Machine learning aided first-principles studies of structure stability of Co₃(Al, X) doped with transition metal elements

Highlights

• E_f of Co₃(Al, X) changed parabolically with the group numbers in the periodic table.

• Co₃(Al, X) (X = Hf, Ta, Ti, Zr, V, Nb) are more stable than Co₃(Al, W).

• Co₃(Al, WX₃) (X = Ti, Hf, Ta, Nb, V, Y, Zr) are more stable than Co₃(Al, W).

• Machine learning prediction are accurate using “Center-Environment” (CE) features.

Abstract

To understand the alloying effects on the stability of Co₃Al precipitate phase in Co-based superalloy, the energetic stability and structure of ternary alloy Co₃(Al, X) doped with the thirty 3d, 4d, and 5d transition metal (TM) elements were studied in this work using first-principles (FP) computation and machine learning (ML) methods. Our FP computation indicated that Hf, Ta, and Ti doping were thermodynamically most stable. Based on the FP computation data, the ML models with three types of chemical composition (CC) and Center-Environment (CE) features, were developed to predict the formation energies and lattice constants of Co₃(Al, X) (X = 3d, 4d, and 5d TM elements). The results show that these ML models all had good prediction accuracy with averaged mean absolute errors (<MAE > ) ~ 0.02 eV/atom for formation energies and ~ 0.01 Å for lattice constants. Then, the effects of important features were discussed on the energy and geometry properties. To study the alloying effects on the stability of Co₃(Al, W) precipitate phase, the ML models of Co₃(Al, X) were found to be equally accurate to predict the untrained structures of Co₃(Al, WX₃) where X = 3d, 4d, and 5d TM elements excluding Co and W. We found that the Co₃(Al, WX₃) structures became most stable when X are IVB and VB group TM elements. This work show that machine learning methods can efficiently extend the capabilities of first-principles predictions on the structure stabilities in multi-component alloy design.

Introduction

The conventional Co-based superalloys exhibit higher melting points, better corrosion resistance, and creep resistance[1], [2], [3], [4], [5], [6], [7] compared with the commercial Ni-based superalloys[8], but they are limited by high temperature strength[9]. Similar to the $\gamma$/${\gamma }^{\text{'}}$ strengthening mechanism in Ni-based superalloys, Sato et. al.[10] found the L1₂ Co₃(Al, W) ternary ${\gamma }^{\text{'}}$phases in Co-based superalloys play a critical role of strengthening but it is not yet very stable. Since then there have been many studies focusing on the improvement of structure stability of Co₃Al or Co₃(Al, W) precipitate phases by doping approach. Kobayashi et. al.[11] studied experimentally the phase stability and thermodynamic stability of ${\gamma }^{\text{'}}$–Co₃(Al, W, Ti) in the quaternary Co-Al-W-Ti system, and constructed the vertical cross section of the phase diagram of Co-9.4Al-9.6 W and Co-16.5Ti. Chen et. al.[12] studied the phase equilibrium of alloying elements in the Co-Al-V based superalloys and found that Ti, Nb, Mo, Ta, W, and Ni were allocated to ${\gamma }^{\text{'}}$phase, while Cr was allocated to $\gamma$ phase. Ti, Nb, Ta, and Ni can increase the solution temperature of ${\gamma }^{\text{'}}$ phase. Xu et. al[13] studied computationally the mechanical stability of Co₃ (M, W) (M = Al, Ge, Ga) and found that the elastic heat-resistant properties of Co₃(Ge, W) phase are lower than the other phases, and the Co₃(Al, W), Co₃(Ge, W), and Co₃(Ga, W) have higher elastic anisotropy. Jin et. al.[14] calculated the mechanical properties and structural stability of Co₃(Al, X) (X = Ti, V, Cr, Zr, Nb, Mo, Hf, Ta, W), most of which L1₂ Co₃(Al, X) have excellent mechanical stability and ductility.

The studies on the phase stabilities of the Co₃Al and Co₃(Al, W) systems doped with various alloying elements are very costly for both experiments and computations so there are still lack of systematic studies on the alloying effects including all transition metal (TM) elements and dual component substitutions. Compared with research methods of experiments and computations, the machine learning (ML)[15], [16] method has obvious advantages. The ML methods are effective data-driven statistical algorithms to construct complex correlations by learning from the experimental or computational source data. The introduction of ML methods can predict efficiently the properties of materials, accelerating the experimental and computational investigation. Materials informatics becomes an emerging field benefited from the hybrid of materials science and computer science. Takahashi et al. [17] discussed the construction of material big data, implementation of machine learning, and platform design in the development of materials science, and put forward the potential solutions. Takahashi et. al. [18] studied the lattice constants of binary body centered cubic crystals with support vector regression algorithm and found that the descriptor can be used to predict the lattice constants of complex crystals. Rajan et al. [19] introduced the concepts of material informatics that study complex multi-scale problems in a high-throughput, statistically robust, and physically significant way, facilitating the development of material design and discovery. Ghiringhelli et. al.[20] discussed the relationship between the descriptor and property, and systematically found meaningful descriptors in the energy difference of sphalerite, wurtzite, and the halite semiconductor.

Recently, the machine learning algorithms commonly used in materials research include Random Forest (RF)[21], Support Vector Regression(SVR)[22], Neural Networks[23] and so on. The RF is an ensemble learning method, which constructs a large number of decision trees[24] during training where each decision trees gives a unit vote for the most popular class at input. With high accuracy, RF can quickly and effectively manage thousands of input variables while reducing the error rate. The Support Vector Machines (SVM)[25] is a machine learning algorithm based on statistical learning theory. It follows the principle of structural risk minimization and minimizes the upper limit of generalization error. The SVM is referred to Support Vector Regression (SVR) when used to solve function approximation and regression problem. The Neural Network (NN) is a collection of interconnected systems connected by basic computing units. The number of these units may be very large and the network architectures are also very complicated. Compared with the other algorithms, the NN usually need a very large amount of training data to get satisfied results that are not readily obtained in materials science. So, we adopted the RF and SVR algorithms in the ML modeling of this work.

In this work, we combined first-principles (FP) density functional theory (DFT) computations, and ML methods to study systematically the alloying effects of the structure stability of the ternary Co₃(Al, X) (X = thirty 3d, 4d, and 5d TM elements) as well as the quaternary dual-substituted Co₃(Al, WX₃) structures (X = 3d, 4d, and 5d TM elements except for Co and W). We first carried out first-principles calculations to study the alloying effects of the stabilities of ternary Co₃(Al, X) alloys, the six types of structures and 180 structures in total, where X are thirty 3d, 4d, and 5d TM elements, including Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd, Ag, Cd, La, Hf, Ta, W, Re, Os, Ir, Pt, Au, and Hg. Then we constructed ML models with three types of chemical composition (CC) and the Center-Environment (CE) features[26], [27] using RF and SVR with Radial Basis Function (SVR_RBF) algorithms to predict the stability of Co₃(Al, X) structures. The prediction accuracy of the ML-CE models was compared with that of the ML-CC models where the three types of CC features were constructed using the features of X element (CC-X), the features of Co, Al, and X elements with equal proportion (CC-EP), and the features of Co, Al, and X elements with the proportions of chemical formula (CC-CF). In the construction of these four types of ML models, we used random split training/test (8/2) data sets and leave-one-out cross validation for untrained group elements. Furthermore, to examine the alloying effects of Co₃(Al, W), we applied the ML models trained for Co₃(Al, X) to study the quaternary dual-substituted structures Co₃(Al, WX₃) where X are 3d, 4d, and 5d TM elements excluding Co and W, namely, Sc, Ti, V, Cr, Mn, Fe, Ni, Cu, Zn, Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd, Ag, Cd, La, Hf, Ta, Re, Os, Ir, Pt, Au, and Hg. The ML predictions on these untrained Co₃(Al, WX₃) structures were evaluated by the additional first-principles computations. The trends of structure stability shown in periodic table and the most stable structures were also discussed.