Thermokinetic Investigation Of The Zr Alloys And Machine Learning Study

CALPHAD?CALculaiton of PHAse Diagram?method has been successfully applied in many fields for property predictions,technological process improvement and materials design,etc..The accurate thermodynamic and kinetic databases are key factors for practical applications,which are obtained by using appropriate models to fit available data.The rapid development of first-principles calculations facilitates the construction of CALPHAD databases by providing the thermophysical properties not only for stable systems but also for metastable systems.Zirconium alloys are widely used in nuclear industry as cladding materials and core structural materials due to their combination of excellent neutron economy,good mechanical properties and significant corrosion resistance.With the development of the new nuclear power plants,the demand for new high-quality Zr alloys is increasing rapidly.The improvement of the Zr alloy properties relies on the addition of trace alloying elements and the subsequent heat treatments,which can lead to desired precipitations.From the perspective of materials design,the CALPHAD method can be used to predict the phase transition process and phase amount after heat treatments,which can accelerate the pace of new materials discoveries.In the present work,we focused on the key sub-ternary Zr-Nb-Cr and Zr-Fe-Nb systems in the Zr-based alloys systems.Based on previous literature,we designed key experiments for Zr-Cr binary system to obtain the composition of phases by electron probe microscopic analysis and X-ray for a better thermodynamic model.The first-principles calculations were employed to predict the phase stability and the enthapies of solution and intermetallic phases,which can be used for the modeling of the end-members.The high-throughput first-principles calculations were utilized to predict the diffusion coefficients of 14 solutes in HCP-Zr based on the eight-frequency model.To identify the dominant diffusion mechanism for each solute,the formation energies of solutes in nine high-symmetry interstitial sites were calculated as well as the migration energies between two octahedral sites at basal plane and along C-axis direction.The diffusion mechanism was analyzed based on the comparison of the vacancy-mediated and interstitial activation energies.Machine learning models have been widely utilized in materials science to discover trends in existing data and then make predictions to generate large databases,providing powerful tools for accelerating materials discovery and design.However,there is a significant need to refine approaches both for developing the best models and assessing the uncertainty in their predictions.In this work,we evaluate the performance of Gaussian kernel ridge regression?GKRR?and Gaussian process regression?GPR?for modeling ab-initio predicted impurity diffusion activation energies,using a database with 15 hosts and 408 host-impurity pairs.We demonstrate the advantages of basing the feature selection on minimizing the Leave-Out-Group?LOG?cross-validation?CV?root mean squared error?RMSE?instead of the random K-fold CV RMSE.The LOG RMSE from GKRR?GPR?model is only 0.148 eV?0.155 eV?and the corresponding 5-fold RMSE is 0.116 eV?0.129 eV?,demonstrating the good accuracy of the model.We also show that the ab-initio impurity migration barrier can be employed as a feature to increase the accuracy of the model significantly while still yielding a significant speedup in the ability to predict new systems.Finally,we propose a way of analyzing the accuracy of any machine learning error estimates?specifically,predicted standard deviation of predicted vs actual data?.The approach defines r as the magnitude of the ratio of the predicted standard deviations to the actual error?residual?in left out data during CV,and compares the distribution of r to a Normal distribution.Deviations can be used to quantify the accuracy of the machine learning error estimates,and tests with GPR generally show that the approach yields accurate error estimates for this diffusion data set,which improve for GPR models with lower CV scores.All the results are:?1?The C14 phase in the Cr-Nb system was unstable based on the phonon calculations,which is consistent with experiments.The solubility of Laves_C15 phase in the Zr-Cr system was investigated experimentally.The SQS-VASP method was used to calculate the enthapies of solution phases in the Cr-Nb and Zr-Cr systems.The thermodynamic description of the Zr-Nb-Cr system was then assessed based on the available experimental data and the first-principles calculations.The calculated phase diagram and thermodynamic properties yield a good agreement with available experimental data.?2?The Fe₂₃Zr₆ phase in the Zr-Fe system was treated as a stable phase based on the analysis of available experimental results.The high-temperature phase Laves_C36was regarded as a metastable phase,thus not included in the stable phase diagram.The Zr-Fe system was reassessed based on new phase equilibria results.The thermodynamic description for the Fe-Nb system was slightly optimized to fit the experimental data of the Laves_C14 phase boundary.The thermodynamic database for Zr-Fe-Nb system was established by fitting the available experimental isothermal sections.?3?The migration barriers and attempt frequencies of 14 solutes in HCP-Zr were calculated by using the CI-NEB method and phonon calculations.The diffusivities were determined by using the eight-frequency model.The interstitial formation energies were calculated together with the migration energies for two octahedral sites at basal plane and along C-axis direction.The dominant diffusion mechanism for each solute was identified based on the comparison of the vacancy-mediated and interstitial activation energies.The Cr,Cu,V,Zn,Mo,W,Al,Au,Ag,Nb,Ta and Ti diffuse dominantly by an interstitial mechanism in HCP-Zr while Hf,Zr and Sn are vacancy-mediated diffusers although the identification of mechanisms for these elements at high-temperature is quite uncertain due to the entropy effect.?4?We evaluated the performance of GKRR and GPR for modeling the impurity activation energies of 408 host-solute systems based on their elemental properties.We demonstrated the advantages of using the LOG RMSE?Root Mean Squared Error?for feature selection as it gave similar errors for 5-fold cross-validation and much lower errors for LOG cross-validation when compared to using the 5-fold RMSE for feature selection.We also demonstrated that forward selection can easily miss key features that lead to overall better models,and that these might need to be added based on intuitions or more comprehensive automated search methods.We also found that some features cause others to be removed from the feature list even when the features have relatively modest correlation.This result suggests we cannot always demand all physically expected features to be selected as the physics may enter through different paths in different features.Based on the features selected by LOG RMSE,the LOG CV RMSE for GKRR and GPR are 0.148 eV and 0.155 eV while the corresponding5-fold RMSE are about 0.116 eV and 0.129 eV,respectively,demonstrating the good predictive capability of the model.The accuracy of the models can be further improved significantly by incorporating the migration barrier as a feature,which can be calculated at a cost significantly less than the full diffusion coefficient model.We also explored a method of assessing the error bars from the GPR by analyzing the distribution of the ratio of the actual errors to the GPR errors during LOG CV.We found the distribution is close to a Normal distribution with a slightly peaked shape and generally about 95%of the true values fell within two standard deviations of the predictions.The thermodynamic assessment of the Zr-Nb-Cr and Zr-Fe-Nb ternary systems is a prerequisite for the Zr-based multiple elements database,which can be used to predict the phase compositions of Zr-based alloys and accelerate the pace of new materials discoveries.The dominant diffusion mechanism of some alloying elements in HCP-Zr was identified to be interstitial diffusion based on the first-principles calculations,which can be beneficial for understanding the alloys behavior under different temperature,irradiation,and alloying conditions.Also,the interstitial diffusion may be much common than it is generally assumed.The machine learning study based on the high-throughput calculated impurity diffusion activation energies demonstrated that the LOG CV outperforms the random Kfold CV.The standard deviation from GPR model is an effective measurement of the predictive uncertainty,which is useful for the predictions based on small data set.