Coupling physics in machine learning to investigate the solution behavior of binary Mg alloys

To Chen,Qin Go,Yun Yun,Tingyu Li,Qin Xi,Tingting Liu,Aito Tng,b,Any Wtson,Fusheng Pn,b

a National Engineering Research Center for Magnesium Alloys,College of Materials Science and Engineering,Chongqing University,Chongqing 400000,PR China

b State Key Laboratory of Mechanical Transmissions,College of Materials Science and Engineering,Chongqing University,Chongqing 400000,PR China

c Faculty of Materials and Energy,Southwest University,Chongqing 400715,PR China

d Hampton Thermodynamics,Hampton TW12 1NL,United Kingdom

Abstract The solution behavior of a second element in the primary phase(α(Mg))is important in the design of high-performance alloys.In this work,three sets of features have been collected:a)interaction features of solutes and Mg obtained from first-principles calculation,b)intrinsic physical properties of the pure elements and c)structural features.Based on the maximum solid solubility values,the solution behavior of elements in α(Mg)are classified into four types,e.g.,miscible,soluble,sparingly-soluble and slightly-soluble.The machine learning approach,including random forest and decision tree algorithm methods,is performed and it has been found that four features,e.g.,formation energy,electronegativity,non-bonded atomic radius,and work function,can together determine the classification of the solution behavior of an element in α(Mg).The mathematical correlations,as well as the physical relationships among the selected features have been analyzed.This model can also be applied to other systems following minor modifications of the defined features,if required.

Keywords:Mg alloys;Solid solubility;Machine learning;First-principles calculation;Diagrammatic method.

1.Introduction

Magnesium(Mg)alloys,as the lightest structural metallic materials,are promising with regard to transport,as electronic 3C materials,and in energy applications owing to their high specific strength,high specific stiffness,good electromagnetic shielding and damping properties[1-5].However,the range of applications of Mg alloys is still limited by their poor room temperature plasticity and corrosion resistance.Although some special treatments can improve the performance of Mg alloys[6-9],alloying is still the most common and efficient approach[10-14].

In principle,the alloying method for‘one-principleelement’materials involves alloying for the formation of precipitates or for the dissolution of the alloying addition in the primary phase.It is found that dissolution of a solute into the primary phase can synergistically improve the properties of Mg alloys.For example,the dissolution of rare earth elements(REs)in the primary Mg phase(α(Mg))can weaken the texture and improve the ductility and corrosion resistance of Mg alloys,simultaneously[15-19].The solidsolution of Mn inα(Mg)can also synergistically enhance the strength and ductility of Mg alloys[16,20,21].Mg alloys with Ca dissolved in theα(Mg)-phase have a high oxidation resistance[22]and good biocompatibility[23].There-fore,understanding the solution behavior of alloying elements inα(Mg)is crucial for the design of high-performance Mg alloys.

In the early 1930s,the relationship between the physical properties of two pure elements and their mutual-solubility was studied by Hume-Rothery et al.and qualitative rules were proposed:the Hume-Rothery rules[24-26].It was proposed that differences in the electronegativity,atomic radius,valance electron,and crystal structure of the two pure elements can influence their mutual solid-solubility[24-26].Subsequently,many researchers have worked on developing a higher-precision method to predict solid solubility limits in binary alloys,e.g.,the diagrammatic method(Darken-Gurry method[27])[28-32]and machine learning methods[33-36].Recently,using the diagrammatic method,Zhang et al.[31]postulated a prediction rule for the solubility limit in binary transition metal alloys that was based on the electrochemical factor(electronegativity)and size factor.Zhou et al.[29]classified the solubility of elements inα(Mg)as≥1 at.% and＜1 at.% based on first principles calculations of total energy and volume difference of the Mg15X(X=Solute)supercell.Moreover,machine learning methods allow the prediction of materials properties from complex and multiple series of raw data without any physical or mathematical formulae[37,38].Zhang et al.[33]predicted the solid solubility of binary alloys and analyzed the weights of different Hume-Rothery rules through an artificial neural network(ANN)that considers atom size factor,valence factor,electrochemical factor,and structure parameter.Li et al.[34]predicted the solid solubility in a 231-alloy binary system,including size factor,valence factor and electronegativity factor,using the support vector machine(SVM)method.Zhang et al.[35]proposed a quantitative structure-property relationship model based on a least-squares analysis and deep neural networks including atomic radius,Mendeleev descriptor,fermi descriptor and electronegativity features to predict solid solubility.Later,Zhang et al.[36]suggested that the Hume-Rothery 15% rule(the solid solubility is extremely limited when the difference in the atomic radii of the elements forming the alloy exceeds 15%)can be applicable to both binary silver-based and copper-based alloys through a fuzzy learning method.All of these results show that the diagrammatic method and machine learning methods are powerful tools for discovering the link between solubility limit and the physical properties of elements.However,the interaction features between solutes and solvent,such as formation energy and volume differences in alloys,have been rarely studied.In addition,there has been little research into the solid solution behavior of Mg alloys.

This work aims to identify prediction criteria of the solidsolubility of elements inα(Mg),using a combination of the physical features of solutes and the elemental interaction features of Mg15X by using a machine learning approach.The elemental interaction features of Mg15X are based on firstprinciples calculations.The connections between these features have been analyzed to provide a physical understanding of the connection mechanism.

2.Computational methods

65 pure elements,including metals and semiconductors,in the first six periods of the periodical table that can replace the Mg atom inα(Mg)(and denoted here as X)have been considered in this work.Their maximum solid solubility(MSS)in Mg and their physical characteristics have been calculated by commercial software or collected from the literature,as follows.

2.1.Solubility calculations

The solubility of Li,Na,Al,Si,K,Ca,Sc,Mn,Fe,Ni,Cu,Zn,Ga,Sr,Y,Zr,La,Ce,Pr,Nd,Sm,Gd,Dy,Ho,Er inα(Mg)were calculated by using the Thermo-Calc2019a[39]software with the TCMG5 database.The solubility calculations in this work were based on chemical equilibrium(Lever rule).As shown in Fig.1,the maximum solid solubility of Al and Mn inα(Mg)is 11.6 at.% and 0.96 at.% at 436°C and 651 °C,respectively.The solubility of the elements that were was not included in TCMG5 were collected from the available literature[40-54].

2.2.First-principles calculations

The first-principles calculations were performed using the commercial Vienna Ab initio Simulation Package(VASP)[55,56],which is based on density functional theory(DFT).The projector augmented wave(PAW)[57]method and generalized gradient approximation(GGA)within the Perdew-Burke-Ernzerhof(PBE)[58]exchange-correlation function was employed in this work.A high-precision optimization of the structural parameters and work function calculations was achieved by minimizing the forces(below 0.00001 eV/?A)and by the selection of a plane wave cut-off energy of 400 eV.15×15×15,11×11×1,and 9×9×6 k-mesh grids generated by Monkhorst-Pack scheme were used to sample the Brillouin zone in the calculations of the optimization of the bulk structural parameters and work functions of the pure elements,and also the Mg15X supercell.

The experimentally measured lattice parameters[59]of Mg were:a=b=3.209?A,c=5.211?A.In this work,the calculated lattice parameters of Mg were:a=b=3.191?A,c=5.180?A.The calculated results are in good agreement with the experimental data.The supercell of Mg15X proposed by Zhang et al.[29]was employed in this work to calculate the total energy,formation energy,c/a ratio,and volume difference.The calculated total energies of Mg16,Mg15Al,and Mg15Y are-24.690,-26.812,and-29.732 eV,respectively,whereas the total energy of Mg16,Mg15Al,and Mg15Y calculated by Zhou et al.[29]are-24.399,-26.502,and 29.383 eV,respectively.The formation energy is calculated as

whereEbulkis the total energy of the Mg15X system andEMg,EXare the energies of the Mg atom and X solute,respectively.After the structural optimization of Mg15X was completed,thec/a ratiowas obtained using the(c)lattice parameter of Mg15X divided by(a).

Fig.1.(a)The Mg–Al phase diagram;(b)The Mg-Mn phase diagram.

Fig.2.The process of the machine learning of solid solubility in binary Mg alloys.

To calculate the average work function of the pure elements(Dy,Er,Ho,Lu,Nd,Tb,Tc,Tm,Yb)in their standard structures,clean surfaces were modeled by using the slab model consisting of at least 11 layers separated by a vacuum of 20?A.During the relaxation process,the upper and lower four layers of the model were allowed to fully relax.The average work functionФcan be defined by the difference between the vacuum region energy(Ev)and the Fermi energy(Ef),as:

2.3.Machine learning methods

The Machine learning algorithm used in this work comes from the scikit-learn library(https://scikit-learn.org/stable/)and is written in Python.The process of machine learning is shown in Fig.2.Firstly,Spearman’s correlation[60],which has been employed in the field of materials research on previous occasions[61,62],was chosen for the present study.Secondly,the random forest method(RF)was applied to select the main features of the correlation and reject those considered to be redundant.The prediction accuracy achieved on using different numbers of features was calculated leading to the‘oob_score’.These scores were obtained by training the RF based on an‘out-of-bag’(oob)estimation.If the oob_score obtained when involving fewer features is better,or at least not much worse than the previous oob_score utilizing more features,the redundant features were then excluded from subsequent fitting processes.Thirdly,when there is no further improvement in the oob_score,the final set of main features was selected.The proportional weighting of each main feature in the determination of the classification of solid solubility of X inα(Mg)was also calculated.Finally,the decision tree algorithm(Classification and Regression)was applied to give a visualized model for the classification and prediction of solid solubility inα(Mg).In the process of decision tree regression modeling,the solid solubility is randomly divided into a training set and a test set,with a ratio of 7:3.

3.Data collection

As shown in Section 2,the solubility of 26 elements inα(Mg)was calculated using the software packageThermo-Calc[39]utilizing the TCMG5 database.The solubilities of Ag,As,Au,Ba,Be,Bi,Cd,Co,Cr,Cs,Eu,Ge,Hf,Hg,In,Ir,Lu,Mo,Na,Nb,Os,Pb,Pd,Pr,Rb,Sb,Sc,Tb,Ti,Tm,V and Yb were collected from the available literature[40-54].Thermodynamic data for Mg-X(X=Os,Pm,Pt,Re,Th,Ru,Ta,Tc,W)were not found.

Fig.3.(a)The unit cell of Mg;(b)The supercell of Mg15X;(c)The classification of solid solubility in α(Mg).

There are various classifications of solid solubility recorded in the literature[24,27,29,63].For example,Hume-Rothery[24]classified“l(fā)imited”and“extensive”solid solubility as＜5 at.% and＞5 at.%,respectively.The 5 at.% criterion was also employed by Darken et al.[27]and Alonso et al.[63].Another common classification criterion is 1 at.%,which was adopted by Alonso et al.[63]and more recently by Zhang et al.[29].Considering the solute substitution in theα(Mg)lattice,the repetition of a supercell in space and the distribution of solubility values of solutes inα(Mg),as shown in Fig.3(c),a classification of the binary solubility in Mg alloy,based on the solute atom amount in the supercell size,is proposed in this work.The classification of the solubility of X inα(Mg)is:Miscible(Mg and Cd,solubility～=1);Soluble(solid solubility＞1/16(6.25 at.%,close to 5 at.%,one solute atom in the 2×2×2 Mg supercell));Sparingly-soluble(1/16＞solid solubility＞1/96(1.04 at.%,close to 1 at.%,one solute atom in the 4×4×3 Mg supercell));and Slightlysoluble(solid solubility＜1/96).Theα(Mg)unit cell model,a 2×2×2 supercell with one solute atom(Mg15X)is shown in Fig.3.

Fig.4.The solid solubility limit of solutes in Mg.

Eleven input features from two categories(the intrinsic features of pure elements and the interaction features between Mg and solutes)are taken into consideration.The basic information on these features has been summarized in Tab.I.Tab.I shows the difference in the non-bonded atomic radius(AR),the electronegativity on the Pauling scale(EL),valency(VA),melting point(MP)and crystal structure of the solutes(CS)obtained from online databases(https://www.rsc.org/periodic-table and https://periodictable.com/index.html).The interaction features,including the difference in volume(VD),c/a ratio(c/a),formation energy(FE)and total energy(TE)of Mg15X were calculated using first-principles.Most work functions(WF)of the pure elements have been reviewed by Michaelson and Hebert[64],but the average work functions of pure elements Dy,Er,Ho,Lu,Nd,Tb,Tc,T and Yb(data missing in the literature)were also provided for the present study using first-principles calculations,as shown in Tab.II.The existence of compounds(CO)between Mg and solute elements are flagged as0(no compounds formed by these two elements)or1(compounds formed).The similarity in crystal structure(CS)of the solute andα(Mg)is also defined as0(different)and1(same).All the solubility and input features have been summarized in Tab.III and Tab.IV.

4.Results

Fig.4 shows the solid solubility limit of 65 solutes inα(Mg).Most solutes have limited solid solubility(＜20 at.%)inα(Mg)and only Cd is reported to be infinitely soluble.It is shown that generally,the most soluble elements come from the IIB,IIIB,IIIA and IVA main groups.

Fig.5 shows the connection between maximum solid solubility and the interaction features calculated from firstprinciples.There are some distinct regions in each sub-figure of Fig.5.For example,the values of the formation energy of Mg15X have a distinct convergence range,where X shows high solid solubility inα(Mg),as shown in Fig.5(a).Fig.5(c)shows that the solute elements that have a high solubility inα(Mg)normally have a small effect on the volume change,where the volume difference of Mg15X with that of Mg16 is less than 5%.Fig.5(d)also shows that there are two distinct regions for elements with a large solubility:one region for the non-RE elements where the c/a ratio is close to that of pure Mg(1.623),and another region for the RE elements where there should be other factors affecting the solid solubility values.

Fig.5.The maximum solid solubility vs interaction features of Mg15X from first-principles calculations,(a)formation energy,(b)total energy,(c)volume difference,(d)c/a ratio.

Fig.6(a)shows the Spearman correlation coefficients of the collected features.A high absolute value of the correlation coefficient means a strong positive/negative correlation between the two features.According to the mutual correlation values,some features are strongly correlated,e.g.the electronegativity and the work function,the electronegativity and the volume difference,the melting point and the total energy,the work function and the volume difference.Moreover,some features may not be relevant,e.g.the atomic radius difference and the melting point,the compound formability and the atomic radius difference,the c/a ratio and the compound formability.

The weights of 11 primary features associated with the solid solubility of X inα(Mg)calculated using the random forest model method are presented in Fig.6(b).The weights of these features are in sequence:formation energy＞volume difference＞total energy＞atomic radius difference＞melting point＞work function＞electronegativity＞c/a ratio＞valence＞crystal structure＞compound formability.This suggests that the formation energy may be the most impactful feature.

Fig.6(c)shows the step-results of the random forest modeling process.The oob_scores of predictions varies with the number of features.Overall,the combination of selected features from the primary 11 features shows good predictions of the solid solubility in Mg(oob_score＞0.8).It should be noted that the selection of the features is programmatically determined with the aim to achieve the maximum oob_score value.As can be seen in Fig.6(c),when the number of initial features is reduced to 4 out of 11,the oob_score obtained shows the highest value at 0.86.The remaining four main characteristic features are electronegativity,formation energy,atomic radius difference,and work function.

Fig.6(d)summarizes the weights of the four main features,electronegativity,formation energy,atomic radius difference,and work function.As shown in Fig.5(d),electronegativity,formation energy,and atomic radius difference should be the crucial features in determining the solid solubility class of the alloys,and work function may play a supporting role in the prediction of the solid solubility of Mg alloys.

Secondly,machine learning was performed using the decision tree classification algorithm,giving the classification shown in Fig.6(e).The conditions for the solid solubility range of solutes inα(Mg)can be quantitatively described.In the first column,it is shown that when the formation energy of the solute in Mg15X is higher than 0.006 eV/atom,the solid solubility of X in Mg can be neglected(＜1/96).In the second column,with FE＜0.006 eV/atom,the solid solubility of the elements can be further classified using electronegativity.The soluble and sparingly-soluble elements occur in the range of electronegativity difference of＜53.1%,with the soluble elements giving an electronegativity difference of＜37.8%.In the fourth column,the classification of the elements belonging to the soluble or sparingly-soluble range can be further determined by atomic radius difference and work function values.

Fig.6.(a)The correlation among the 11 primary features;(b)The weights of 11 primary features in RL results(Total weights=1);(c)The prediction performance with the features decreased;(d)The weights of the last 4 features in RL results(Total weights=1);(e)The results of the decision tree classification algorithm model.

Once these four important features had been determined,machine learning using the decision tree regression algorithm was also performed based on these four features,initially over 10,000 times and a highly accurate prediction score R2＞0.995 was obtained.Two parity plots comparing the solubility of elements from training set and testing set with the prediction values are shown in Fig.7.It is seen the decision tree regression algorithm modeling performs well in the quantitative prediction of the solid solubility of binary Mg alloys.The solubility limit of Os,Pt,Re,Rh,Ru,Ta,Tc,and W elements inα(Mg)can then been predicted using the obtained regression method.The solid solubility of Ru inα(Mg)ispredicted to be 0.049 at.% and the solid solubility of other elements(Os,Pt,Re,Rh,Ta,Tc,and W)inα(Mg)are close to zero.

Fig.7.The parity plots for(a)training set,(b)testing set.

Fig.8.The change in feature as function of group number:(a)formation energy,(b)electronegativity,(c)atomic radius,(d)work function.

Fig.8 shows the periodicity of the formation energy of Mg15X,the electronegativity,atomic radius and work function of the pure elements.It can be seen that these four features show strong relationships with periodicity.The formation energy values of Mg15X show maximum positive values with X in group IA and show maximum negative values with X in group VIIIB.The electronegativity of pure elements coming from B groups and groups IIIA-VA are higher than that of Mg.The atomic radii of elements show a general monotonous decrease from group IA to VA,where most elements have a larger atomic radius than that of Mg,except Be.The periodicity of the work functions of pure elements shows a similar behavior to that of electronegativity,with high values coming from the VIIIB group elements.Detailed discussion is given in section 5.

Fig.9 shows the properties of elements from the La series.Since they are in the same main-group and period,their physical properties are similar.Their work functions are around 3.0 eV.The atomic radius decreases with increasing atomic number,whereas the electronegativity of La series elements shows the opposite trend.The formation energy of Mg15X is around 0 eV for La series elements except for that of Eu and Yb.For the solid solubility limit of solutes inα(Mg),the solid solubility of elements shows a monotonous increase with increasing atomic number,except for Eu,Tm and Yb,where the formation energy of Mg15X with these three elements showing anomalous extreme values.

The connections between the main features of the solid solubility of X inα(Mg)can also be studied using diagrammatic methods.Fig.10 shows the connections between formation energy,electronegativity difference,atomic radius difference,and work function with solid solubility of X.It is shown that the soluble and sparingly-soluble elements cluster in zones on the maps showing the relationships between two of the variables,where the red-zone contains the soluble elements and the blue-zone the sparingly-soluble elements.However,the slightly-soluble elements seem to be just randomly distributed in the maps.In more detail,the soluble and sparingly-soluble elements are sited in the zone of FE∈[-0.025 eV/atom,0.006 eV/atom]in Fig.10(a)(electronegativity difference vs formation energy),which is consistent with the results of the decision tree classification algorithm.The linear relationship between the electronegativity difference and work function of the elements is observed in Fig.10(e).

5.Discussion

5.1.The periodicity of solubility and physical properties of the elements

Both the solid solubility limit of solutes in Mg and the four features,e.g.,formation energy,electronegativity,atomic radius and work function of solutes,show periodicity within the periodic table.

Solid solubility is shown in Fig.4,and the soluble elements belong mainly to the IIB,IIIB,IIIA,IVA main-groups.The sparingly-soluble elements lie around the soluble elements and belong mainly to the IB,IIB,IIIB,IIIA,IVA and VA main-groups,whereas the slightly-insoluble transition metals are coming from the IVB,VB,VIB,VIIB,and VIIIB main-groups.

For the formation energy of Mg15X,where the element X lies in the IA,IIA,IIIB,VIIIB,IB,IIB,IIIA,IVA or the VA main-groups is below 0.006 eV/atom,as shown in Fig.10(a).The soluble and sparingly-soluble elements also belong to these main-groups indicating that the formation energy is a crucial feature for the solubility behavior of solutes inα(Mg).Many elements from the VIIIB,IB,IIB,IIIA and VA main-groups result in a formation energy of Mg15X less than-0.025 eV.The values of electronegativity and work functions of some elements in IA,IIA,IIIB,IVB,IIB and IIIB main-groups are close to those of Mg.In addition,most elements have a larger atomic radius than that of Mg.

5.2.The correlations between collected features

The collected 11 primary physical features can be classified further as interaction factors(formation energy,compoundor-none,the total energy of Mg15X),electrochemical factors(electronegativity,valence,work functions,and melting point),and structural factors(atomic radius,crystal structure of solutes,volume difference of Mg15X,and c/a ratio).These features are also correlated with each other.For example,Miedema et al.[65]suggested that the heat of formation depends mainly on the electronegativity,the atomic size and the electronic density at the boundary of solutes and the matrix.In our work,as shown in Fig.6(a),the correlation coefficients were calculated,and the related mechanisms can be illustrated as follows.

Firstly,the correlation coefficient between formation energy and compound formability is calculated as-0.64,which indicates that a Mg-X system having a large formation energy of Mg15X favors no compound formation.According to the data collected,it is observed that amongst the 18 elements giving a formation energy for Mg15X larger than 0.011 eV,only 4 elements(Ba,Sr,Co,Cu)form compounds with Mg,as shown in Fig.11.Conversely,of the elements that give a formation energy of Mg15X less than 0.011 eV,only Li does not form compounds with Mg.

Secondly,it is observed that the electronegativity,work function,and melting point of the elements depend on their valence electronic structure,where the related correlation coefficients are above 0.49.The ability of an atom in a molecule to attract electrons,named electronegativity,has been defined by Pauling[66].Miedema[67]proposed a quantitative relationship between the charge transferred per atom and the difference in electronegativity,ΔZ=1.2(1-Ca)Δχ,whereΔZ,Ca,Δχare charge transferred per atom,electron concentration of metal and electronegativity difference,respectively.Moreover,it has been reported that there is a linear relationship between the work function,the minimum energy required for an electron moving from the inside of the solid to the surface of the object,and the electronegativity of the element[65],which is also verified in this work,as shown in Fig.10(e).Furthermore,the melting points can also be pre-dicted using information on the valence electronic structure[68].

Fig.10.The distinction of solid solubility of elements in Mg alloy 2-Degree maps with features of:(a)electronegativity difference and formation energy;(b)formation energy and atomic radius difference;(c)electronegativity difference and atomic radius difference;(d)formation energy and work function;(e)electronegativity difference and work function;(f)atomic radius difference and work function.

Fig.11.The distribution of the compound formability(compound-or-none)feature of elements in the electronegativity vs formation energy map.

Thirdly,the correlations between atomic radius difference with electronegativity difference,work function,volume difference and c/a ratio are also presented in Fig.6(a).The negative correlations between atomic radius and electronegativity can be illustrated as the stronger chemical bond between adjacent atoms corresponding to a smaller spacing between elements[15].The atomic radius difference,volume difference and c/a ratio have been classified into structural factors in this work.A large atomic radius difference between the solute atom and Mg atom tends to cause a large lattice distortion[69].In addition,the charge transfer between elements has a contribution to the volume of supercells.For example,when electrons are transferred from Mg to Hg(an element with a high electronegativity),the Mg atom/ion shrinks and the Hg atom/ion expands[67].Liu et al.[15]reported that there is a secondary polynomial relationship between covalent atomic radius and the work function of a solid solute atom.They[15]also investigated the relationship between c/a ratio and electron work function for solutes in Mg alloys.It was shown that when solute species come from the D-block of the periodic table,the c/a ratio values of the resulting Mg alloys were lower than that of pure crystalline Mg because of an increased density of electrons present between the Delectron solute and the Mg solvent atoms[15].In conclusion,the features coming from the three classifications(interaction factors,electrochemical factors,and structural factors)are closely related and have a complex effect in determining the solid solubility of elements in Mg.

Fig.12.The properties of Ru,Os,Pt,Re,Rh,Ru,Ta,Tc,and W elements.

Fig.13.The relation between the work function and their maximum solid solubility for solutes with a maximum solid solubility＞0.1 at.%.

Fourthly,the crystal structure of the elements in their primary phase status may not correlate significantly with other features(correlation coefficient＜0.6).According to our results,the similarity between the crystal structure of element X in its primary phase and the crystal structure ofα(Mg)has no strong connection with the solid solubility classification of X inα(Mg),as shown in Fig.6(b).Moreover,Zhang et al.[33]suggested that the crystal structure doesn’t play a very important role in the prediction of solid solubility.

Overall,the features collected initially were reduced to four main features that were obtained from the random forest method modeling.Interaction factors,electrochemical factorsand structural factors considered together can determine the solid solubility classification of an element inα(Mg).

5.3.The classification and regression results

The effects of 11 features,including the intrinsic physical properties of solutes and the interaction features calculated from Mg15X,on the solid solubility classification of X inα(Mg)have been analyzed using the random forest methodand the decision tree algorithm method in this work.The formation energy,electronegativity difference,atomic radius difference and work function together determine the solid solubility classification of Mg binary alloys.

These four main features have been reported in other works.Based on results from ANN,Zhang et al.[33]proposed that the prediction of solubility can involve different combinations of atomic radius,valence,structure and electronegativity,the best combination of features being atomic size,valence and electronegativity,where the electronegativity and atomic radius show higher impacts than other features.Li et al.[34]investigated influence of these factors on the solid solubility in binary alloys by using the SVM method.Their results[34]show that three of the features,i.e.,size factor,valence factor and electronegativity factor,are the main features affecting solid solubility in binary alloys.Their conclusions are similar to ours.It is worth mentioning that features such as formation energy and work function,were not considered by Zhang et al.[33].In the present work,it is suggested that four physical features,i.e.,electronegativity,formation energy,atomic radius,and work function determine the classification of solid solubility of solutes inα(Mg).

The Darken-Gurry mappings of the binary Mg-alloy systems are shown in Fig.10.Our analysis shows only two features that cannot completely define the solid solubility region of the element,even though the distribution of elements does have aggregation zones in the two-dimension map.Fig.10(d)shows two elements(Mn and Pr)that have been included in the light blue range while their solid solubility values are less than 1/96.The maximum solid solubility of Mn inα(Mg)is 0.96 at.% and that of Pr is 0.064 at.%.These exceptional cases can be included through consideration of other features.Mn has a relatively positive formation energy that results in a small solubility,as shown in Fig.10(b).Although the formation energy and work function of Pr tend to support Pr having a sparing solubility inα(Mg)while the atomic radius difference between Pr and Mg is too large(38.7%)to result in a large mutual solid solubility.In this work,quantitative criteria for the classification of solubility limit inα(Mg)is obtained by a multi-dimensional features set using the machine learning method.

Furthermore,the solid solubility of solutes that have not been found in the literature can then be predicted using the obtained decision tree regression model.Elements Ru,Os,Pt,Re,Rh,Ru,Ta,Tc,and W are slightly-soluble elements.As shown in Fig.12,these elements have a larger electronegativity and atomic radius than Mg.In addition,their formation energy cannot meet the requirement of a soluble solution class,apart from W.However,W has a large atomic radius and work function.

5.4.The effects of physical properties of x elements on the solid solubility of x in α(Mg)

Formation energy is a key parameter in the CALPHAD(Computer Coupling of Phase Diagrams and Thermochemistry)method to describe the interaction behavior of elements[70,71].A structure with a negative enthalpy of formation is generally more stable[67,72,73].Our results show that when the formation energy of the solute in Mg15X is larger than 0.006 eV/atom or less than-0.025 eV/atom,the solid solubility of X inα(Mg)is small(＜1/96).Guo and Liu[74]suggested that for high entropy alloys a solid solution phase can be formed if-22 kJ/mol≤ΔHmix(enthalpy of mixing)≤7 kJ/mol.Juan et al.[75]also investigated the formation rules of laser clad high entropy alloy coatings,and reported that most of the solid solution phases are formed in the region of-14.5 kJ/mol≤ΔHmix≤6.5 kJ/mol.Another finding is that the stability of compounds formed between Mg and X metals depends on the formation energy.Most elements hardly form compounds when their formation energy is larger than 0.011 eV/atom and vice versa,as shown in Fig.11.Hence,a large absolute value(no matter negative or positive)of formation energy(formation enthalpy)is not conducive to the formation of solid solutions.

According to the decision tree obtained,it is shown that,excluding the interaction feature,the electronegativity difference is the most important feature of the decision factor of the solution behavior of metals.The importance of electronegativity on the mutual solubility of metals has also been reported in the literature.It has been reported that a small difference in the electronegativity between the solute and solvent element tends to result in the formation of a large solid solution in a binary alloy[26,33].And it is also believed that a large difference in the component electronegativity favors the formation of stable compounds[76].Singh et al.[77]also suggested that electronegativity can provide an important clue regarding the formation of compounds in high-entropy alloys.Our results show that the formation energy is also an important feature relating to the compound formation behavior,as shown in Fig.11.

The atomic radius difference between elements is another critical criterion in the design of high-entropy alloys since a substituted atom with a large size difference with that of the primary element can result in a large lattice distortion[78,79].Generally,a large misfit is an unfavorable condition for metals to form a solid solution[80].Zhou et al.[29]reported that the condition for the element to have a solid solubility larger than 1 at.% inα(Mg)is that the volume difference of Mg15X should be in the range of[-4.698%,4.415%].Hume-Rothery proposed the difference in the atomic diameters between two elements should be less than 14%(or 15%)to favor the formation of a soluble binary system.Our work shows that the difference in atomic radius of X with that of Mg should be less than 35.8% for sparingly-soluble and soluble elements inα(Mg).

Work function plays a supporting role in the process of classification of the solid solution behavior of X inα(Mg).The solid solubility can be further classified based on the work function values with the pre-conditions as FE＜0.006 eV/atom andΔEL in the region of[37.8% 53.1%].A solute with a small atomic radius and large work function tends to lead to a large solubility if the formation energy and its electronegativity meet the requirements,as shown in Fig.10(c,e,f).The effect of electronegativity and work function on solid solubility in binary Mg alloys can be attributed to the fact that electronegativity and work function affect the electron transfer behavior and ion radius in the crystal structure[65].

5.5.Connection between physical properties and solid solubility of La series elements

La series elements,included in the RE elements of maingroup IIIB,are usually considered to have similar physical properties to one another[81].The effect of the four features on the solid solubility of La series elements inα(Mg)can be demonstrated.In general,the solid solubility of the elements in the La series increase with their atomic number,except for Eu,Tm,and Yb,where increasing solubility is seen with a decrease in atomic radius.Secondly,Eu and Yb have higher formation energy than that of their neighboring elements,which results in small solid solubility values,as shown in Fig.9.The formation energy of the Mg15X supercell with elements from the La series can be analyzed from the viewpoint of electrochemistry.The 4-f electron orbital of Eu is half-filled whereas the electron orbital of Yb is fully filled.Thirdly,Tm has the largest work function,which results in a smaller solid solubility inα(Mg)than that of its adjacent element Er.In conclusion,formation energy,electronegativity,atomic radius,and work function together determine the solid solubility of the solute inα(Mg).

Fig.9.The properties of La series elements.

5.6.The connection between the characteristics of solutes and the properties of Mg alloys

It has been reported that there are many connections between the physical characteristics of solutes and the properties of Mg alloys[12,22,82].For example,the work functions of solutes have been proved to be linked with the Young’s modulus,yield strength,plasticity and hardness in alloys[15,83,84].Hua and Li[83]reported that there is a relationship between the Young’s modulus(E)and the work function(Φ)of pure metals:E≈0.022Φ6.Liu et al.[15]reported that there is a secondary polynomial relationship between the covalent atomic radius of solid solutes and work function.Moreover,the strengthening potency of solutes inα(Mg)can be predicated based on work function values.A large work function difference between solute and Mg suggests a large solid solution strengthening potential.Furthermore,a solute inα(Mg)can enhance both strength and ductility of Mg alloys when the work function of the solute is smaller than that of Mg[15].Lu et al.proposed that the hardness of alloys increases with increasing difference in work function[84],since the solid solution hardening is attributed mainly to elastic modulus and atomic size misfit.From another aspect,the addition of a solute with an intrinsic low work function toα(Mg)may decrease the work function of the Mg alloys.For example,Y additions decrease the work function of the Mg(0001)surface layer,whereas Al additions increase the work function of Mg(0001)surface layer[85].Work function is also closely related to the corrosion potential of alloys[86,87].In conclusion,it is suggested that the solute which combines a small work function with a large solubility limit is recommended in the design of single-phase Mg alloys resulting in a material with a good balance of properties,as shown in Fig.13(Tables 1–4).

Table 1The basic information of features.

Table 2Work function of different low-index faces of pure elements(Mg,Dy,Er,Ho,Lu,Nd,Tb,Tc,Ce,Yb).

Table 3The solubility of solutes and the interaction features between Mg and solutes(X=solutes).

Table 3(continued)

Table 4The physical properties of solutes.

Table 4(continued)

6.Conclusions

In this work,the solubility limits inα(Mg)of more than 50 elements have been collected and presented.11 features have been considered in the machine learning and diagrammatic methods.The interaction features between solutes andα(Mg),such as formation energy,have been systematically analyzed using first-principles calculation.The major conclusions are as follows:

a)The formation energy of the Mg15X supercell has a strong connection with the solid solubility of X inα(Mg)and the formation of Mg-X compounds.

b)The maximum solid solubility of 56 elements inα(Mg)can be classified into four categories and the decision tree algorithm model obtained can describe the classification criteria.

c)Four physical features,electronegativity,formation energy,atomic radius,and work function are shown to be the main features of the classification of solid solution behavior of solutes inα(Mg).

The feature selection procedure and the derivation of the prediction model discussed in this work can be applied to other binary alloy systems.

Data availability

The datasets generated during and/or analyzed during the current study are available on request.

Declaration of Competing Interest

On behalf of all authors,the corresponding author states that there is no conflict of interest.

Acknowledgment

The authors are grateful for the financial support from the National Natural Science Foundation of China(51971044 and U1910213),Natural Science Foundation of Chongqing(cstc2019yszx-jcyjX0004),Fundamental Research Funds for the Central Universities(2020CDJDPT001).