Multi-label phase-prediction in high-entropy-alloys using Artificial-Neural-Network

Journal Pre-proofs Multi-label Phase-prediction in high-entropy-alloys using Artificial-NeuralNetwork Shrey Dixit, Vineet Singhal, Abhishek Agarwal, A.K. Prasada Rao PII: DOI: Reference:

S0167-577X(20)30311-6 https://doi.org/10.1016/j.matlet.2020.127606 MLBLUE 127606

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Materials Letters

Received Date: Revised Date: Accepted Date:

23 December 2019 21 February 2020 4 March 2020

Please cite this article as: S. Dixit, V. Singhal, A. Agarwal, A.K. Prasada Rao, Multi-label Phase-prediction in high-entropy-alloys using Artificial-Neural-Network, Materials Letters (2020), doi: https://doi.org/10.1016/ j.matlet.2020.127606

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Multi-label Phase-prediction in high-entropy-alloys using Artificial-Neural-Network Shrey Dixit1, Vineet Singhal1, Abhishek Agarwal1, A.K. Prasada Rao1* 1School

of Engineering and Technology, BML Munjal University, Gurgaon, India- 122413

Abstract A novel Artificial-Neural-Network architecture has been developed, for the first time, which predicts eight coexisting-phases present in high-entropy-alloys. The model considers composition and processing-route to compute physical and thermodynamic-parameters of the alloy. The Artificial-Neural-Network uses these parameters and predicts the phases in a given alloy. Validation reveals prediction accuracy of 87.083%. This modeling technique can find potential application in designing new high-entropy-alloys and choosing their processing-routes. Keywords: High-entropy-alloys; Phase-transformation; Simulation and modeling Introduction High-entropy-alloys (HEAs) are multicomponent systems having five or more elements, each element is about 5-35% in atomic-composition [1]. HEAs show superior thermal, mechanical, chemical and electromagnetic properties that lend their applications in structural, aerospace, food and energy industries [2-5]. The properties are sensitive to the phases in an HEA. In HEAs, BCCphase result in high yield-strength and very low plasticity, while the FCC-phase contributes to low yield-strength and high-plasticity and BCC+FCC phases lead to high-strength and good-ductility [6-8]. Phase-prediction is crucial as the properties of HEA depend on the former. Few prediction methods viz. Ab-initio, Molecular-Dynamics-simulation, and CALPHAD are known [9]. In recent

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Corresponding author: Dr. A.K. Prasada Rao, Professor, BML Munjal University, Gurgaon, India – 122413. e-Mail: [email protected]; Phone: +91-8295 96 3823

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years, computationally-intelligent techniques have been proposed for phase-prediction. Huang et.al. [10] used Machine-learning for predicting the solid-solutions/ intermetallics in HEAs with an accuracy of 70%. Agarwal et.al. [11] predicted FCC and BCC phases in HEAs based on both their elemental-composition and thermodynamic-parameters through artificial-intelligence. But the scope of all these independent research works was limited to the prediction of major-phases such as solid-solutions, intermetallics, FCC and BCC. i.e., the approaches reported were unsuccessful in predicting all the co-existing phases. Present work is an Artificial-Neural-Network approach for the phase-prediction in an HEA formed by random-combination of elements chosen from a set of 58 (supplementary data file-1) for a given composition and processing route with 87.083% accuracy. Given composition and its processing-route, proposed model predicts all the coexisting-phases (FCC, BCC, FCC1+FCC2, BCC1+BCC2, B2, Laves, C14 Laves and sigma) in a HEA. Materials and Methods Experimental data used for training of the model was collected from literature [1]. A total of 452 HEAs and seven parameters were chosen for the modeling. The parameters considered were ΔSmix, mixing-entropy; VEC, average-valence-electronic-configuration; processing-route; ∆Hmix, enthalpy-of-mixing; δ, atomic-size-difference; Ω, a combination-effect of ΔSmix and ∆Hmix; χ, difference in electronegativity. These parameters were calculated using Eq. (1-7) [12-14]. 𝑛

4Δ𝐻𝑚𝑖𝑥 𝐴𝐵 𝑐𝑖𝑐𝑗

(1)

∆𝑆𝑚𝑖𝑥 = ― 𝑅∑𝑖 = 1(𝑐𝑖 ln 𝑐𝑖)

(2)

∆𝐻𝑚𝑖𝑥 = ∑𝑖 = 1,

𝑖≠𝑗 𝑛

(

𝑛

𝛿 = ∑𝑖 = 1𝑐𝑖 1 ―

2

𝑟𝑖 2 𝑟

)

(3)

Ω=

𝑇𝑚Δ𝑆𝑚𝑖𝑥 Δ𝐻𝑚𝑖𝑥 𝑛

𝑇𝑚 = ∑𝑖 = 1𝑐𝑖(𝑇𝑚)𝑖 𝑛

𝑉𝐸𝐶 = ∑𝑖 = 1𝑐𝑖𝑉𝐸𝐶𝑖 𝑛

2 𝜒 = ∑𝑖 = 1𝑐𝑖(𝜒𝑖 ― 𝜒)

ci and cj 𝜟𝑯𝒎𝒊𝒙 𝑨𝑩 R 𝐫𝐢 𝐧

𝐫=

∑𝐜 𝐫

(4) (5) (6) (7)

atomic-parentages of ith and jth components respectively enthalpy-of-mixing of liquid binary-alloys [15] universal-gas-constant. atomic-radius of ith element average atomic-radius

𝐢 𝐢

𝐢=𝟏

𝑻𝒎 (𝐓𝐦)𝐢 𝐕𝐄𝐂𝐢 𝛘𝐢 𝒏

𝝌=

∑𝒄 𝝌

melting-temperature of n-element alloy, calculated using rule of mixtures melting-temperature of ith element Valence-electron-concentration of ith element Pauling-electronegativity of an element average Pauling-electronegativity

𝒊 𝒊

𝒊=𝟏

The processing-routes considered in this work were arc-melting (AM); Bridgmansolidification (BS); injection-casting (IC); induction-melting (IM); laser-cladding (LC); mechanical-alloying (MA); Magnetron-sputtering (MaS); Melt-spinning (MeS); Suction-casting (SC) and Sputtering. During modeling, it is recommended to check the interdependency of parameters, as they measure essentially the same property of an HEA and can decrease the accuracy of the model. Therefore, highly-correlated parameters should be removed if they do not contribute towards phase-prediction. A hierarchical-cluster-analysis [16] was used to study the correlations between parameters which groups the features together based on their Spearman's-Rank-CorrelationCoefficients. The Spearman's-Rank-Correlation-Coefficient [17] is a measure of the strength of 3

the association that exists between two variables. The magnitude of the correlation-coefficient signifies the strength of the correlation, which lies in the range of 0 to 1. Where ‘0’ indicates nocorrelation and ‘1’ indicates the perfect degree of association. A hierarchical-cluster-analysis approach was used to group the parameters based on their mutual-correlations. Hierarchical-cluster-analysis is an algorithm that finds two most-correlated variables and replaces them with a new single-variable obtained from their average. This is an iterative process, and the iteration continues until two variables are left. The results of this analysis are shown in a dendrogram (Fig.1).

Fig .1 Hierarchical-parametric-clustering based on Spearman's-Rank-Correlation-Coefficient The horizontal axis in Fig.1 shows the extent of correlation between two parameters, i.e., the shorter the dendrogram branch between a pair of parameters, the stronger their correlation is. The short-dendrogram branch of Ω and ΔHmix means that these two parameters are highlycorrelated. Similarly, a significantly longer-dendrogram branch indicates poor-correlation among a parameter-pair (eg. Ω and χ). The understanding of the correlations was further used in improving the accuracy of the model by eliminating the redundant-parameter. The Artificial-Neural-Network (ANN) model presented in this work has two hidden-layers with 20 neurons in the first-one and 12 in the second-one. Each hidden-layer is followed by a Rectified-Linear-Unit activation (ReLU) and Batch-Normalization [18]. The batch-normalization 4

layer normalizes the outputs of the previous-layer by subtracting the mean and dividing by the standard-deviation. The output-layer has 8 neurons corresponding to 8 phases in our dataset. The output from each neuron is squeezed between 0 and 1 after applying the sigmoid-function, which is shown in its mathematical-form in Eq. (8). This function was used to interpret the output-value of a neuron as a probability of the phase (corresponding to that neuron) present in a given HEA. 1

(8)

𝜎(𝑥) = 1 + 𝑒 ―𝑥

The model was trained on 362 HEAs and validated on 90 HEAs. The dataset has been provided in the supplementary data file-2. The authors have developed a python-program which automatically calculates all the above six-parameters for any random-combination of elements chosen from a set of 58 for a given composition. The accuracy of the model was calculated using the Hamming-score which is also known as labelbased accuracy. Hamming-accuracy is the fraction of labels that are correctly predicted multiplied by 100. The Eq. (9) represents the formula for calculating the Hamming score [19]. 1

|𝑁|

|𝐿|

Accuracy, (A) = [1 ― |𝑁|.|𝐿|∑𝑖 = 1∑𝑗 = 1𝑋𝑂𝑅(𝑦𝑖,𝑗, 𝑧𝑖,𝑗) ] ∗ 100

(9)

N = number of examples; L = number of labels; y = actual labels; z = predicated labels Trials were carried out to decide the threshold value of the output-neurons. And it was found that the threshold value of 0.2 corresponds to the highest accuracy. It means that if a neuron in the output-layer outputs a value bigger than 0.2, the model will predict that phase corresponding to that neuron. At 0.2 threshold, the authors achieved Hamming-Accuracy of 82.5%. In order to remove redundant-parameters, the importance of each of the parameters on the accuracy of the model was calculated by the following steps: Step-1 Random shuffling of the values of a parameter in the validation dataset Step-2 Calculation of the accuracy using the dataset obtained from Step-1 5

Step-3 Calculation of the difference between the accuracy obtained from Step-2 and 82.5 Step-4 Repeat from Step-1 for each of the parameters The differences in the accuracy obtained from the above process have been identified as the importance of the respective parameters on the accuracy of the model, which has been shown in Fig 2. An important observation that can be made from Fig. 2 is that the entropy has a very less effect on Phase-prediction. This is because the entropy acts only as a determining factor for an alloy to be HEA or not. On the other hand, δ plays the most important role in Phase-prediction than the parameters considered, which is followed by χ. A similar study to understand the importance of parameters on Phase-prediction of solid solutions and intermetallics was carried out by Huang et.al. [10], and it was observed that δ had the most important role. This is coherent with the results obtained in the current work, which shows that δ has the highest importance. Further, present work can be used in alloy design as one can synthesize any number of virtual alloys until the desired phases are predicted, and further use the findings to synthesis the real alloy.

Fig 2. Relative Importance of each parameter on the accuracy of the model From Fig. 2, it can also be seen that Ω has the least influence on the accuracy of the model. This can be attributed to its close correlation with ∆Hmix. Therefore, Ω was not considered in the current modeling. 6

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Results and Discussion The model was retrained without Ω and the accuracy achieved was 87.083 %. The model has been demonstrated using two screenshots as shown in Fig. 3(a-b). If the user provides an HEA and its processing route as inputs, the model predicts the phases of the HEA successfully, which also reconfirm the experimental reports by Kao et. al [20] and Yeh et. al [21] respectively.

(a) (b) Fig.3 Demonstration of the model for (a) CoFeMnTi2.5V1.9Zr2.4 and (b) Al1.2CoCrCuFeNi The model has been validated by comparing the predicted-phases with those of the experimental ones reported in the literature for a processing-route (supplementary data file-3). The model will fail to predict if the given alloy forms an HEA or not. Given a non-HEA, it will still give some predictions which may or may not be correct. The model also fails to predict any other phases except the eight phases it was trained on. The authors believe that if given sufficient data on other phases, they can retrain the model so that it works for other phases too i.e, the limitation is not the ability of the model but the availability of the data. Conclusion An Artificial-Neural-Network model has been developed which is capable of predicting all the coexisting phases in an HEA. It can also predict the phases in an HEA formed from a random combination of elements from a bunch of 58, with an accuracy of 87.083 %. Hence it is possible to synthesize any number of virtual alloys until the desired phases are predicted, thereafter synthesize the real alloy. Therefore, it finds a potential scope in the design of an HEA and selection of its processing route. It has been found that δ plays the most important role in Phase-prediction, followed by χ. 8

References [1] B.S. Murty, J.W. Yeh, and S. Ranganathan, High-Entropy Alloys, 1st ed., London: Butterworth-Heinemann, 2014 [2] M.R. Chen, S.J. Lin, J.W. Yeh, M.H. Chuang, S.K. Chen, and Y.S. Huang, Metall. Mater. Trans. A 37 (2006) 1363. [3] C.Y. Hsu, W.R. Wang, W.Y. Tang, S.K. Chen, and J.W. Yeh, Adv. Eng. Mater. 12, (2010) 44. [4] S.G. Ma and Y. Zhang, Mater. Sci. Eng. A 532 (2012) 480. [5] Y. Zhang, T.T. Zuo, Z. Tang, M.C. Gao, K.A. Dahmen, P.K. Liaw, and Z.P. Lu, Prog. Mater Sci. 61, (2014) 1. [6] Y. Zhang, T.T. Zuo, Z. Tang, M.C. Gao, K.A. Dahmen, P.K. Liaw, et. al., Prog. in Mater. Sci., 61, (2014)1. [7] M. Yao, K. Pradeep, C. Tasan, D. Raabe, Scripta Mater.72, (2014)5. [8] Y. Hsu, C-C. Juan, W-R. Wang, T-S. Sheu, J-W. Yeh, S-K. Chen: Mater. Sci. Eng., A 528, (2011)3581. [9] C. Chattopadhyay, A. Prasad, and B.S. Murty, Acta Mater. 153, (2018) 214. [10] W. Huang, P. Martin, and H.L. Zhuang, Acta Mater. 169, (2019) 225. [11] A. Agarwal and A.K.P Rao, JOM, 71, (2019) 3424. [12] C. Liu, Physical metallurgy and mechanical properties of ductile ordered alloys (Fe, Co, Ni) 3V, Intl. Metals Rev. 29 (1984) 168. [13] M.C. Gao, J.W. Yeh, P.K. Liaw, Y. Zhang, High-entropy-alloys: Fundamentals and Applications, Springer, Switzerland, 2016

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[14] D.B. Miracle and O.N. Senkov, Acta Mater. 122, (2017) 448. [15] A.Takeuchi, A. Inoue, Mater. Trans (JIM), 46(12), (2005)2817. [16] D. Mullner, Modern Hierarchical, Agglomerative clustering algorithms, arXiv:1109.2378v1 [stat.ML], 2011 [17] Spearman Rank Correlation Coefficient. In: The Concise Encyclopedia of Statistics. Springer, New York, 2008 [18] S. Ioffe and C. Szegedy, Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift, Computer Science, Mathematics ICML 2015. [19] S.R. Khade, S.R.Balwan, Int. J. Comp. Applications, 159(9), (2017)0975. [20] Y.F. Kao, S.K. Chen, J.H. Sheu, J.T. Lin, W.E. Yeh, et. al., Int. J. Hydrogen Energy, 35, (2010) 9046. [21] J.W. Yeh, S.K. Chen, J.Y. Gan, S.J. Lin, T.S.Chin, T.T. Shun, et. al., Metall. Mater. Trans. A, 35, (2004)2533. AUTHORS’ CONTRIBUTION

Shrey Dixit Vineet Singhal Abhishek Agarwal A.K. Prasada Rao

Computational Methodology, Creating model, validation, generating the results, writing the initial draft Conducting a research and investigation process, data collection, writing the initial draft Analysing the results and Preparation of the initial draft Conceptualization, Editing and Writing the final manuscript, and overall supervision

[22] Declaration of interests

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☒ The authors declare that they have no known conflict of interests or personal relationships that could have appeared to influence the work reported in this paper.

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

[23] Declaration of interests

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

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[24] Highlights 

Artificial Neural Network predicts all the phases in a given HEA and processing route.



Alloy composition and processing route of HEA are sufficient inputs for phase-predictions.



A python programme calculates ∆Smix, ∆Hmix, δ, VEC, Ω, χ and ANN uses these parameters for phase-predictions.

 [25]

Present work can be used in alloy design.

[26]

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[27]

[28]

[29]

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