Stability and structure prediction of cubic phase in as cast high entropy alloys

Introduction

High entropy alloys, which are one class of the emerging structural materials, have attracted more and more interest of people because of their high hardness, wear resistance, high temperature softening resistance and oxidation resistance.^1–3 Comparing to traditional alloys, high entropy alloys are designed by the strategy of equal or near equal atomic ratio, typically composed of more than five metallic elements, and usually form simple body centred cubic (bcc) or face centred cubic (fcc) solid solutions rather than multiple intermetallic compounds.^4–6 Based on such solid solutions structure, high entropy alloys present particular microstructure and properties, which can offer many potential applications on tools, moulds and mechanical parts, etc. ⁷

For high entropy alloys, the most common constituent elements are Fe, Cr, Mo, V, Mn with bcc structure, Cu, Al, Ni with fcc structure and Ti, Co with hexagonal close packed structure. By suitable alloy formula design, high entropy alloys with different structure and properties are formed when these constituent elements are mixed with different combination and different amounts of certain elements. For example, as the amount of Al increases, the structure of as cast Al_xCoCrCuFeNi (x = 0·5–3·0) high entropy alloys can change from fcc to fcc+bcc structures, and finally transform into bcc structure. ⁸ Furthermore, the hardness and strength of alloys also increase with the rise of Al content. Thus, the structure of high entropy alloys has an important influence on the properties of alloys, and a reasonable and stable structure can make the high entropy alloys have more excellent properties. According to previous studies,^1–8 the high entropy of mixing caused by equiatomic ratio mixing of multiple alloying elements is the main reason that leads to the formation of the solid solution structure. However, there is lack of systematic research for the cause of structure stability in high entropy alloys. In addition, in order to further develop high entropy alloys, it is necessary to implement an effective method to predict the structure of high entropy alloys.

In recent years, due to the development of computer aided design in the field of materials science, the artificial neural network (ANN) technology has been widely used in materials research and achieved many good effects, especially the back propagation artificial neural network (BP-ANN) technology.^9–12 The ANN method can automatically summarise hidden rules from the existing experimental data by simulating brain to learn the external environment and be good at dealing with the problem of complicated non-linear relation between input and output elements. In the present study, the aim of this paper is to investigate the relationship between stability of cubic structure and physical parameters of solid solutions in as cast high entropy alloys, and then implement a method of predicting the simple cubic structure of as cast high entropy alloys using the BP-ANN.

Analysis of stability of cubic phase

According to the Hume–Rothery rule, ¹³ two main factors can affect the formation of the solid solution in binary alloys. One is the atomic size difference δ. It is most probable to form a substitution solid solution when the atomic size difference between components is <15. The atomic size difference of component in alloys is calculated by the following equation ¹⁴ 1 where c_i and r_i refer to atomic percentage and radius of the ith component respectively. Zhang et al. ¹⁴ found that when the value of atomic size difference δ ranges from 1 to 6, the alloys tends to form substitution solid solution.

The second factor based on the Hume–Rothery rule is the chemical compatibility between components, including the mixing enthalpy ΔH_mix, electronegativity difference Δχ and electron concentration. The larger the mixing enthalpy is, the more likely the alloys form compounds rather than solid solutions. Zhang et al. ¹⁴ found that the alloys tend to form solid solutions in the region delineated by −0·16≤ΔH_mix≤0·1 eV. The mixing enthalpy ΔH_mix of alloys is calculated as follows 2

Here, c_i is atomic percentage, and is the mixing enthalpy of binary AB alloys. In the present paper, the value of was calculated using the first principles calculation. Regarding the electronegativity difference Δχ, the Pauling electronegativity difference for a multicomponent alloy system can be calculated by equation (3) ¹⁵ 3 where c_i is atomic percentage, and χ_i refers to the Pauling electronegativity of the ith component. Moreover, for the effect of electron concentration, because the high entropy alloys are usually composed of transition metals, the bonding in alloys comes mainly from the contribution of valence electrons. Therefore, for convenience, the valence electron concentration VEC was introduced to study the effect of electron concentration on the structure stability in high entropy alloys and can be calculated by the following equation ¹⁶ 4

Here, c_i is atomic percentage, and (VEC)_i is the valence electron concentration of the ith component. Table 1 shows the required physical parameters for the common alloying elements in high entropy alloys.

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Table 1

Physical parameters of elements in high entropy alloys

Elements	Atomic radius/Å	Pauling electronegativity	Valence electron concentration
Al	1·432	1·61	3
Co	1·251	1·88	9
Cr	1·249	1·66	6
Fe	1·241	1·83	8
Mn	1·350	1·55	7
Ni	1·246	1·91	10
Ti	1·462	1·54	4
V	1·316	1·63	5
Mo	1·363	2·16	6
Cu	1·278	1·90	11

By summarising as cast high entropy alloys systems with cubic structure reported from previous studies,^17–32 the relationship between four physical parameters (the atomic size difference δ, mixing enthalpy ΔH_mix, electronegativity difference Δχ and valence electron concentrate VEC) in these alloys was obtained and shown in Fig. 1. In order to make the relationship between these physical parameters more clearly, ΔH_mix, Δχ and VEC are all plotted as the function of δ.

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1

Relationship between physical parameters of high entropy alloys with cubic structure (δ atomic size difference, ΔH_mix mixing enthalpy, Δχ electronegativity difference, VEC valence electron concentration)

From Fig. 1, it can be seen that for these as cast high entropy alloys with cubic structure, the calculated physical parameters of alloys are all concentrated in certain areas. The value of atomic size difference δ ranges from 3·5 to 7·5, which results in small lattice distortion appearing in the formation of as cast high entropy alloys. The mixing enthalpy ΔH_mix of as cast high entropy alloys ranges from −0·6 to 0·2 eV. The electronegativity difference Δχ and valence electron concentration VEC meet the conditions delineated by 0·11≤Δχ≤0·17 and 6·0≤VEC≤8·5 respectively. From the view of energy, the mixing enthalpy ΔH_mix is the only effective parameter to define the stability of solid solution. A stable solid solution would form when −0·16≤ΔH_mix≤0·1 eV based on the conclusions from Zhang et al., ¹⁴ and in our present work, the value of mixing enthalpy ΔH_mix for as cast high entropy alloys with cubic structure ranges from −0·6 to 0·2 eV, as shown in Fig. 1a. Obviously, the solid solution would be more stable once ΔH_mix is smaller (−0·6≤ΔH_mix≤−0·16 eV). When ΔH_mix of alloys ranging from 0·1 to 0·2 eV presents a higher value, although it is higher than 0·1 eV, the effect of high mixing entropy originated from multiple principal elements mixing in high entropy alloys would offset this higher part of mixing enthalpy ΔH_mix. Thus, these solid solutions are also stable. It is also the main characteristic that high entropy alloys differ from traditional alloys. Furthermore, these as cast high entropy alloy systems, which have higher value of mixing enthalpy ΔH_mix, all have lower atomic size differences (3·5≤δ≤6·0). Hence, the lattice distortion in these alloys is small when the substitution solid solutions are formed, which is favourable for the formation of stable solid solution.

In addition, as shown in Fig. 1a and b, it is worth noting that, except Mo containing as cast high entropy alloys, the mixing enthalpy ΔH_mix of alloys presents a downward trend with the increase in atomic size difference δ, while there is a slightly rise of electronegativity difference Δχ, which indicates that the mixing enthalpy ΔH_mix of as cast high entropy alloys declines when the lattice distortion of alloys increases. This part of additional decreasing energy could become a potential driving force of phase transformation for alloys transforming to a more stable structure with smaller distortion energy. For example, in order to reduce the distortion energy of solid solution, alloys usually transform from fcc to bcc structure, which has a lower atomic stacking effect. ²⁴ Moreover, the effect of alloying elements segregation is enhanced with the rise of electronegativity difference Δχ, which also provides favourable conditions for the phase transformation in high entropy alloys.

Structure prediction of cubic phase

The information about the stability of the solid solution in as cast high entropy alloys is clearly shown from Fig. 1. However, it is not clear which phases (or structure of solid solution) would form in as cast high entropy alloys. As reported by Guo et al., ³³ VEC can be used to quantitatively predict the phase forming in as cast high entropy alloys, especially for AlCoCrCuFeNi high entropy alloys system. Sole bcc structure forms when VEC<6·87; sole FCC structure forms when VEC≥8·0; and when 6·87≤VEC<8, as cast high entropy alloys will form mixed fcc and bcc structures. Although this criterion is successful to a certain extent, it fails to predict the structure of Mo and Mn containing as cast high entropy alloys, as shown in Fig. 1c. It suggests that the formation of simple cubic phases in as cast high entropy alloys does not depend only on single factor. Many works have tried to achieve structure prediction in as cast high entropy alloys,^33–35 and all found that the structure of solid solution in as cast high entropy alloys is related to various physical parameters. Based on the results about stability of cubic phase discussed above and previous study, ³³ for as cast high entropy alloys with cubic structure, the atomic size difference δ and mixing enthalpy ΔH_mix play important roles in phase transformation, and electronegativity difference Δχ and valence electron concentrate VEC have influences on element segregation and stacking character of structure respectively. Obviously, the formation of different structures in as cast high entropy alloys depends mainly on these four physical parameters. It can be easily seen that the change of type or content of alloying elements in as cast high entropy alloys would make these physical parameters with different values, which possibly results in the formation of different structures. However, it is a pity that the relationship between formation of cubic phase and physical parameters is complex and has no quantitative conclusions so far. Thus, structure prediction of as cast high entropy alloys has not yet been reached. In our present work, for investigating such complex and non-linear relationship, ignoring the internal mechanism, we provide a simple solution to predict the cubic structure of as cast high entropy alloys using the BP-ANN and hope that it can provide ideas for further research about high entropy alloys.

A typical BP-ANN usually consists of an input layer, an output layer and one or more hidden layers. The learning process of BP-ANN is composed of positive transfer of information and back propagation of error. ³⁶ Taking a two-layer BP-ANN for example,^36,37 mathematically, setting the input is P_j, output is a and target vector is t. There are r neurons in input layer and S1 neurons in hidden layer. The activation function is set as f₁. There are S₂ neurons in output layer, and corresponding activation function is f₂. In the process for positive transfer of information, the output of ith neuron in hidden layer and kth neuron in output layer can be calculated by equations (5) and (6) respectively 5 6

In which the error function E can be obtained as follows 7

In the process for back propagation of error, the change of weight from the input of ith neuron to the output of kth neuron can be expressed as 8 where , . This equation can be changed by deviation method as 9

The change of weight in hidden layer can be calculated by equation (10) 10

Here, , . Similarly, equation (10) also can be changed by deviation method as 11

During the network training, the system error E reaches the minimum via the calculation for back propagation of error and correction of weight w_kj. At this time, recording the weight w_kj and predicting the output. Before training the BP-ANN model, both input and output variables should be normalised within the range from 0 to 1 to obtain a usable form for the network to read. ³⁷ In our present work, the normalisation method of variables was selected as following equation 12 where X_i is the sample data, and is the normalised data. X_min and X_max are the minimum and maximum values of X respectively.

The number of neurons in hidden layer is important for the precision of predicted values and was determined by the trial and error procedure. ³⁸ The number of hidden layer, learning rate and target error values and so on were also determined at the same time. For predicting the cubic structure of as cast high entropy alloys, a three-layer feed forward BP-ANN was introduced after repeated adjustment and training of ANN, and the schematic of BP-ANN is shown in Fig. 2. In this BP-ANN model, the input parameters are four physical parameters, the atomic size difference δ, mixing enthalpy ΔH_mix, electronegativity difference Δχ and valence electron concentration VEC, and the output is the structure parameter of as cast high entropy alloys. In order to parameterise the output data, when as cast high entropy alloys form sole fcc structure, sole bcc structure and mixed fcc and bcc structures, the output data were defined as ‘1’, ‘2’ and ‘3’ respectively. The established BP-ANN model contained a hidden layer and had a size of 8×6×1. The training function of network was chosen as trainscg. The target error values and learning rate were set as 0·001 and 0·2 respectively.

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2

Schematic of back propagation ANN

In the present paper, for obtaining reasonable BP-ANN model, random two-thirds of the data (49 samples) were chosen for training, and the rest one-third (25 samples) for test. After training the BP-ANN, the performance was evaluated. The best results from the trained BP-ANN were further analysed by the way of linear regression analysis between the network outputs and corresponding targets, as shown in Fig. 3. It is obvious that the predicted values of training and test agree well with the experimental data, and the correlation coefficient R for training and test datasets are 0·988 and 0·824 respectively, which indicates that this performance of BP-ANN has achieved a good result. In addition, Fig. 4 shows the comparison of the experimental and predicted structure parameters for training and test datasets. It can be easily seen that there is a good fitting between output of the network and experimental results, and >95 of the absolute value of relative error for test datasets is within 10 (in which foregoing 49 samples are compared with the results of network training output, and the final 25 samples are the results of test output). The results indicate that the established BP-ANN model has a satisfactory precision and can be used to predict. Therefore, corresponding output for structure parameter of as cast high entropy alloys would be obtained when non-sample data within the scope of selected sample data (as shown in Fig. 1) are input into this network. In our present work, for predicting the cubic structure of as cast high entropy alloys, we can first calculate the four physical parameters, the atomic size difference δ, mixing enthalpy ΔH_mix, electronegativity difference Δχ and valence electron concentration VEC of alloys, and then obtain the corresponding structural parameters using the BP-ANN model, and finally further predict the structure of as cast high entropy alloys.

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3

Linear regression analysis between network predicted and experimental values for a training and b test datasets

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4

Comparison of predicted and experimental structure parameters for training and test datasets

Discussion

The effect of four physical parameters (atomic size difference δ, mixing enthalpy ΔH_mix, electronegativity difference Δχ and valence electron concentration VEC) on the phase stability and structure prediction has been investigated above. However, there is lack of mechanism about phase transformation between FCC and BCC structures in as cast high entropy alloys. Taking as cast high entropy alloys Al_xCoCrCuFeNi (x = 0·5–3·0) for example, ⁸ the structures and calculated physical parameters of this alloys system are listed in Table 2. It can be seen that the content of Al in the as cast Al_xCoCrCuFeNi alloys can tune the crystal from sole fcc to mixed fcc and bcc structures, and to fully bcc structure. Because Al atom has a larger atomic radius 1·432 Å (listed in Table 1), the atomic size difference δ of alloys obviously rises with the increase in Al amount, resulting in bigger lattice distortion in alloys. Thus, it is favourable that alloys transform from fcc to bcc structure with a lower atomic stacking effect in order to reduce the lattice distortion energy of solid solution. ²⁴ Moreover, the bigger lattice distortion quantity is, the more content of bcc phase is. From the view of mixing enthalpy ΔH_mix, there is a downward trend for mixing enthalpy ΔH_mix of alloys with the increase in Al amount, and this part of declining mixing enthalpy could be a potential support for phase transformation occurring from fcc to bcc structure. Regarding electronegativity difference Δχ, as low electronegativity element Al atom is added, electronegativity difference of alloys gradually increases; thus, the segregation effect between alloying elements in as cast high entropy alloys would be more remarkable, which makes bcc phase forming in the matrix of fcc phase become possible. The valence electron concentration VEC of solid solution plays an important role on stacking character of structure, ³³ especially for structure with long range ordered character. With the decrease in VEC by adding more content of Al, alloys tend to form a low atomic stacking character bcc structure instead of fcc structure with higher atomic stacking effect. It can be proven from this example that the structures of as cast high entropy alloys are indeed affected by these four physical parameters (atomic size difference δ, mixing enthalpy ΔH_mix, electronegativity difference Δχ and valence electron concentration VEC), and it is reasonable to predict cubic structures of as cast high entropy alloys using these physical parameters, because each of different as cast high entropy alloy systems corresponds to four certain values of physical parameters (δ, ΔH_mix, Δχ and VEC), and a certain value of structural parameter can also be predicted using this BP-ANN model developed in our work. Based on the BP-ANN model evaluated above, for test datasets, >95 of the absolute value of relative error is within 10, which suggests that the predicted value of structural parameter would be quite close to that of real structure. Hence, it is believable that prediction of structure for as cast high entropy alloys can really be implemented.

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Table 2

Physical parameters of as cast high entropy alloys Al_xCoCrCuFeNi (x = 0·5–3·0) (δ atomic size difference, ΔH_mix mixing enthalpy, Δχ electronegativity difference, VEC valence electron concentration)

Alloys	Structure	δ	ΔHmix/eV/atom	Δχ	VEC
Al0·5CoCrCuFeNi	fcc	4·169	0·14	0·109	8·27
Al0·8CoCrCuFeNi	fcc+bcc	4·922	0·04	0·116	8·00
AlCoCrCuFeNi	fcc+bcc	5·281	−0·01	0·119	7·83
Al1·5CoCrCuFeNi	fcc+bcc	5·892	−0·12	0·125	7·46
Al2CoCrCuFeNi	fcc+bcc	6·257	−0·20	0·128	7·14
Al2·5CoCrCuFeNi	fcc+bcc	6·478	−0·25	0·131	6·86
Al3CoCrCuFeNi	bcc	6·610	−0·29	0·132	6·62

Although the stability and structure of cubic phases in as cast high entropy alloys can be investigated via these physical parameters, clear judgment formula cannot be given in the current study. A more in depth research is needed to investigate the internal mechanism about the formation of fcc and bcc structures in as cast high entropy alloys. It is also worth noting that there is a big demand for raw input data in BP-ANN, the more input data are, the higher prediction accuracy of network is. Therefore, it is necessary to set up a large database for collecting structural parameters of as cast high entropy alloys, and the prediction of structure can be turned into reality using the BP-ANN. More works along these directions are under way.

Conclusions

In summary, the stability and structure prediction of cubic phase in as cast high entropy alloys have been investigated by introducing four physical parameters of solid solution in the present work, including the atomic size difference δ, mixing enthalpy ΔH_mix, electronegativity difference Δχ and valence electron concentration VEC, and the following research outcomes have been realised.