Zr₂O₇ (RE = Sc, Y, La, Ce, and Pr) oxides and (5RE_0.2)₂Zr₂O₇ HEPOs. A set of KPPs has been selected to unlock heat transfer mechanisms in muti-components HEPOs, which are broadly rooted in the domain knowledge from lattice engineering/distortion criteria. One of the compounds with the low lattice thermal conductivity at 300 K is identified and validated by the first-principles method and synthesized by experiment which may be good TBCs. Accordingly, the results and trends from the detailed DFT calculations and experimental studies together speak to the validity of lattice engineering and distortion design principles and have led to the rapid development of low 𝜅 HEPOs. Meanwhile, severe lattice distortion could cause strong phonon scatting and local phase/charge transformation, which have been further fed back to the domain knowledge. It could be envisioned that this present strategy has demonstrated its proficiency in designing ultralow κ HEPOs using the limited atomic and electronic KPPs which can be extracted from structural information. The entire workflow will not only unlock the coupling effects of lattice vibrations and charges on heat transfer but also foreshadow designer HEPOs with κ optimized for specific operating environments.

Datasets and Feature Matrix

The A₂B₂O₇ type HEPOs and their binary, tripartite and quaternary derivatives studied in the present work have accumulated amount of data from well-controlled experiments and simulations so far, which is beneficial for data mining. Since all the data are taken from the publications, the influence of different conditions has been declined by excluding data exhibiting large deviations for the same component. The training data consists of 171 samples with known ${\kappa }$ values at room temperature, which are given in the Supplementary Information (Table S1).

The feature matrix consists of crystal features and electronic features, as have been used previously in several studies on predicting thermal conductivity ^5,10,19,26. Features in this work have been further divided into basic structural information and advanced DFT information, which can be extracted automatically from a material system and collected from the results of a DFT calculation, respectively. Table 1 summarized the feature sets with names and symbols. Additionally, an explanation of the features is informed in Supplementary Information.

Table 1

Various data characters are used in this work and divided into crystal and electronic features by their basic physical meanings.
	Crystal features	Electronic features
Structural information	radius disorder parameter (r*)	electronegativity (E_neg)
	the radius of cations in A/B site (r_A/r_B)	electronegativity (E_neg)
	average mass (m_avg)	valence electron concentration (VEC)
	mass disorder parameter (m*)	valence electron concentration (VEC)
DFT information	lattice parameter (a)	equilibrium energy (E₀)
		Fermi energy (E_fermi)
		the band gap (E_gap)
	equilibrium volume (V₀)	bulk modulus (B)
	equilibrium volume (V₀)	the first derivative of B with pressure (B՛)

Machine learning models

This work employed the extremely randomized trees classifier (ET), random forest model (RF), XGBoost model, Gaussian Naive Bayes model (GNB), k-nearest neighbor model (KNN), logistic regression model (LR), decision tree model (DT), and support vector machine model (SVM) methods for regression ^37,50. Shapley additive explanations (SHAP) package is utilized for interpreting ML models ⁵¹. To acquire acceptable accuracy and stability, the 10-fold cross-validation and a grid-search technique is utilized to avoid overfitting and evaluate the feasibility of the models. The original dataset is divided randomly into a nonoverlapping training set (90%) and a validation set (10%). The predictive performance of models is evaluated using the coefficient of determination (R²) and root mean square error (RMSE) based on the actual and predicted values. Since the R² is calculated using${R}^{2}\text{=}\frac{{\sum }_{i}({\widehat{y}}_{i}\text{-}{\text{y}}_{i}{)}^{2}}{{\sum }_{i}({y}^{2}\text{-}{\text{y}}_{i}^{2}{)}^{2}}$, the small R² hits the reliability of predicted results. Here, RMSE is produced in a ten-fold cross-validation process (10 RMSE data generated) using $RSME\text{=}\sqrt{\frac{1}{n}\sum _{i=1}^{n}({\text{y}}_{i}\text{-}{\widehat{y}}_{i}{)}^{2}}$, where (y_i) is the true value and $\left({\widehat{y}}_{i}\right)$ the predicted value. The same cross-validation folds are used for ET, RF, XGBoost, GNB, KNN, LR, DT, and SVM models to cross-compare their performance. Therefore, the XGBoost algorithm resulted in the smallest RSME and the biggest R² is chosen for predicting the fundamental properties of HEPOs.

First-principles calculation

The first-principles calculations are carried out by utilizing the Perdew-Burke-Ernzerhof exchange-correlation functional, the projector augmented wave and with generalized gradient approximation method provided by the Vienna Ab initio simulation package (VASP) ⁵². Based on the density functional theory with the Hubbard correction algorithm, the Hubbard U correction with U = 4.0 eV for Zr 3d and 6.0 eV for RE 4f is applied. The electronic self-consistency convergence threshold is 10^− 6 eV atom^− 1 and sampled by a 4 × 4 × 4 k-point grid. The plane wave cutoff energy is set to 1.4 times the maximum energy of each element. The physical properties such as equilibrium volume (V₀), bulk modulus (B) and its first derivative concerning pressure (B′) can be obtained by the four-parameter Birch-Murnaghan equation of states ⁵³. The complex electron environment induced by RE atoms and high entropy design criterion can be revealed by bonding charge density (Δρ) ^44,54,55.

Lattice thermal conductivity

The quantitative expression of theoretical lattice thermal conductivity in acoustic modes has been proposed to benchmark experimental observations ⁵⁶. At the temperature range where heat is only conducted by acoustic phonons and Umkplapp processes dominate, the following formula can be obtained ⁵⁷:

$${{\kappa }}_{\left(1\right)}=A\frac{{M}_{\text{a}v}{{\Theta }}^{3}{V}^{\frac{1}{3}}}{T{\gamma }^{2}}$$

where A is given by A = $\frac{2.43\times {10}^{-8}}{1-0.514{{\gamma }}^{-1}+0.228{{\gamma }}^{-2}}$. This equation predicts lattice thermal conductivity at Debye temperature for an acoustic phonon branch $\varTheta$ using the Grüneisen parameter, $\gamma$, primitive cell volume, V, and the atomic mass, ${M}_{\text{a}v}$.

In order to simplify Eq. (1), Slack put $\gamma$ ≈ 2 in this expression and extended the model to crystals with n atoms per unit cell, thus Eq. (1) can be rewritten as:

$${\kappa }_{\left(2\right)}=A\frac{{M}_{av}{\varTheta }^{3}{{V}_{0}}^{\frac{1}{3}}}{{n}^{2/3}T{\gamma }^{2}}$$

where A is a constant equal to 3.1 × 10^− 6 for ${\kappa }_{ph}$ in W/mK, ${M}_{av}$in amu, and ${{V}_{0}}^{\frac{1}{3}}$ in Angstroms. The implementation is based on a combinatorial approach (as shown in Figure S7) where the lattice thermal conductivity is calculated with two variable Grüneisen parameter ${\gamma }^{\left(n\right)}$ and Debye temperature ${\varTheta }_{\left(m\right)}$. Based on the quasi-harmonic approximation, different formulations are employed and then combine to obtain different values for ${{\kappa }}_{\left(i\right)}({{\Theta }}_{\left(m\right)},{\gamma }^{\left(n\right)})$, where i = 1 ~ 2 for lattice thermal conductivity calculated using Eq. (1) or Eq. (2), m = 1 ~ 2 for Debye temperature calculated using Eq. (3) or Eq. (4), and n = 1 ~ 5 represent five Grüneisen parameter models.

The value of Debye temperature ${\varTheta }_{\left(m\right)}$ can thus be calculated in the assumption that the RE₂Zr₂O₇ pyrochlore is isotropic, by Eq. (1) Debye equation and Eq. (2) Debye–Grüneisen Eq. 5⁸:

$${{\Theta }}_{\left(1\right)}^{ }={s\left(\nu \right)AV}^{1/6}{n}^{1/3}{\left(\frac{{B}_{S}}{{M}_{\text{a}v}}\right)}^{1/2}$$

$${{\Theta }}_{\left(2\right)}^{ }=s\left(\nu \right){AV}_{0}^{1/6}{\left(\frac{{B}_{0}}{{M}_{\text{a}v}}\right)}^{1/2}{\left(\frac{{V}_{0}}{V}\right)}^{\gamma }$$

Here, B_S is the adiabatic bulk modulus, A is a constant equal to $\frac{ħ}{{k}_{B}}{\left(6{\pi }^{2}\right)}^{1/3}$, B₀ is the bulk modulus at 0 K, V₀ the equilibrium volume, $s\left(\nu \right)$ is the Debye temperature scaling factor given by:

$$s\left(\nu \right)={3}^{5/6}{\left[{4\sqrt{2}\left(\frac{1+\nu }{1-2\nu }\right)}^{3/2}+{\left(\frac{1+\nu }{1-\nu }\right)}^{3/2}\right]}^{-1/3}$$

The Poisson ratio $\nu$ for RE₂Zr₂O₇ is 0.25 for the lattice thermal conductivity calculation. While the bulk modules at constant pressure are calculated using the GIBBS2 software ⁵⁹.

It is essential to mention that Debye's temperature ${\varTheta }_{\left(2\right)}^{ }\left({\gamma }^{\left(n\right)}\right)$ depends on the Grüneisen parameter ${\gamma }^{\left(n\right)}$. The Grüneisen parameter is mostly temperature independent and could be adopted to describe the anharmonicity of crystal, and various groups have used different models to predict this quantity ^17,22. There are various definitions of the Grüneisen parameter, including the thermal or macroscopic formulation (${\gamma }^{th}$), Slater (${\gamma }^{S}$), Dugdale–MacDonald (${\gamma }^{DM}$), Vaschenko–Zubarev (${\gamma }^{VZ}$) and the modified free-volume (${\gamma }^{mfv}$) expressed as ⁴¹

$${\gamma }^{\left(1\right)}={\gamma }^{th}=-\frac{dIn\varTheta \left(V\right)}{dInV}\text{=}-\frac{\alpha {B}_{T}V}{{C}_{V}}$$

$${\gamma }^{\left(2\right)}={\gamma }^{S}=\frac{1}{2}{\left(\frac{dB}{dP}\right)}_{T=0K}-\frac{1}{6}$$

$${\gamma }^{\left(3\right)}={\gamma }^{DM}=\frac{1}{2}{\left(\frac{dB}{dP}\right)}_{T=0K}-\frac{1}{2}$$

$${{\gamma }^{\left(4\right)}=\gamma }^{VZ}=\frac{1}{2}{\left(\frac{dB}{dP}\right)}_{T=0K}-\frac{5}{6}$$

$${{\gamma }^{\left(5\right)}=\gamma }^{mfv}=\frac{1}{2}{\left(\frac{dB}{dP}\right)}_{T=0K}-0.95$$

Acknowledgments

This work was financially supported by National defense basic scientific research (Grant No. 211-CXCY-N103-03-04-00 and 2022-JCKY-JJ-1086). First-principles calculations were carried out on the clusters at the Northwestern Polytechnical University.

Author contributions

YZ conducted the high-throughput first-principles calculations advised by WYW and JSL. XYG and HFS developed the SAE code and constructed the multicomponent systems for first-principles calculations. RHY finished the machine-learning study. Experimental designs and tests were completed by KR, JW, YGW, XBL, and JSL. This work was designed by WYW, YGW, XBL and JSL. All authors contributed to the data analysis, interpretation of results and the writing of the manuscript.

Competing interests

The authors declare no competing interests.

Data availability

The data that support the findings of the work are in the manuscript and Supplementary Information. Additional data will be available upon reasonable request.

N P Padture, M Gell & E H Jordan. Thermal barrier coatings for gas-turbine engine applications. Science 296, 280-284, (2002).
C Oses, C Toher & S Curtarolo. High-entropy ceramics. Nat. Rev. Mater. 5, 295-309, (2020).
D R Clarke & S R Phillpot. Thermal barrier coating materials. Mater. Today 8, 22-29, (2005).
E L Pang, G B Olson & C A Schuh. Low-hysteresis shape-memory ceramics designed by multimode modelling. Nature 610, 491-495, (2022).
A Sarkar et al. High-entropy oxides: Fundamental aspects and electrochemical properties. Adv. Mater. 31, 1806236, (2019).
J L Braun et al. Charge-induced disorder controls the thermal conductivity of entropy-stabilized oxides. Adv. Mater. 30, 1805004, (2018).
C Toher et al. High-entropy ceramics: Propelling applications through disorder. MRS Bull. 47, 194-202, (2022).
Q M Hu & R Yang. The endless search for better alloys. Science 378, 26-27, (2022).
Z Rao et al. Machine learning-enabled high-entropy alloy discovery. Science 378, 78-85, (2022).
K C Pitike, A Macias, M Eisenbach, C A Bridges & V R Cooper. Computationally accelerated discovery of high entropy pyrochlore oxides. Chem. Mater. 34, 1459-1472, (2022).
D R Clarke. Materials selection guidelines for low thermal conductivity thermal barrier coatings. Surf. Coat. Technol. 163, 67-74, (2003).
N Marzari, A Ferretti & C Wolverton. Electronic-structure methods for materials design. Nat. Mater. 20, 736-749, (2021).
J G He et al. Accelerated discovery and design of ultralow lattice thermal conductivity materials using chemical bonding principles. Adv. Funct. Mater. 32, 2108532, (2021).
A J Wright et al. Size disorder as a descriptor for predicting reduced thermal conductivity in medium- and high-entropy pyrochlore oxides. Scripta Mater. 181, 76-81, (2020).
C L Wan et al. Glass-like thermal conductivity in ytterbium-doped lanthanum zirconate pyrochlore. Acta Mater. 58, 6166-6172, (2010).
H Liu et al. Copper ion liquid-like thermoelectrics. Nat. Mater. 11, 422-425, (2012).
W Qiu et al. Part-crystalline part-liquid state and rattling-like thermal damping in materials with chemical-bond hierarchy. PNAS 111, 15031-15035, (2014).
C Wan, Z Qu, A Du & W Pan. Order–disorder transition and unconventional thermal conductivities of the (Sm_1−xYb_x)₂Zr₂O₇ series. J. Am. Ceram. Soc. 94, 592-596, (2011).
A J Wright et al. Single-phase duodenary high-entropy fluorite/pyrochlore oxides with an order-disorder transition. Acta Mater. 211, 116858, (2021).
Z Teng et al. Preparation and phase evolution of high-entropy oxides A₂B₂O₇ with multiple elements at A and B sites. J. Eur. Ceram. Soc. 41, 3614-3620, (2021).
K E Sickafus et al. Radiation tolerance of complex oxides. Science 289, 748-751, (2000).
M D Nielsen, V Ozolins & J P Heremans. Lone pair electrons minimize lattice thermal conductivity. Energy. Environ. Sci. 6, 570-578, (2013).
S Lee et al. Resonant bonding leads to low lattice thermal conductivity. Nat Commun 5, 3525, (2014).
X Qian, J Zhou & G Chen. Phonon-engineered extreme thermal conductivity materials. Nat Mater 20, 1188-1202, (2021).
T M Tritt. Thermal conductivity: Theory, properties, and applications. (Kluwer Academic/Plenum Publishers, 2004).
B Liu et al. Application of high-throughput first-principles calculations in ceramic innovation. J. Mater. Sci. Technol. 88, 143-157, (2021).
S Curtarolo et al. The high-throughput highway to computational materials design. Nat. Mater. 12, 191-201, (2013).
J-P Poirier. Introduction to the physics of the earth's interior. 2 edn, (Cambridge University Press, 2012).
M I Jordan & T M Mitchell. Machine learning: Trends, perspectives, and prospects. Science 349, 255-260, (2015).
B Kailkhura, B Gallagher, S Kim, A Hiszpanski & T Y J Han. Reliable and explainable machine-learning methods for accelerated material discovery. NPJ Comput. Mater. 5, 108, (2019).
M K Li, G H Cao, Y F Luo, C Y Sheng & H J Liu. Predicting the lattice thermal conductivity of alloyed compounds from the perspective of configurational entropy. NPJ Comput. Mater. 8, 75, (2022).
R Jaafreh, Y S Kang & K Hamad. Lattice thermal conductivity: An accelerated discovery guided by machine learning. ACS Appl. Mater. Interfaces 13, 57204-57213, (2021).
B Mortazavi et al. Machine-learning interatomic potentials enable first-principles multiscale modeling of lattice thermal conductivity in graphene/borophene heterostructures. Mater. Horiz. 7, 2359-2367, (2020).
X Qian, S Peng, X Li, Y Wei & R Yang. Thermal conductivity modeling using machine learning potentials: application to crystalline and amorphous silicon. Mater. Today. Phys. 10, (2019).
R Kumar & A K Singh. Chemical hardness-driven interpretable machine learning approach for rapid search of photocatalysts. NPJ Comput. Mater. 7, 197, (2021).
C X Zou et al. Integrating data mining and machine learning to discover high-strength ductile titanium alloys. Acta Mater. 202, 211-221, (2021).
X Zhong et al. Explainable machine learning in materials science. NPJ Comput. Mater. 8, 204, (2022).
W Y Wang et al. Atomic and electronic basis for the serrations of refractory high-entropy alloys. NPJ Comput. Mater. 3, 23, (2017).
Y Gu et al. Engineering the oxygen vacancies of rocksalt-type high-entropy oxides for enhanced electrocatalysis. Nanoscale 14, 515-524, (2022).
C G Levi. Emerging materials and processes for thermal barrier systems. Curr. Opin. Solid. St. & Mater. Sci. 8, 77-91, (2004).
L Vočadlo, J P Poirer & G D Price. Grüneisen parameters and isothermal equations of state. Am. Min. 85, 390-395, (2000).
X Q Cao, R Vassen & D Stoever. Ceramic materials for thermal barrier coatings. J. Eur. Ceram. Soc. 24, 1-10, (2004).
L Su et al. Direct observation of elemental fluctuation and oxygen octahedral distortion-dependent charge distribution in high entropy oxides. Nat Commun 13, 2358, (2022).
W Y Wang et al. Atomic and electronic basis for solutes strengthened (010) anti-phase boundary of L1₂Co₃(Al, TM): A comprehensive first-principles study. Acta Mater. 145, 30-40, (2018).
J Zhang et al. Plastic deformation in silicon nitride ceramics via bond switching at coherent interfaces. Science 378, 371-376, (2022).
J P Heremans. The anharmonicity blacksmith. Nat. Phys. 11, 990-991, (2015).
H Raebiger, S Lany & A Zunger. Charge self-regulation upon changing the oxidation state of transition metals in insulators. Nature 453, 763-766, (2008).
A Gupta, A Kumar, U V Waghmare & M S Hegde. Activation of Oxygen in Ce₂Zr₂O_7+x across Pyrochlore to Fluorite Structural Transformation: First-Principles Analysis. J. Phys. Chem. C 121, 1803-1808, (2017).
Y Wang, F Yang & P Xiao. Glass-like thermal conductivities in (La_1-x1Y_x1)₂(Zr_1-x2Y_x2)₂O_7-x2 (x=x₁+x₂, 0⩽x⩽1.0) solid solutions. Acta Mater. 60, 7024-7033, (2012).
K Choudhary et al. Recent advances and applications of deep learning methods in materials science. NPJ Comput. Mater. 8, 59, (2022).
A E Roth. Lloyd Shapley (1923-2016). Nature 532, 178, (2016).
Y Wang & J P Perdew. Correlation hole of the spin-polarized electron gas, with exact small-wave-vector and high-density scaling. Phys. Rev. B 44, 13298-13307, (1991).
F Birch. Finite Elastic Strain of Cubic Crystals. Phys. Rev. 71, 809, (1947).
Y Zhang et al. Revealing the local microstates of Fe–Mn–Al medium entropy alloy: A comprehensive first-principles study. Acta Metall. Sin-Engl. 34, 1492-1502, (2021).
Y Zhang et al. When a defect is a pathway to improve stability: a case study of the L1₂ Co₃TM superlattice intrinsic stacking fault. J. Mater. Sci 54, 13609-13618, (2019).
D G Cahill, S K Watson & R O Pohl. Lower limit to the thermal conductivity of disordered crystals. Phys Rev B 46, 6131-6140, (1992).
K Watari & S L Shinde. High thermal conductivity materials. Mrs Bulletin 26, 440-441, (2001).
X L Liu, B K VanLeeuwen, S-L Shang, Y Du & Z-K Liu. On the scaling factor in Debye–Grüneisen model: A case study of the Mg–Zn binary system. Comp. Mater. Sci. 98, 34-41, (2015).
A Otero-de-la-Roza & V Luaña. Gibbs2: A new version of the quasi-harmonic model code. I. Robust treatment of the static data. Comp. Phys. Comm. 182, 1708-1720, (2011).
B Liu, J Y Wang, Y C Zhou, T Liao & F Z Li. Theoretical elastic stiffness, structure stability and thermal conductivity of La₂Zr₂O₇ pyrochlore. Acta Mater. 55, 2949-2957, (2007).
R Vassen, X Q Cao, F Tietz, D Basu & D Stover. Zirconates as new materials for thermal barrier coatings. J. Am. Ceram. Soc. 83, 2023-2028, (2000).
J Feng, B Xiao, R Zhou & W Pan. Thermal conductivity of rare earth zirconate pyrochlore from first principles. Scripta Mater. 68, 727-730, (2013).
K Ren, Q K Wang, G Shao, X F Zhao & Y G Wang. Multicomponent high-entropy zirconates with comprehensive properties for advanced thermal barrier coating. Scripta Mater. 178, 382-386, (2020).
C L Wan et al. Effect of point defects on the thermal transport properties of(La_xGd_1−x)₂Zr₂O₇: Experiment and theoretical model. Phys. Rev. B.74, (2006).

(Not answered)

Download PDF

Version 1

posted

You are reading this latest preprint version

Discovering the Ultralow Thermal Conductive High-Entropy Pyrochlore Oxides Through the Hybrid Knowledge-Assisted Data-Driven Machine Learning

Status:

Version 1

Abstract

Figures

Introduction

Results

Hybrid knowledge-assisted ML Design criteria

Smart design HEPOs via addressed KPPs

Thermodynamic combinatorial models for cross-validation

Atomic and electronic basis for the domain knowledge design criteria

Discussion

Conclusion

Methodology

Datasets and Feature Matrix

Machine learning models

First-principles calculation

Lattice thermal conductivity

Declarations

References

Additional Declarations

Supplementary Files

Status:

Version 1