







































MFB. Noor et al. /Future Sustainability                                                                                February 2024| Volume 02 | Issue 01 | Pages 
47-58 

47 

 

 

 

Article 

Machine learning in high-entropy alloys: phase 

formation predictions with artificial neural 

networks 
Md Fahel Bin Noor*, Nusrat Yasmin, Tiglet Besara 

Department of Physics, Astronomy and Materials Science, Missouri State University, USA 

               A R T I C L E   I N F O 
 

Article history: 
Received 01 November 2023  
Received in revised form 
30 November 2023 
Accepted 06 December 2023 
 
Keywords:  
Phase formation prediction, High entropy alloys, 
Artificial neural networks, Machine learning 
 
*Corresponding author 
Email address: 
mdfahelbin1@missouristate.edu 
 
 
DOI: 10.55670/fpll.fusus.2.1.5 
 

A B S T R A C T 
 

Due to their complex compositions, high entropy alloys (HEAs) offer a diverse 

range of material properties, making them highly adaptable for various 

applications, including those crucial for future sustainability. Phase engineering 

in HEAs presents a unique opportunity to tailor materials for environmentally 

friendly technologies and energy-efficient solutions. However, the challenge of 

predicting phase selection, a key aspect in harnessing the full potential of HEAs 

for sustainable applications, is compounded by the limited availability of HEA 

data. This study presents a distinctive approach by using a precisely produced 

and selected dataset to train an artificial neural network (ANN) model. This 

dataset, unlike prior studies, is uniquely constructed to contain an equal 

amount of training data for each phase in HEAs, which includes single-phase 

solid solutions (SS), amorphous (AM), and intermetallic compounds (IM). This 

methodology is relatively unexplored in the field and addresses the imbalanced 

data issue common in HEA research. To accurately assess the model's 

performance, rigorous cross-validation was employed to systematically adapt 

the model's hyperparameters for phase formation prediction. The assessment 

includes metrics such as phase-wise accuracy (AM 86.67% SS 81.25% & IM 

82.35%), confusion matrix, and Micro-F1 score (0.83), all of which collectively 

demonstrate the effectiveness of this approach. The study highlights the 

importance of feature parameters in phase prediction for HEAs, shedding light 

on the factors influencing phase selection. Its balanced dataset and training 

method notably advance machine learning in HEA phase prediction, providing 

valuable insights for material design amidst challenges and data scarcity in the 

field. 

 

1. Introduction 
Recently, Multi-principle element alloys (MPEAs) have 

been different from conventional metal alloys, as these alloys 

consist of an equal proportion of individual principal 
elements [1]. MPEA is commonly mentioned interchangeably 

with high entropy alloy (HEA) in the literature [2–4]. Due to 

its remarkable properties, high entropy alloys are 
characterized as novel and promising materials class. These 

alloys tend to have complex chemical compositions 
containing several components [5–8]. Nonetheless, the HEA 

definition limits the number of species to a minimum of four. 

In comparison, only two species of identical atomic 

concentrations can comprise an MPEA. We opt to constantly 
use the term HEA in this paper because of its broader 
classification [3, 9]. Phase engineering is a strategic approach 

that employs various phase structures found in HEAs to 
achieve remarkable performance configuration [10]. This 

approach offers an abundance of potential to modify HEAs for 

specific applications, producing materials that are precisely 
tuned to meet various technological requirements. HEAs can 

exhibit a wide range of desirable characteristics, including 
elevated strength for high load-bearing capacity, increased 

hardness for improved durability, heightened ductility for 

improved deformability, robust wear resistance against 

Future Sustainability 

Open Access Journal 

https://doi.org/10.55670/fpll.fusus.2.1.5 

 

 

 

 

 

 

 

 

 

 

 

 

 

February 2024| Volume 02 | Issue 01 | Pages 47- 58 

Journal homepage: https://fupubco.com/fusus 

 
ISSN 2995-0473 

mailto:mdfahelbin1@missouristate.edu
https://doi.org/10.55670/fpll.fusus.2.1.5
https://fupubco.com/fusus


MFB. Noor et al. /Future Sustainability                                                                                February 2024| Volume 02 | Issue 01 | Pages 47-58 

48 

 

abrasion, immense environmental corrosion resistance,  and 
exceptional catalytic characteristics that enable various 

chemical reactions [11–23]. Specific mechanical properties 

can be targeted utilizing the phases present in HEAs. These 

phases consist of amorphous (AM), intermetallic compounds 
(IM), single-phase solid solutions (SS), and hybrid SS and IM 
phases [6,24–26]. Predicting phase selection is essential for 

designing HEA, though the mechanism behind predicting 
phase selection is crucial in tailored HEA design, yet the 

mechanisms underlying phase formation are uncertain. 
Additionally, the properties of HEAs are significantly 
impacted by the phase structure, and despite improvements, 

designing HEA phases is still challenging and time-consuming 
[1, 5, 6].  

Machine learning has become an important tool to help 
with material design [27–29]. Machine learning, inclusive of 
deep learning, requires extracting features from large 

datasets to recapitulate the relationships, which also offers 
the chance to predict the phase formation of HEAs focusing on 

existing research data by using a range of deep learning. 

Several studies reveal intriguing outcomes in phase formation 
prediction by compiling data on HEAs and developing deep 

learning algorithms [30, 31]. For predicting phase formation, 
Zhu et al. [5] have introduced a deep neural network (DNN) 

architecture using a residual network (ResNet), which 

achieves 81.9% overall accuracy. An ANN model is utilized in 
Islam et al.'s study for phase prediction, with 99% accuracy 

on training data, while the practical prediction accuracy was 
below 80%. [1]. Several algorithms were employed, including 

logistic regression, decision tree, support vector machine 
(SVM) classifier, random forest, gradient boosting classifier, 

and  ANN in Y.V. Krishna et al.’s research work [32], ANN has 

demonstrated the best accuracy of more than 80% for the test 
data among these algorithms. New alloys were synthesized 

and characterized to validate the predictions that ANN is the 
most accurate prediction method in the studied alloy system. 

K-nearest neighbors (KNN), SVM, and ANN are the three 
machine learning algorithms used in the study by Huang et al. 
[33], and ANN exhibits superior testing accuracy than other 

models for predicting phases in new HEAs. In Uttam et al.'s 
[31] study, the use of a neural network (NN) model is 

introduced for the first time to predict the hardness of a 

refractory high entropy alloy (RHEAs), and the prediction is 
verified through experimental synthesis and microstructural 

analysis. This model successfully applies to various alloys to 
predict hardness, which is consistent with available 

experimental results. In another recent study [24], the 

predictive accuracy of an ANN model in determining phase 
selection across three distinct alloy types is evaluated. It 

emphasizes the extensive impact of atomic size differences 
(δ) on the phases within AM, SS, and IM alloys. The research 

effectively forecasts the phases in two novel alloys by 
leveraging this learning model in conjunction with a 

combination of three or four key parameters with 

confirmation through X-ray diffraction. This approach 
provides a potentially promising tool for advancing the 

composition design and phase selection of novel alloys. While 
the field is extending, challenges continue in predicting HEA 

phases, such as advancing deep learning algorithms and 

dealing with a lack of experimental data. Given the vast 
unexplored compositional design space, developing efficient 

machine learning algorithms based on existing data becomes 
crucial for precise HEA phase prediction. 

This work presents an ANN model that tackles the 

challenge of training with balanced data for each phase in 

HEAs, contrasting the frequent problem of unbalanced 
datasets in past research. The objective is to determine the 
hyperparameters that maximize predictive accuracy and 

generality when predicting phase selection in new HEAs, 
utilizing a balanced dataset. In this study for HEA phase 

formation prediction, an ANN model architecture is optimized 
for the current balanced dataset and fine-tuned 
hyperparameters such as batch size, learning rate, epochs, 

and dropout rate. The model's performance is evaluated on 
the final test set by measuring phase-wise accuracy, 

originating a confusion matrix, and determining the Micro-F1 
score, and the results are then compared with prior studies. 
The study also analyses the significance of feature parameters 

in phase prediction outcomes, clarifying the relative 
importance of physical parameters influencing phase 

selection. 

2. Computational Methods 

The HEA dataset underwent preprocessing and 
preparation using conventional data science techniques prior 
to being utilized to train the model. Three distinct datasets 

from various earlier studies [34–36] were selected and 

employed to construct the ANN-based model. A dataset of 240 

HEAs was obtained after the elimination of redundant 
samples and sections with incomplete or duplicated data. An 
instance of randomly selected five rows of the dataset is 

presented in the Pandas DataFrame format in Table 1. In this 
dataset, there are an equal number of 80 data points for each 

of the AM, SS, and IM phases. The dataset consists of 240 
instances and 6 features, including 1 categorical feature 

designating different phases (AM, SS, or IM) and 5 numeric 
features representing valence electron concentration (VEC), 
the difference in electronegativity difference (Δχ), atomic size 

difference (δ), mixing enthalpy (ΔHmix), and mixing entropy 
(ΔSmix). 

 
Table 1. A glimpse of Pandas displaying random 5 instances of the 

data employed in this study. The units for ΔHmix and ΔSmix are 
kJmol−1 and JK−1mol−1, respectively  

 

The labels were encoded into integers and assigned the 
values 0 for AM, 1 for SS, and 2 for IM phases, respectively, to 
denote the alloy phases in the ANN model. The formulas 

provided below are used to compute the numeric values for 

the five features [37–40]. 

𝑉𝐸𝐶 =  ∑ 𝑐𝑖𝑉𝐸𝐶𝑖
𝑛
𝑖=1                                      (1) 

∆𝜒 = √∑ 𝑐𝑖 (𝜒𝑖 − 𝜒)2𝑛
𝑖=1                             (2) 



MFB. Noor et al. /Future Sustainability                                                                                February 2024| Volume 02 | Issue 01 | Pages 47-58 

49 

 

𝛿 = 100 𝑋 √∑ 𝑐𝑖 (1 −  𝑟𝑖 𝑟⁄ )2𝑛
𝑖=1                                             (3) 

Δ𝐻𝑚𝑖𝑥 = ∑ 4𝐻𝑖𝑗
𝑛
𝑖=1,𝑖<𝑗 𝑐𝑖𝑐𝑗                            (4) 

Δ𝑆𝑚𝑖𝑥 =  −𝑅 ∑ 𝑐𝑖 𝑙𝑛𝑐𝑖
𝑛
𝑖=1                              (5) 

Here, ci (where 0 < ci < 1) denotes the atomic concentrations 

of the i-th element, while n represents the total number of 
components within a HEA. VECi and ri signifies the atomic 

concentration, VEC, and atomic radius of each species of the i-

th element and R denotes the gas constant. Using Miedema's 
model, the enthalpy of atomic pairs, is calculated [41]. 𝜒 and 

𝑟 refer for the weighted Pauling electronegativity and atomic 
radius, respectively, written as follows.  

𝜒 =  ∑ 𝑐𝑖𝜒𝑖
𝑛
𝑖=1                                               (6) 

𝑟 =  ∑ 𝑐𝑖𝑟𝑖
𝑛
𝑖=1                                                   (7) 

The data undergo preprocessing for feature values before 

training the architecture. Using the Pandas library [42], these 
values are normalized and scaled them between 0 and 1, as 

shown below:  

𝑋𝑛𝑒𝑤 = 
𝑋𝑖 −𝑋𝑚𝑖𝑛,𝑖

𝑋𝑚𝑎𝑥,𝑖− 𝑋𝑚𝑖𝑛,𝑖
                           (8) 

Where, Xnew represents to the normalized feature, Xi refers to 

the actual feature information, Xmin,i and Xmax,i stand for the 
minimum and maximum values respectively. Through this 

normalization procedure, dimensionless numeric features 
are generated, which ensures effective uniform numeric 
scaling and consistent treatment of all features.  

A layer of neurons performs computational task with the ANN 
model. the output of each neuron within the hidden layer is 

denoted by aj, as expressed in the following equation. 

𝑎𝑗 = ∑ 𝑥𝑖
𝑛
𝑖=1  𝑊𝑖𝑗 + 𝑏𝑗                                  (9) 

Where 𝑏𝑗 designates the bias coefficients and  𝑊𝑖𝑗 
corresponds to the weights of each input parameter 𝑥𝑖 . 

Google's TensorFlow [43] is a well-recognized library in this 
field and based on that, the machine learning neural network 

architecture is used. Figure 1 shows the architecture used in 
this study which encompasses backpropagation functions 
and several hidden layers. Eq. (9) is used to calculate the value 

of 𝑎𝑗 for each neuron which is related with connection-

specific weights. The activation function takes it as an input 

value. Five features of parameters are encompassed as input 

and the three neurons denoting different phases are included 

in the output layer.  
The leaky Rectified linear unit (LReLU) activation 

function has been applied within the hidden layers. The 

Rectified Linear Unit (ReLU) [44], illustrated in Figure 2, is a 
common and popular activation function in Neural Networks 

(NNs).  By compelling precise tuning of the learning rate, it 
can extend past predefined bounds during the NN training 
process because of its easiness and subsequent reduction in 

training computation time. Due to this issue, the activation 
function remains inactive for the neurons within the negative 

region during the training process. By assigning a small 
constant value, like 0.2, to the negative region, Leaky ReLU 

(LReLU) [45], solves this, as presented in Figure 2. Three 
nodes of the model’s output layer represent the alloy phases 

which receive input from the final hidden layer and then these 

nodes employ activation functions to predict the phase.  The 

broadly used activation function, SoftMax, illustrated in 
Figure 3, was utilized in the output layer for this classification. 
The probability of the input belonging to different classes is 

illustrated by this normalized exponential function illustrates. 
Generally, the SoftMax function [46] is expressed as: 

𝜎(𝑦𝑖
′) =

𝑒 𝑦𝑖
′

∑ 𝑒𝑦𝑖
′

𝑛
𝑖=1

                                  (10) 

Here, 𝜎(𝑦𝑖
′) denotes the subsequent probability and the 

prediction vector is referred as 𝑦𝑖
′ . Consequently, the model's 

output was contrasted with the target labels to assess 
network's error. Here, cross-entropy [47] is utilized as cost 

(loss) function, which resembles to the negative logarithm of 

probability, and the following equation represents the 
function. 

𝐻𝑦(𝑦 ′) =  − ∑ 𝑦 log(𝜎(𝑦𝑖
′))𝑛

𝑖=1        

Here, 𝑦 ′ refer to the prediction and y stand for one of the three 
target vectors. The neural network’s final output is converted 

into a probability, and then, using cross-entropy, it is utilized 

to calculate the loss. The deviation between the actual 
distribution and the model's expected output distribution is 

computed by Cross-entropy. Afterward, the gradient descent 
algorithm is used, utilizing a learning rate of 0.013, to convey 
back this error through the network. The weights and bias are 

initiated randomly in the beginning of the training process. 
The loss function is minimized by adjusting them at each 

epoch. The accuracy of the network is quantified by the 
number of successful determinations of the target.  The 
hyperparameter configuration for the artificial neural 

network (ANN) model encompassed a range of values and 
architectures, contributing to the systematic tuning process. 

For the number of hidden layers, the model was 
experimented with settings ranging from 3 to 5 layers, 
exploring the impact of network depth. Regularization  

techniques, such as L1 and L2, were introduced within the 

ranges of 0.01 to 0.025, allowing for the assessment of their 

influence on model generalization. Similarly, the learning 
rate, an essential factor in optimization, was varied between 

0.001 and 0.013 to identify the optimal balance between 
convergence and avoiding local minima. Dropout rates, a 

regularization method to mitigate overfitting, were adjusted 

within the range of 0.1 to 0.4. Different batch sizes, ranging 
from 8 to 120, were examined to evaluate their effect on 

model training efficiency and convergence. These diverse 
configurations and their corresponding results constituted a 
comprehensive exploration of the ANN's hyperparameters to 

achieve the best predictive performance and phase wise 
accuracy. After training the model with the training dataset, a 

distinct, unseen test set, which was preserved during training, 
was used to evaluate the model. The feedback from validation 
provides guidance to adjust the parameter. The best model is 

chosen after demonstration of the optimum validation result. 
The hyperparameters are described in Table 2 and the test set 

is used to evaluate this model.  



MFB. Noor et al. /Future Sustainability                                                                                February 2024| Volume 02 | Issue 01 | Pages 47-58 

50 

 

 

 

 
 
 

 

Figure 2. Rectified linear unit (ReLU) and leaky ReLU (LReLU) 

 
 

3. Results and discussion 

3.1 Data analysis 

Comprehending the dataset of 240 records is an essential 
preliminary stage before applying a machine learning 
algorithm. Then, we generate a scatter matrix plot using the 

Seaborn package and then compute the correlation matrix of 

the features using Pandas library. These two matrices aid in 

comprehending feature relationships within the curated HEA 
dataset and offer both qualitative and quantitative 

 

 

 
 
 

 
interconnection estimates. Our prediction pertains to the 

phases, with a specific emphasis on five quantitative features 
of the HEA compositions. To visualize the data, we employ a 5 
× 5 scatter matrix plot, as depicted in Figure 4. The diagonal 

subfigures illustrate histograms of phase distributions, 
considering individual utilization of each of the five features. 

All histograms within subfigures overlap, suggesting no 
isolated feature for complete alloy phase classification.  
Correlations among the five features influence phase 

selection in HEAs, which is evident in off-diagonal subfigures 
of Figure 4. 

 
Figure 3. SoftMax function 

 

Figure 1. Illustration of the artificial neural network (ANN) architecture designed for predicting phase formations in High-Entropy Alloys 
(HEAs). For clarity, only five neurons (illustrated as circles) within the hidden layers are depicted. Empty squares symbolize input features 

and output values. The AM, SS, and IM phases are encoded as vectors 0, 1, and 2, respectively. 



MFB. Noor et al. /Future Sustainability                                                                                February 2024| Volume 02 | Issue 01 | Pages 47-58 

51 

 

Table 2. Hyperparameters of ANN 

Hyperparameter Value  

Number of hidden layers 5  

Number of hidden neurons 150 neurons 

Regularization L1: 0.025, L2: 0.01 

Activation function LeakyReLU (alpha= 0.1) 

Dropout rate 0.4 

Batch size 65 

Learning rate 0.013 (Adam optimizer) 

Epochs 100 

Loss function Categorical Crossentropy 

 
 

We also calculate the Pearson correlation coefficient for 

features x and y to provide a quantitative description of their 

correlations [48]. 

𝑟𝑥𝑦 =  
1

𝑛−1

∑ (𝑥𝑖−𝑥)(𝑦𝑖− 𝑦)𝑛
𝑖=1

𝑆𝑥𝑆𝑦
                 (12) 

 

 

Here, x and y represent the mean values of two features, while 

𝑆𝑥 and 𝑆𝑦 are their respective standard deviations. 

Correlation values can vary between -1 and 1, indicating 
negative or positive relationships. The computed correlation 

matrix elements are presented in Figure 5.  Centering on the 
correlation between two distinct features, the matrix 

elements range from -0.61 to 0.72. Out of the ten distinct 

correlation matrix elements, seven exhibit negativity, while 
the remaining are positive. This outcome also exhibits 

resemblance to a prior study [1]. In the correlation matrix and 
scatter plot, of electronegativity difference (Δχ) and atomic 
size difference (δ), a positive correlation is observed, meaning 

that Δχ tends to increase with higher values of δ. Additionally, 
both Δχ and δ show negative correlations with valence 

electron concentration (VEC) and mixing enthalpy (ΔHmix). In 
general, the correlation matrix elements exhibit moderate 
magnitudes, allowing all five features to be employed 

collectively as input for our neural network architecture. 

 

 

 

 
 

Figure 4. The scatter plots presented in the off-diagonal sections reveal the correlations among the values of the five distinct features. Within 
the diagonal panels, the histograms illustrate the distributions of the three phases based on the five features. Each phase is represented using 

varying shapes and colors: a yellow circle signifies amorphous (AM), a blue diamond represents solid solution (SS), and a red square denotes 
intermetallic (IM). 



MFB. Noor et al. /Future Sustainability                                                                                February 2024| Volume 02 | Issue 01 | Pages 47-58 

52 

 

 
Figure 5. Correlation matrix heatmap of the five features 

 

3.2 Artificial Neural Network (ANN) Results 
The development of all Python network models is 

accomplished using the Keras framework with a TensorFlow 
backend for ANNs. Subsequently, the hyperparameter values 

are adjusted and the optimal model and hyperparameter 

settings for the ANN model are determined through a 3-fold 
cross-validation process as depicted in Figure 6. The best 

parameter resulted in an average cross-validation accuracy of 
86.46%, as detailed in Table 2. 

Afterwards, the ANN architecture is employed to train on 

80% of the developed balanced dataset and then applied to 
test the remaining 20% of the dataset. This process is 

visualized in Figure 7(a), (b) and (c) illustrating the 
progression of training loss and validation loss across number 

of epochs for three separate folds.  

Figure 6. Training and testing process of the ANN model 

Notably, both the training loss and validation loss curves 
exhibit a similar trajectory, demonstrating that the 

optimization algorithm consistently updates the weights of 

hidden layer neurons to minimize the loss and enhance the 

learning process at each epoch without overfitting the data. 
Across the three-fold training set, the loss converges to 1 after 
40 epochs, maintaining a consistent trend thereafter. 

Furthermore, it's important to highlight that when the model 
is evaluated on the final set of data, there is a noticeable 

decrease in the loss value. This reduction brings the loss down 
to 0.5 after approximately 30 epochs of the training process. 
This trend is visually represented in Figure 7(d), where the 

curve illustrating the loss for the final test set which closely 
resembles the trajectory observed during the training 

process. This indicates that the model's performance on the 
test data doesn't show any signs of overfitting. This consistent 
reduction in loss and the convergence of the curves 

emphasize the model's ability to generalize well and perform 
effectively on new, unseen data for each of the HEA phases.  In 

Figure 8(a), a visual representation is provided for the three-

fold cross-validation process that was employed to assess the 
model’s performance, showcasing the accuracy achieved for 

each individual fold along with average accuracy while 
training the data. Notably, the calculated average validation 

accuracy across all three folds was determined to be 86.46%. 

This approach of three-fold cross-validation ensures proper 
evaluation of the model's effectiveness across different 

subsets of the data. Additionally, the prediction of phase-wise 
accuracy on the training datasets is depicted in Figure 8(b). 

Remarkably, the final validation set attains a prediction rate 
nearing 83.33%, affirming the strong performance of the 

developed ANN model and its favorable generalization 

capabilities.  
 

 

 



MFB. Noor et al. /Future Sustainability                                                                                February 2024| Volume 02 | Issue 01 | Pages 47-58 

53 

 

Figure 9 illustrates the accuracy comparison between the 
training and validation processes for the final test set, 

revealing a positive correlation between them and overall 

accuracy improvement with epochs. Observing the training 

outcomes depicted in Figure 9, it's evident that the training 
set accuracy incrementally rises as iterations progress. After 
approximately 30 epochs, the accuracy stabilizes, suggesting 

effective convergence of the model.  The effectiveness of the 
ANN model in predicting each of the phases is displayed in 

Figure 10. In contrast to some other studies that typically 
report overall accuracy, it is equally crucial to highlight phase-
wise accuracy to illustrate the model's competence and its 

ability to predict various phases effectively. The results 
underscore the model's proficiency in accurately predicting 

distinct phases, and notably, the phase-wise accuracy reaches 
impressive levels, with AM achieving 86.67%, SS reaching 
81.25%, and IM attaining 82.35%. These results exemplify the 

ANN model's effectiveness in predictive performance, 
particularly when it is trained on a balanced dataset for each 

phase. Using Micro-F1 to evaluate the prediction outcomes, 

the test set is employed to validate the effectiveness of the 
ANN model. The necessary equations for calculating the 

Micro-F1 score are provided below [5,49]:Using Micro-F1 to 
evaluate the prediction outcomes, the test set is employed to 

validate the effectiveness of the ANN model.  

 

 
 

 
 
 

The necessary equations for calculating the Micro-F1 
score are provided below [5,49]: 

𝑅𝑒𝑐𝑎𝑙𝑙𝑚𝑖𝑐𝑟𝑜 =
∑ 𝑇𝑃𝑖

𝑛
𝑖=1

∑ 𝑇𝑃𝑖
𝑛
𝑖=1 + ∑ 𝐹𝑁𝑖

𝑛
𝑖=1

                      (13) 

𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛𝑚𝑖𝑐𝑟𝑜 =
∑ 𝑇𝑃𝑖

𝑛
𝑖=1

∑ 𝑇𝑃𝑖
𝑛
𝑖=1 + ∑ 𝐹𝑃𝑖

𝑛
𝑖=1

                (14) 

𝐹1𝑚𝑖𝑐𝑟𝑜 = 2
𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛𝑚𝑖𝑐𝑟𝑜∗𝑅𝑒𝑐𝑎𝑙𝑙𝑚𝑖𝑐𝑟𝑜

𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛𝑚𝑖𝑐𝑟𝑜+𝑅𝑒𝑐𝑎𝑙𝑙𝑚𝑖𝑐𝑟𝑜
         (15) 

Here, true positive (𝑇𝑃𝑖) means positive cases correctly 

identified as positive cases of the i element, false positive 

(𝐹𝑃𝑖) means negative cases is incorrectly identified positive 
cases of the i element, true positive (𝐹𝑁𝑖) means positive 

cases is incorrectly identified negative cases of the i element. 
𝑅𝑒𝑐𝑎𝑙𝑙𝑚𝑖𝑐𝑟𝑜 measures the accuracy of correctly predicting 
actual positive samples within the sample space. 

𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛𝑚𝑖𝑐𝑟𝑜 quantifies the accuracy of forecasting positive 
predictions. F1micro is the aggregated average that considers 

both 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛𝑚𝑖𝑐𝑟𝑜 and 𝑅𝑒𝑐𝑎𝑙𝑙𝑚𝑖𝑐𝑟𝑜. The Micro F1 Score on 
final test set is 0.83. To assess the predictive performance of 

the ANN model for each of the HEA classes within the dataset, 

a confusion matrix was generated using a testing dataset 
comprising 48 samples, as illustrated in Figure 11. 

 
 

 
 

 
 
 

 

 

 

 

0

1

2

3

4

5

6

7

0 50 100

L
o

ss

Epochs

Training Loss

Validation Loss

0

1

2

3

4

5

6

7

0 50 100

L
o

ss

Epochs

Training Loss
Validation Loss

0

1

2

3

4

5

6

7

0 50 100

L
o

ss

Epochs

Training Loss
Validation Loss

0

1

2

3

4

0 20 40 60

L
o

ss

Epochs

Training Loss

Validation Loss

Figure 7. Comparing the training and validation loss of the ANN model for a) Fold 1, b) Fold 2, c) Fold 3, and d) Final test set. 



MFB. Noor et al. /Future Sustainability                                                                                February 2024| Volume 02 | Issue 01 | Pages 47-58 

54 

 

 

 
Figure 9. Comparing the accuracy of the ANN model between the 

training set and final validation set 

 

 

 
Figure 10. Comparing phase-wise accuracy and average accuracy 
found for the final test set 

 
The confusion matrix indicates notably high precision 

and recall values across all three classes of HEA, affirming the 
model's robustness and generalization. The variance between 

the cross-validation accuracy and the confusion matrix 

accuracy arises because the former represents the average of 
all validation accuracies, while the latter directly reports 

 

 
accuracy for the testing dataset that has been kept separate 
during the model development process.  The developed ANN 

model aims to predict the phase of previously unseen data, 
and its performance was benchmarked against other 

alternative methods, as illustrated in Figure 12. Among the 
various machine learning algorithms assessed for accuracy, 

our developed ANN model demonstrates the highest accuracy 
of 83.33%. Notably, while Islam et al.  [1] and Krishna et al. 
[32] also utilized an ANN model, their datasets exhibit 

unequal proportions of data across different HEA phases. In 
contrast, our ANN model maintains consistency by employing 

the same number of instances for AM, IM, and SS phases, and 
this uniformity contributes to the model's robust 
performance, allowing it to achieve the noteworthy accuracy 

of 83.33%, surpassing the accuracy of other methods 

[5,33,50], including those that employed differing dataset 

compositions. The developed ANN model aims to predict the 
phase of previously unseen data, and its performance was 
benchmarked against other alternative methods, as 

illustrated in Figure 12. Among the various machine learning 

algorithms assessed for accuracy, our developed ANN model 

demonstrates the highest accuracy of 83.33%.  
 

 
Figure 11. Confusion matrices of ANN model used in amorphous, 

solid solution, and intermetallic phase prediction on Final Test Set. 

 

  
 

(b) (a) 

Figure 8. Comparing (a) fold-wise accuracy and (b) average phase-wise accuracy found for the cross-validation data employing the ANN model. 



MFB. Noor et al. /Future Sustainability                                                                                February 2024| Volume 02 | Issue 01 | Pages 47-58 

55 

 

 
Figure 12. Evaluating the Precision of Machine Learning Algorithms  

 
 

Notably, while Islam et al.  [1] and Krishna et al. [32] also 
utilized an ANN model, their datasets exhibit unequal 

proportions of data across different HEA phases. In contrast, 

our ANN model maintains consistency by employing the same 
number of instances for AM, IM, and SS phases, and this 

uniformity contributes to the model's robust performance, 
allowing it to achieve the noteworthy accuracy of 83.33%, 

surpassing the accuracy of other methods [5,33,50], including 
those that employed differing dataset compositions.  

3.3 Relative feature impact assessment 
The ANN architecture was used to assess the relative 

significance of the five input features used to train the model. 

To investigate this, a series of five experiments were carried 
out, with each experiment systematically omitting one feature 

while keeping the remaining four. This procedure entailed 
retraining the model and making predictions in order to 
thoroughly investigate the impact on the test set accuracy. 

The results of these experiments can be seen in Figure 13, 
which depicts the decline in accuracy across the five scenarios 

mentioned above. This trend highlights an important 

observation: removing any of the five features consistently 
resulted in a decrease in the accuracy of the model, 

highlighting the significant influence that each feature has on 
test accuracy [5,25]. When compared to other features, 

differences in atomic sizes and the concentration of valence 

electrons have a greater influence on the accuracy of the 

model. Notably, it has been determined that the key design 
parameters derived from the current ANN approach—the 
atomic size difference and the valence electron 

concentration—align closely with the preexisting parametric 
guidelines for HEA phase formation.  

 
 

 

 

 
 
 

 
 

 
 
 

 
 

 
 
 

 
 

 

 
 

 
 

 

 
 

 
 

 
Figure 13. Effect on the test set accuracy upon the removal of 
individual features. 

 

This convergence highlights an intriguing correlation, 
confirming the developed ANN method's reliability. 

Consistent with Hume-Rothery principles, the atomic size 

difference plays a crucial role in phase formation of HEA, 
especially in case of solid solution (SS) phase [5,38,51,52]. 

Furthermore, the Hume-Rothery principles show that the 
number of valence electrons per atom is critical in 

determining the stability of solid solutions in metal binary 
systems [5,52–54], and this stability in the mentioned 

systems hinges on electron density, specifically where peaks 

in the density of states occur, coinciding with the point where 
the Fermi sphere intersects the Brillouin zone boundary. As a 

result, the structure becomes stable at a specific electron 
concentration level. While atomic radius and 



MFB. Noor et al. /Future Sustainability                                                                                February 2024| Volume 02 | Issue 01 | Pages 47-58 

56 

 

electronegativity differences are not always conclusive 
predictors of outcomes, they're both highly indicative 

parameters in the design of HEA compositions [51–54], 

underscoring the significance of considering electronic 

structure alongside other material properties when designing 
HEAs. 

 

4. Conclusion 
In this study, a carefully developed ANN model was 

introduced to address the persistent challenge of imbalanced 
datasets when predicting phase selection in HEAs. Through a 
rigorous optimization process encompassing various 

hyperparameters, the ANN model was developed using a 
balanced dataset, resulting in excellent predictive 

performance. Using a three-fold cross-validation strategy, the 
model's effectiveness was carefully evaluated. The results 
showed an impressive average validation accuracy of 86.46% 

across all three folds and eventually led to a high prediction 
rate of nearly 83.33% on the final test set, highlighting the 

model's robustness and capacity for generalization. This 

study also emphasized the importance of phase-wise 
accuracy, with the ANN model achieving remarkable accuracy 

levels for the studied HEA phases (86.67% for AM, 81.25% for 
SS, and 82.35% for IM). A detailed confusion matrix analysis 

also confirmed the model's robustness across all classes, 

highlighting its precision and recall balance, while 
comparison against alternative methods demonstrated its 

superior accuracy, and the Micro-F1 score validated the 
model's effectiveness with a score of 0.83 on the final test set. 

Notably, this was accomplished by maintaining dataset 
balance for each phase, which distinguished this approach 

from previous studies that frequently used imbalanced 

datasets and didn't mention phase-wise accuracy which is 
very important to showcase the model's ability for 

generalization. Furthermore, the study investigated the 
relative importance of input features, identifying that atomic 

size difference and valence electron concentration played 
critical roles in test accuracy, in line with established 
guidelines for HEA phase formation and reinforcing the 

developed ANN method's reliability. It should also be noted 
that including more data for each phase can contribute to 

even higher model performance, providing an exciting 

potential for further predictive accuracy improvement. 
Overall, this study not only provides an effective solution to 

an existing issue in materials science, but it also provides 
critical insights into the impact of physical parameters on 

phase selection, making it invaluable for future alloy design 

and engineering efforts. 

Ethical issue 
The authors are aware of and comply with best practices in 
publication ethics, specifically with regard to authorship 
(avoidance of guest authorship), dual submission, 
manipulation of figures, competing interests, and compliance 
with policies on research ethics. The authors adhere to 
publication requirements that the submitted work is original 
and has not been published elsewhere. 

Data availability statement 
The manuscript includes existing data, and additional data 

that support the findings of this study are openly available in 
High-Entropy-Alloy-Phase-Prediction-Using-Balanced-

Dataset at https://github.com/fahel-bin-noor/High-Entropy-
Alloy-Phase-Prediction-Using-Balanced-Dataset. 

Conflict of interest 

The authors declare no potential conflict of interest.  

References 
[1]     N. Islam, W. Huang, H.L. Zhuang, Machine learning for 

phase selection in multi-principal element alloys, 
Computational Materials Science. 150 (2018) 230–

235. 

https://doi.org/10.1016/j.commatsci.2018.04.003. 
[2] D.B. Miracle, O.N. Senkov, A critical review of high 

entropy alloys and related concepts, Acta Materialia. 
122 (2017) 448–511. 

https://doi.org/10.1016/j.actamat.2016.08.081. 

[3] J.-W. Yeh, S.-K. Chen, S.-J. Lin, J.-Y. Gan, T.-S. Chin, T.-T. 
Shun, C.-H. Tsau, S.-Y. Chang, Nanostructured High-

Entropy Alloys with Multiple Principal Elements: 
Novel Alloy Design Concepts and Outcomes, 

Advanced Engineering Materials. 6 (2004) 299–303. 
https://doi.org/10.1002/adem.200300567. 

[4] B. Cantor, I.T.H. Chang, P. Knight, A.J.B. Vincent, 

Microstructural development in equiatomic 
multicomponent alloys, Materials Science and 

Engineering: A. 375–377 (2004) 213–218. 
https://doi.org/10.1016/j.msea.2003.10.257.  

[5] W. Zhu, W. Huo, S. Wang, X. Wang, K. Ren, S. Tan, F. 

Fang, Z. Xie, J. Jiang, Phase formation prediction of 
high-entropy alloys: a deep learning study, Journal of 

Materials Research and Technology. 18 (2022) 800–
809. https://doi.org/10.1016/j.jmrt.2022.01.172. 

[6] Y. Zhang, T.T. Zuo, Z. Tang, M.C. Gao, K.A. Dahmen, P.K. 

Liaw, Z.P. Lu, Microstructures and properties of high-
entropy alloys, Progress in Materials Science. 61 

(2014) 1–93. 
https://doi.org/10.1016/j.pmatsci.2013.10.001. 

[7] B. Gludovatz, A. Hohenwarter, D. Catoor, E.H. Chang, 

E.P. George, R.O. Ritchie, A fracture-resistant high-
entropy alloy for cryogenic applications, Science. 345 

(2014) 1153–1158. 
https://doi.org/10.1126/science.1254581.  

[8] W. Huo, F. Fang, X. Liu, S. Tan, Z. Xie, J. Jiang,  
Remarkable strain-rate sensitivity of nanotwinned 

CoCrFeNi alloys, Applied Physics Letters. 114 (2019) 

101904. https://doi.org/10.1063/1.5088921.  
[9] E.G. Reineke, O.T. Inal, Crystallization behavior of 

amorphous Ni50Nb50 on continuous heating, 
Materials Science and Engineering. 57 (1983) 223–

231. https://doi.org/10.1016/0025-5416(83)90212-

4. 
[10] X. Chang, M. Zeng, K. Liu, L. Fu, Phase Engineering of 

High-Entropy Alloys, Advanced Materials. 32 (2020) 
1907226. 
https://doi.org/10.1002/adma.201907226.  

[11] W. Huo, X. Liu, S. Tan, F. Fang, Z. Xie, J. Shang, J. Jiang, 
Ultrahigh hardness and high electrical resistivity in 

nano-twinned, nanocrystalline high-entropy alloy 
films, Applied Surface Science. 439 (2018) 222–225. 
https://doi.org/10.1016/j.apsusc.2018.01.050. 

[12] Z. Li, K.G. Pradeep, Y. Deng, D. Raabe, C.C. Tasan, 
Metastable high-entropy dual-phase alloys overcome 

https://github.com/fahel-bin-noor/High-Entropy-Alloy-Phase-Prediction-Using-Balanced-Dataset
https://github.com/fahel-bin-noor/High-Entropy-Alloy-Phase-Prediction-Using-Balanced-Dataset


MFB. Noor et al. /Future Sustainability                                                                                February 2024| Volume 02 | Issue 01 | Pages 47-58 

57 

 

the strength–ductility trade-off, Nature. 534 (2016) 
227–230. https://doi.org/10.1038/nature17981.  

[13] W. Huo, H. Zhou, F. Fang, X. Hu, Z. Xie, J. Jiang, Strain-

rate effect upon the tensile behavior of CoCrFeNi 

high-entropy alloys, Materials Science and 
Engineering: A. 689 (2017) 366–369. 
https://doi.org/10.1016/j.msea.2017.02.077.  

[14] W. Huo, H. Shi, X. Ren, J. Zhang, Microstructure and 
Wear Behavior of CoCrFeMnNbNi High-Entropy Alloy 

Coating by TIG Cladding, Advances in Materials 
Science and Engineering. 2015 (2015) e647351. 
https://doi.org/10.1155/2015/647351.  

[15] Y. Shi, L. Collins, R. Feng, C. Zhang, N. Balke, P.K. Liaw, 
B. Yang, Homogenization of AlxCoCrFeNi high-

entropy alloys with improved corrosion resistance, 
Corrosion Science. 133 (2018) 120–131. 
https://doi.org/10.1016/j.corsci.2018.01.030. 

[16] S. Wang, W. Huo, F. Fang, Z. Xie, J.K. Shang, J. Jiang, 
High entropy alloy/C nanoparticles derived from 

polymetallic MOF as promising electrocatalysts for 

alkaline oxygen evolution reaction, Chemical 
Engineering Journal. 429 (2022) 132410. 

https://doi.org/10.1016/j.cej.2021.132410.  
[17] Md.F. bin Noor, B. Mallick, A. Habib, Heat storage 

system: A modern way to reuse and recycle energy to 

reduce thermal pollution, in: International 
Conference on Mechanical, Industrial and Energy 

Engineering, 2018. 
[18] A.S. Lakhnot, R.A. Panchal, J. Datta, V. Mahajani, K. 

Bhimani, R. Jain, D. Datta, N. Koratkar,  Intercalation 
Hosts for Multivalent-Ion Batteries, Small Structures. 

4 (2023) 2200290. 

https://doi.org/10.1002/sstr.202200290.  
[19] J. Datta, N. Koratkar, D. Datta, Open-tunneled oxides 

as intercalation host for multivalent ion (Ca and Al) 
batteries: A DFT study, (2023). 

https://doi.org/10.48550/arXiv.2303.12301.  
[20] Z.U. Mahmud, S. Karmakar, A. Haque, K.C. Ghosh, A 

study of fabrication and characterization of NaxMnO2 

as a cathode material for sodium-ion battery, MRS 
Advances. (2023). https://doi.org/10.1557/s43580-

023-00611-4. 

[21] M.S. Uddin, R.A. Mayanovic, M. Benamara, On the 
synthesis and characterization of bimagnetic 

CoO/NiFe2O4 heterostructured nanoparticles, AIP 
Advances. 13 (2023) 025314. 

https://doi.org/10.1063/9.0000561.  

[22] M.T. Islam, M.S. Rabbi, M.S. Uddin, Noise reduction of 
helicopter rotor blades by using spoiler, AIP 

Conference Proceedings. 2121 (2019) 040014. 
https://doi.org/10.1063/1.5115885.  

[23] N. Yasmin, M.F.B. Noor, T. Besara, Structure and 
Magnetism of the New Cage-structured Compound 

HfMn2Zn20, (2023). 

https://doi.org/10.48550/arXiv.2306.01146.  
[24] L. Wang, P. Li, W. Zhang, F. Wan, J. Wu, L. Yong, X. Liu, 

Prediction of phase selection of amorphous alloys and 
high entropy alloys by artificial neural network, 

Computational Materials Science. 223 (2023) 112129. 

https://doi.org/10.1016/j.commatsci.2023.112129.  

[25] S.Y. Lee, S. Byeon, H.S. Kim, H. Jin, S. Lee, Deep 
learning-based phase prediction of high-entropy 

alloys: Optimization, generation, and explanation, 

Materials & Design. 197 (2021) 109260. 

https://doi.org/10.1016/j.matdes.2020.109260.  
[26] B. Mallick, R. Das, S. Banik, Md.F. bin Noor, A. Habib, 

Performance Enhancement of an Automobile 

Radiator by Using a Nozzle Arrangement, 
International Journal of Innovative Technology and 

Exploring Engineering. X (2019).  
[27] F. Aydin, R. Durgut, Estimation of wear performance 

of AZ91 alloy under dry sliding conditions using 

machine learning methods, Transactions of 
Nonferrous Metals Society of China. 31 (2021) 125–

137. https://doi.org/10.1016/S1003-
6326(20)65482-6. 

[28] J. Datta, D. Datta, V. Sharma, Transferable and Robust 

Machine Learning Model for Predicting Stability of Si 
Anodes for Multivalent Cation Batteries, J Mater Sci. 

58 (2023) 11085–11099. 

https://doi.org/10.1007/s10853-023-08705-y. 
[29] K.G. Naik, B.S. Vishnugopi, J. Datta, D. Datta, P.P. 

Mukherjee, Electro-Chemo-Mechanical Challenges 
and Perspective in Lithium Metal Batteries, Applied 

Mechanics Reviews. 75 (2023). 

https://doi.org/10.1115/1.4057039.  
[30] R. Machaka, Machine learning-based prediction of 

phases in high-entropy alloys, Computational 
Materials Science. 188 (2021) 110244. 

https://doi.org/10.1016/j.commatsci.2020.110244.  
[31] U. Bhandari, Md.R. Rafi, C. Zhang, S. Yang, Yield 

strength prediction of high-entropy alloys using 

machine learning, Materials Today Communications. 
26 (2021) 101871. 

https://doi.org/10.1016/j.mtcomm.2020.101871.  
[32] Y.V. Krishna, U.K. Jaiswal, R.M. R, Machine learning 

approach to predict new multiphase high entropy 
alloys, Scripta Materialia. 197 (2021) 113804. 
https://doi.org/10.1016/j.scriptamat.2021.113804.  

[33] W. Huang, P. Martin, H.L. Zhuang, Machine-learning 
phase prediction of high-entropy alloys, Acta 

Materialia. 169 (2019) 225–236. 

https://doi.org/10.1016/j.actamat.2019.03.012. 
[34] S. Guo, C.T. Liu, Phase stability in high entropy alloys: 

Formation of solid-solution phase or amorphous 
phase, Progress in Natural Science: Materials 

International. 21 (2011) 433–446. 

https://doi.org/10.1016/S1002-0071(12)60080-X. 
[35] C.E. Precker, A. Gregores Coto, S. Muíños Landín, 

Materials for Design Open Repository. High Entropy 
Alloys, (2021). 

https://doi.org/10.5281/zenodo.6403257.  
[36] R. Machaka, Dataset for High-Entropy Alloys Phases, 

3 (2021). https://doi.org/10.17632/7fhwrgfh2s.3. 

[37] S. Fang, X. Xiao, L. Xia, W. Li, Y. Dong, Relationship 
between the widths of supercooled liquid regions and 

bond parameters of Mg-based bulk metallic glasses, 
Journal of Non Crystalline Solids. 321 (2003) 120–

125. https://doi.org/10.1016/S0022-

3093(03)00155-8. 



MFB. Noor et al. /Future Sustainability                                                                                February 2024| Volume 02 | Issue 01 | Pages 47-58 

58 

 

[38] Y. Zhang, Y. Zhou, J. Lin, G. Chen, P. Liaw, Solid‐
Solution Phase Formation Rules for Multi‐component 

Alloys, Advanced Engineering Materials. 10 (2008) 

534–538. https://doi.org/10.1002/adem.200700240. 

[39] C.T. Liu, Physical metallurgy and mechanical 
properties of ductile ordered alloys (Fe, Co, Ni)3 V, 
International Metals Reviews. 29 (1984) 168–194. 

https://doi.org/10.1179/imtr.1984.29.1.168.  
[40] J.H. Zhu, P.K. Liaw, C.T. Liu, Effect of electron 

concentration on the phase stability of NbCr2-based 
Laves phase alloys, Materials Science and 
Engineering: A. 239–240 (1997) 260–264. 

https://doi.org/10.1016/S0921-5093(97)00590-X. 
[41] Classification of Bulk Metallic Glasses by Atomic Size 

Difference, Heat of Mixing and Period of Constituent 
Elements and Its Application to Characterization of 
the Main Alloying Element, (n.d.). 

https://www.jstage.jst.go.jp/article/matertrans/46/
12/46_12_2817/_article (accessed August 25, 2023).  

[42] W. McKinney, Python for Data Analysis: Data 

Wrangling with Pandas, NumPy, and IPython, second 
edition, O’Reilly Media, Inc., 2017.  

[43] M. Abadi, P. Barham, J. Chen, Z. Chen, A. Davis, J. Dean, 
M. Devin, S. Ghemawat, G. Irving, M. Isard, M. Kudlur, 

J. Levenberg, R. Monga, S. Moore, D.G. Murray, B. 

Steiner, P. Tucker, V. Vasudevan, P. Warden, M. Wicke, 
Y. Yu, X. Zheng, TensorFlow: a system for large-scale 

machine learning, in: Proceedings of the 12th USENIX 
Conference on Operating Systems Design and 

Implementation, USENIX Association, USA, 2016: pp. 
265–283. 

[44] M.M. Lau, K. Hann Lim, Review of Adaptive Activation 

Function in Deep Neural Network, in: 2018 IEEE-
EMBS Conference on Biomedical Engineering and 

Sciences (IECBES), 2018: pp. 686–690. 
https://doi.org/10.1109/IECBES.2018.8626714. 

[45] A.L. Maas, Rectifier Nonlinearities Improve Neural 
Network Acoustic Models, in: 2013. 
https://www.semanticscholar.org/paper/Rectifier-

Nonlinearities-Improve-Neural-Network-
Maas/367f2c63a6f6a10b3b64b8729d601e69337ee3

cc (accessed August 30, 2023). 

[46] E. Byvatov, U. Fechner, J. Sadowski, G. Schneider, 
Comparison of support vector machine and artificial 

neural network systems for drug/nondrug 

classification, J Chem Inf Comput Sci. 43 (2003) 

1882–1889. https://doi.org/10.1021/ci0341161.  
[47] L. Devroye, L. Györfi, G. Lugosi, A probabilistic theory 

of pattern recognition, 3. print, Springer, New York, 

NY, 2008. 
[48] J.D. Kelleher, B.M. Namee, A. D’Arcy, Fundamentals of 

Machine Learning for Predictive Data Analytics, (n.d.). 
[49] P. Vateekul, M. Kubat, Fast Induction of Multiple 

Decision Trees in Text Categorization from Large 

Scale, Imbalanced, and Multi-label Data, in: IEEE 
Computer Society, 2009: pp. 320–325. 

https://doi.org/10.1109/ICDMW.2009.94. 
[50] A. Oñate, J.P. Sanhueza, D. Zegpi, V. Tuninetti, J. 

Ramirez, C. Medina, M. Melendrez, D. Rojas, 

Supervised machine learning-based multi-class phase 
prediction in high-entropy alloys using robust 

databases, Journal of Alloys and Compounds. 962 

(2023) 171224. 
https://doi.org/10.1016/j.jallcom.2023.171224. 

[51] A. Takeuchi, A. Inoue, Classification of Bulk Metallic 
Glasses by Atomic Size Difference, Heat of Mixing and 

Period of Constituent Elements and Its Application to 

Characterization of the Main Alloying Element, 
Materials Transactions. 46 (2005) 2817–2829. 

https://doi.org/10.2320/matertrans.46.2817. 
[52] M.G. Poletti, L. Battezzati, Electronic and 

thermodynamic criteria for the occurrence of high 
entropy alloys in metallic systems, Acta Materialia. 75 

(2014) 297–306. 

https://doi.org/10.1016/j.actamat.2014.04.033. 
[53] U. Mizutani, H. Sato, T.B. Massalski, The original 

concepts of the Hume-Rothery rule extended to alloys 
and compounds whose bonding is metallic, ionic, or 

covalent, or a changing mixture of these, Progress in 
Materials Science. 120 (2021) 100719. 
https://doi.org/10.1016/j.pmatsci.2020.100719. 

[54] S. Guo, C. Ng, J. Lu, C.T. Liu, Effect of valence electron 
concentration on stability of fcc or bcc phase in high 

entropy alloys, Journal of Applied Physics. 109 (2011) 

103505. https://doi.org/10.1063/1.3587228. 
  

 This article is an open-access article 

distributed under the terms and conditions of the Creative 
Commons Attribution (CC BY) license 

(https://creativecommons.org/licenses/by/4.0/). 

 

https://creativecommons.org/licenses/by/4.0/
https://creativecommons.org/licenses/by/4.0/

