An materials informatics approach to Ni-based single crystal superalloys lattice misfit prediction

Computational Materials Science 143 (2018) 295–300

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An materials informatics approach to Ni-based single crystal superalloys lattice misfit prediction Xue Jiang ⇑, Hai-Qing Yin ⇑, Cong Zhang, Rui-Jie Zhang, Kai-Qi Zhang, Zheng-Hua Deng, Guo-quan Liu, Xuan-hui Qu Collaborative Innovation Center of Steel Technology, University of Science and Technology Beijing, 30 Xueyuan Road, Haidian District, 100083 Beijing, PR China

a r t i c l e

i n f o

Article history: Received 22 August 2017 Accepted 11 September 2017

Keywords: Materials informatics Machine learning Ni-based superalloys Lattice misfit Alloy design

a b s t r a c t The lattice misfit between c and c0 phase in Ni-based single crystal superalloys plays a critical role for microstructural stability and high-temperature creep and fatigue resistance. Making predictions of the lattice misfit rapidly and accurately is therefore of much practical importance, especially for costly and time-consuming material design by trial and error. In this study, we provide a machine learning approach to predict misfit using relevant material descriptors including the chemical composition, dendrite information and measurement temperature and so on. We perform support vector regression, sequential minimal optimization regression and multilayer perceptron algorithms with linear and poly kernels on experimental dataset for appropriate model selecting, and multilayer perceptron model works well for its distinguished prediction performance with high correlation coefficient and low error values. The approach is validated by comparing the predicted lattice misfit with a widely used empirical formula and experimental observation with respect to prediction accuracy. Ó 2017 Elsevier B.V. All rights reserved.

1. Introduction Ni-based single crystal superalloys are widely used in turbine blades and vanes of the most advanced aero engines and industrial gas turbines due to their remarkable high-temperature creep and fatigue resistance [1]. The high creep resistance is induced by the precipitation of ordered cuboid L12 c0 phase in a continuous FCC c phase matrix with completely coherent {1 0 0} interfaces [2]. During creep deformation, the directional coarsening of c0 phase, also named ‘‘c0 rafting”, depends on the sign and magnitude of natural c/c0 lattice misfit between the two phases [2,3]. Therefore, the misfit, as a parameter in mechanical behavior modeling [4], is critical to superalloys development because of its strong effect on internal stresses and deformation mechanisms of superalloys. This so-called natural misfit is defined as:

d¼2

ac0  ac ac0 þ ac

ð1Þ

where ac0 and ac are the free lattice parameters of c and c phase respectively. It is strongly affected by the concentration of solute elements in c and c0 phase [1].

The lattice misfit can be obtained by measurement or by calculation. Neutron diffraction [5], X-ray diffraction [6], and convergent beam electron diffraction (CBED) [7,8] can give relatively accurate measurement for lattice parameters of c and c0 phase, which are used to calculate the misfit by Eq. (1). However, during alloy design, these methods are costly and time-consuming to get desired misfit by try and error experiments. For Ni-based superalloys, there are more than 10 kinds of alloy elements, so the number of candidate composition can be more than 1020, with each element concentration varying by 1%. Experimental measurement cannot satisfy with the increasing speed of elements number, for the exponential growth of chemical space dimension. WATANABE (1957) proposed an alloy design model combining theoretical analysis and experimental examination, based on lattice constant and constitution of c/c0 phase [9]. WATANABE (1957) model is defined as:

ac ¼ aNi þ

X

V i xi ;

ac0 ¼ aNi3 Al þ

X

V 0i x0i

ð2Þ

0

⇑ Corresponding authors. E-mail addresses: [email protected] (X. Jiang), [email protected] (H.-Q. Yin). https://doi.org/10.1016/j.commatsci.2017.09.061 0927-0256/Ó 2017 Elsevier B.V. All rights reserved.

where aNi and aNi3 Al are the lattice constant of Ni and Ni3 Al, xi and x0i are the mole fraction of alloy element i, V i and V 0i are the Vegard coefficient of alloy element i in Ni and Ni3 Al, respectively [1,9]. The phase constitution in this model can be obtained by measurement and thermodynamics calculation currently. Thermodynamics calculation is based on equilibrium state and there exists

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discrepancy with actual constitution, especially for orientated solidification superalloys with more than 10 alloy elements. Wang (2006) proposed a computational model integrated with first-principles calculations, existing experimental data and phenomenological modeling [10]. Wang (2006) model is defined as:

ac;c0 ¼ a0 þ bT þ cT 2 þ

XX s ki ysi s

ð3Þ

i

where a0 is the calculated lattice parameter of Ni or Ni3 Al, b and c are thermal expansion coefficients [11], s indicates different sublats tices, ysi and ki are the atomic fraction and effect coefficient of element i in sublattice s respectively. Thermal expansion coefficients in this model, which crucially affects the model uncertainty, are not convenient to obtain. It prefers to be given experimentally for greater certainty desired, such as diffraction measurements, instead of by first-principles linear-response theory [12]. Furthermore, other factors, such as specimen thickness and dendrite information, are ignored in both these models, to which lattice misfit is also sensitive. Therefore, a more efficient and comprehensive approach is needed for lattice misfit during alloy design and we prefer a more rapid and low-cost way to alloy design based on the Materials Genome Initiative [13] (MGI) and Integrated Computational Materials Engineering [14] (ICME) methodology. Data-driven machine learning approaches are gradually penetrating into materials design and development, and have been applied in some excellent studies including predictions of thermal hysteresis [15], transformation temperature [16], solvothermal synthesis [17], and specific surface area [18] and so on. However there is no work on superalloys misfit prediction by this method currently. In the present work, we propose an accelerated method to estimate the lattice misfit by machine learning algorithm. It shows that our misfit model based on multilayer perceptron algorithm, using chemical composition, dendrite information, specimen thickness and measurement temperature as descriptors, works exceedingly well in misfit prediction. We compare the predicted misfits by our data-driven model with both experimental observation by CBED and the predicted by WATANABE (1957) model. The result shows our model works better than WATANABE (1957) model especially with respect to the sign (positive or negative) of misfit prediction. 2. Dataset All the data used in this work is collected manually from published studies of Le Graverend J-B (2015) [2], Völkl R (1998) [8], Schulze C (2000) [19], and Pyczak F (2004) [20]. The dataset consists of 136 instances, covering 11 kinds of alloy grand and 4 generations. The statistic information of the 136 instances is listed in Table 1. The composition of superalloys in this dataset contains 13 elements, like Ni, Al, Co, Cr, Mo and so on, where Ni is dominated. The lattice misfits of the 136 instances are all measured by CBED after standard heat treatment of the specified alloy grand. The natural lattice misfit of c and c0 phase is depicted in Fig. 1(a). As it is illustrated in the published literatures [2,8,19,20], lattice misfit is definitely different in dendrite center and interdendrite, so we consider the measurement position of dendrite as an factor for lattice misfit during data collection, as shown in Fig. 1(b). The specimen thickness also makes effects on misfit for the differences in the elastic strains between thin foil and bulk [8,19], and at the same time the temperature can influence lattice volume and the distribution of alloying elements in c and c0 phases. So the features we select for the input of statistical learning model include chemical composition, measurement position of dendrite, specimen thickness and temperature. Here we ignore the difference of stan-

Table 1 Grand and generation information of superalloys in the dataset. Grand

Generation

Number

CMSX-4 [8,20] SRR99 [8] SC16 [8] CMSX-10 [19] IN792DS [20] ExAl7 [20] ExAl8 [20] ExAl9 [20] ExAl10 [20] CMSX-6 [20] AM1 [2]

2nd 1st 1st 3rd 1st 2nd 4th 1st 2nd 1st 1st

19 2 2 4 17 17 17 18 18 17 5

Total

{1st, 2nd, 3rd, 4th}

136 instances

dard heat treatment for different alloys to minimize the dependence on processing conditions; otherwise the data dimension will be too high for statistical learning model training based on the limited 136 training samples. To guide future alloy design with the collected data, we developed a web-access database (http://www.materdata.cn/search2. php?ztmc = 899) to facilitate data accumulation of ongoing experiment and literature collection, which is subjected to National Materials Scientific Data Sharing Network (http://www.materdata.cn). All the data in our database are validated by the affiliation and literature derivation information to assure the quality reliability. 3. Data driven models 3.1. Data preprocessing Data preprocessing is necessary before training models by machine learning algorithm. Data discretization and normalization, as common and crucial processes during preprocessing, can reduce the effect of extreme and abnormal values to make the model more stable, and some algorithms engine can only support the input dataset after discretization and normalization [21]. Feature selection is finished manually on the basis of theoretic knowledge in Section 2. For discretization process, we use the chemical weight percent of 13 nominal elements directly as the chemical composition features, so the feature types are all numeric. As for the feature ‘‘measurement position of dendrite”, we use number {0,1,2} to represent three dendrite conditions, where 0 and 1 indicate dendrite center and interdendrite respectively and 2 presents no records captured on it. The measurement temperature arranges from 25 °C to 1200 °C, so this feature type is numeric. Some of the specimens are foil while others are bulk, and we use ‘‘0” and ‘‘1” to describe feature specimen thickness correspondingly. The dataset with selected features is shown in Supplementary Table 1. Data normalization can improve the accuracy and efficiency of distance-based regression and classification algorithm. So we perform min–max normalization, as Eq. (4), on our training dataset. Here MinU and MaxU are the minimum and maximum values for attribute U, and a linear transformation is applied on the original data mapping a value u of U to u0 belonging to [MinU 0 , MaxU0 ] [22].

u0 ¼

u  MinU  ðMaxU0  MinU0 Þ þ MinU0 MaxU  MinU

ð4Þ

3.2. Machine learning models In order to build misfit prediction model, a quantitative relationship between lattice misfit and the selected features is needed. The relationship can be described as Y ¼ f ðXÞ; f may be linear or

X. Jiang et al. / Computational Materials Science 143 (2018) 295–300

297

Fig. 1. (a) Lattice misfit of c and c0 phase, (b) Location difference between interdendrite and dendrite center.

nonlinear, Y is material property (lattice misfit), and X is the feature vector containing sensitive factors to Y (chemical composition, dendrite position, specimen thickness, temperature). This is a regression problem, and a very effective framework for regression tasks involving features is provided by statistical learning theory. So we perform several regression algorithms on the training set by Weka3.8, including Support Vector Regression with linear kernel (SVR/lin) and poly kernel (SVR/poly), SMO regression with normalizedpoly kernel (SMOreg/norpoly) and poly kernel (SMOreg/ poly), and Multilayer Perceptron (MLP). Support Vector Regression (SVR) can achieve generalized performance through minimizing the generalized error bound, instead of the observed training error [23]. SMO algorithm, called sequential minimal optimization, is the most commonly used algorithm for numerical solutions of the dual problem in support vector learning [24]. Multilayer Perceptron is one type of neural network to model highly non-linear functions, without making prior assumptions concerning the data distribution, and it can be accurately generalized when dealing with new, unseen data [25]. Kernel function provides a map from input space to high dimension space to reduce the complexity of algorithms and optimizes regression process. To minimize the effect of over fitting in nonlinear regression, we use k-fold cross-validation to avoid contingency during model training. Cross-validation, as a model validation technique, can be used for assessing how the results of a statistical analysis will generalize to an independent data set [26]. Here we set k = 10. Let D be our training set, and we split D into 10 mutually exclusive folds D1 , D2 ; . . . ; D10 of approximately equal size with 13 or 14 instances randomly. The regression algorithms are performed 10 times, and for each time 9 folds are used for training and leave one fold for testing. The performance of model training is shown in Fig. 2 with measured misfit on horizontal axis and predicted misfit on vertical axis. The predicted misfit values generated by the above five models are plotted as a function of the measured. The more closely the plots align along the 45° diagonal line, the

more consistent the predicted misfits are with the measured. We can see that the MLP model performs a perfect fitting intuitively by Fig. 2(e). Here we use correlation coefficient (CC), root mean squared error (RMSE) and mean absolute error (MAE) to evaluate the regression performance quantitatively. CC is a measure of the linear correlation between input features and output targets, and can be described as the formation in Eq. (5) [27].

qX;Y ¼

cov ðX; YÞ

rX rY

ð5Þ

Here qX;Y is CC, cov ðX; YÞ ¼ E½ðX  lx ÞðY  ly Þ is the covariance of X and Y, E is the expectation, lx and ly are the mean of X and Y, rX and rY are the standard deviation of X and Y. We use CC to evaluate the correlation between our input features (chemical composition, measurement position of dendrite, specimen thickness and temperature) and the lattice misfit. It is more correlated if the CC is higher. Fig. 3 shows the correlation coefficient of the five machine learning models, and the MLP model has the highest value with 0.9794. The MAE and RMSE are means of difference evaluation between two continuous variables [28], given in Eqs. (6) and (7) respectively where jei j is the standard difference between the measured misfit and the predicted. Here MAE and RMSE are used to evaluate the difference between the measured and the predicted misfit. The lower MAE and RMSE are, the more consistent the measured and the predicted misfit are with each other, and the more accurate the learning model is. The values of MAE and RMSE are depicted in Fig. 4, where the blue1 bars represent RMSE and the green bars are MAE. We can see the MLP model has the lowest MAE and RMSE value with 0.178 and 0.2307. So we choose MLP as 1 For interpretation of color in Fig. 4, the reader is referred to the web version of this article.

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X. Jiang et al. / Computational Materials Science 143 (2018) 295–300

Fig. 2. The performance of machine learning models on the training set. (a) Support Vector Regression with linear kernel (SVR/lin); (b) Support Vector Regression with poly kernel (SVR/poly); (c) SMO regression with normalizedpoly kernel (SMOreg/norpoly); (d) SMO regression with poly kernel (SMOreg/poly); (e) Multilayer Perceptron (MLP).

Fig. 4. The MAE and RMSE of machine learning models on training set. Fig. 3. The correlation coefficient of machine learning models on training set.

the selected model to predict lattice misfit on Ni-based single crystal superalloys quantitatively, for its distinguished correlation degree and minimal error.

RMSE ¼ ½n1

n X

1=2

jei j2 

ð6Þ

i¼1

MAE ¼ n1

n X jei j

ð7Þ

i¼1

4. Model validation To validate the accuracy of our MLP misfit model, we compare the predicted misfits with the measured misfits and the misfits cal-

culated by WATANABE (1957) formula. The validation dataset is from the published work of Schulze C [19] and Shi Q [29], which is not contained in our training set with 136 instances. The validation dataset consists of 4 instances (1#, 2#, 3# and 4#), listed in Table 2. There are two reasons why we pick these instances for validation. One is for the large scale variability of chemical composition these 4 instances have; we can test the sensitivity capability of processing different compositions, especially for Co (from 5.7 to 15.8), Re (from 2.1 to 6) and Ru (from 0 to 2) whose contents highly affect creep and fatigue properties. The other is that these 4 instances involve room temperature (25 °C) and high temperature (800 °C) conditions, different dendrite locations (dendrite center, interdendritic, and no records provided) and two distinct specimen thicknesses (foil and bulk); we can test the prediction capability under multi-dimensional constraints of our MLP misfit model. The actual misfits in Table 2 are measured by CBED, illustrated in the published works of Schulze C [19] and Shi Q [29].

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X. Jiang et al. / Computational Materials Science 143 (2018) 295–300 Table 2 The instances for MLP misfit model validation (composition in wt.%) [19,29]. Instance

Alloy

Ni

Al

Co

Cr

Mo

Re

Ru

Ta

W

Hf

Nb

Ti

Ir

Temperature °C

Dendrite location

Specimen thickness

Measured misfit (103)

1#

CMSX-10 [19] CMSX-10 [19] AlloyA [29] AlloyB [29]

69.57

5.7

3

2

0.4

6

0

8

5

0.03

0.1

0.2

0

800

Foil

2.7

69.57

5.7

3

2

0.4

6

0

8

5

0.03

0.1

0.2

0

800

Dendrite center Interdendritic

Foil

1.2

61.6 54

6.2 6

7.9 15.8

3.3 3.4

2.6 2.6

2.1 2.1

2.0 2.0

8.3 8.0

6 6.1

0 0

0 0

0 0

0 0

25 25

No record No record

Bulk Bulk

1.39 0.56

2# 3# 4#

Table 3 The mole fractions of alloy element i in c and c0 phase in validation set (at.%) [19,29]. Alloy

Phase

Ni

Al

Co

Cr

Mo

Re

Ru

Ta

W

Hf

Nb

Ti

Ir

CMSX-10-Dendrite [19]

c c0

67.71 74.36

2.58 15.68

9.35 2.06

8.05 1.32

0.56 0.18

8.72 0.63

0 0

0.36 3.73

2.6 1.68

0 0

0 0

0.08 0.37

0 0

CMSX-10-Interdendrite [19]

c c0

68.51 76.33

2.6 13.52

9.24 2.27

7.92 1.39

0.56 0.17

8.12 0.53

0 0

0.43 3.83

2.53 1.6

0 0

0 0

0.09 0.37

0 0

AlloyA [29]

c c0

64.7 66.04

10.59 15.97

9.87 7.43

5.42 2.99

2.4 1.23

1.33 0.23

1.55 1.08

1.78 3.17

2.35 1.86

0 0

0 0

0 0

0 0

AlloyB [29]

c c0

54.25 57.36

10.37 14.99

19.99 15.99

6.15 3.77

2.43 1.43

1.36 0.27

1.54 1.17

1.62 3.04

2.29 1.98

0 0

0 0

0 0

0 0

Table 4 The Vegard coefficient of alloy element i in Ni and Ni3 Al (103 Å=at:%). Phase

Al [1,30–32]

Co [1,30–32]

Cr [1,30–32]

Mo [1,30–32]

Re [1,30–32]

Ru [1,30–32]

Ta [1,30–32]

W [1,30–32]

Hf [1]

Nb [1]

Ti [1]

Ir [1]

c c0

1.790 –

0.196 0.01762

1.100 0.04

4.780 2.080

4.410 2.620

3.125 1.335

7.000 5.000

4.440 1.940

10.356 7.788

6.653 4.426

4.222 2.520

4.734 4.417

Fig. 5. Misfits comparison of measured by CBED, WATANABE (1957) calculated and MLP predicted.

We perform the same data transformation on the validation set as the training set, and treat the preprocessed validation set (excluding measured misfit) as the input of our MLP misfit model. It takes almost 0 s to run the model and come up with the predicted results, which are 5.125, 1.551, 0.396 and 0.75 sequentially. We also use WATANABE (1957) model to calculate the corresponding misfits. The WATANABE (1957) model is described in Section 1 Eq. (2). The lattice constants of Ni or Ni3 Al are aNi ¼ 3:524 Å and aNi3 Al ¼ 3:57 Å [1], the mole fractions of alloy ele-

ment i in c and c0 phase are listed in Table 3 provided by the published works of Schulze C [19] and Shi Q [29], and the Vegard coefficients of alloy element i in Ni and Ni3 Al are given in Table 4, respectively [1,9]. So after filling in Eq. (2) with the lattice constants, the mole fractions of alloy element in c and c0 phase and the Vegard coefficients, we can obtain the WATANABE (1957) results, which are 0.003, 0.731, 0.52 and 0.77 sequentially. The actual measured misfits by CBED, the calculated misfits by WATANABE (1957) model and our predicted misfits by MLP are compared in Fig. 5. It is obviously that for instance 2#, 3# and 4#, our MLP misfit model achieves the same accuracy as WATANABE (1957), both highly consistent with the measured misfit actually. And for instance 1#, the difference between our MLP predicted misfit and the measured value is 2.425, smaller than 2.703 between WATANABE (1957) and the measured value, and our MLP model has a high agreement with the measured in misfit sign (positive or negative) prediction, which is better than WATANABE (1957) model. The factors like thickness and dendritic position are not considered in WATANABE (1957) model, while in our MLP model these factors are introduced to get more consistent with the actual situation. The further improvement on accuracy can be achieved by increasing the number of samples with thickness and dendritic position as well as considering more factors related to lattice misfit. Therefore, we successfully propose an effective informatics approach for lattice misfit prediction of Ni-based single crystal superalloys. It has accelerated misfit prediction dramatically just based on the initial information of chemical composition, temperature, dendrite location and specimen thickness, without any experiment and measurement during alloy design. At the same time, our machine learning model performs as well as the wellknown empirical formula given by WATANABE (1957) in prediction error, and better in positive or negative prediction of misfit.

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5. Conclusion The informatics approach integrated with machine learning algorithms provides a novel methodology during material design, especially for lattice misfit of Ni-based single crystal superalloys. In this paper, we accumulate relevant dataset manually from open access literatures and construct a dedicated Ni-based single crystal superalloys database for data reuse. After data preprocessing is performed, we train misfit models by SVR, SMOreg and MLP machine learning algorithms with linear and poly kernels for appropriate model selecting. Finally we choose MLP model for its distinguished prediction performance with high correlation coefficient and low MAE and RMSE values. In addition, we compare the prediction accuracy between our MLP misfit model and the well-known empirical formula, and our method performs better. Therefore, machine learning assisted approach accelerates the misfit prediction procedure quantitatively with little experiment and measurement. It is of great importance to reduce the time and cost during alloy design. In the present study, we have considered the effects from composition, dendrite information, shape and size of specimen and temperature. However, other factors, including the effects of different heat treatment parameters and stress constraints, which are also critical to determine misfit, are to be resolved in the follow-on work. The explicit rules of factors to the desired misfit, which is suitable for creep and fatigue properties, are also expected to serve for inverse design of Ni-based single crystal superalloys. Acknowledgements This work is financially supported by the National Key Research and Development Program of China (Grant No. 2016YFB0700503), National High Technology Research and Development Program of China (Grant No. 2015AA034201), Municipal Science and Technology Program of Beijing (Grant No. D161100002416001) and Beijing Key Laboratory of Materials Genome Initiative.

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