Phase stability,electronic,elastic and thermodynamic properties of Al-RE intermetallics in Mg-Al-RE alloy:A frst principles study

H.L.Chen*,L.Lin,P.L.Mao,Z.Liu

School of Materials Science and Engineering,Shenyang University of Technology,Shenyang 110870,China

Phase stability,electronic,elastic and thermodynamic properties of Al-RE intermetallics in Mg-Al-RE alloy:A frst principles study

H.L.Chen*,L.Lin,P.L.Mao,Z.Liu

School of Materials Science and Engineering,Shenyang University of Technology,Shenyang 110870,China

Electronic structure and elastic properties of Al2Y,Al3Y,Al2Gd and Al3Gd phases were investigated by means of frst-principles calculations from CASTEP program based on density functional theory(DFT).The ground state energy and elastic constants of each phase were calculated, the formation enthalpy(ΔH),bulk modulus(B),shear modulus(G),Young’s modulus(E),Poisson’s ratio(ν)and anisotropic coeffcient(A)were derived.The formation enthalpy shows that Al2RE is more stable than Al3RE,and Al-Y intermetallics have stronger phase stability than Al-Gd intermetallics.The calculated mechanical properties indicate that all these four intermetallics are strong and hard brittle phases,it may lead to the similar performance when deforming due to their similar elastic constants.The total and partial electron density of states(DOS),Mulliken population and metallicity were calculated to analyze the electron structure and bonding characteristics of the phases.Finally,phonon calculation was conducted,and the thermodynamic properties were obtained and further discussed.

First-principles calculation;Phase stability;Electronic structure;Elastic properties;Thermodynamic properties

1.Introduction

Because of their low density,good stiffness and the highest strength-to-weight-ratio of all structural metals[1,2],and especially the signifcant advantage of recycling,magnesium alloys are becoming more and more attractive for lightweight structural applications[3,4].Mg-Al alloys are the well-known series of precipitation hardening alloys which signifcantly improve the mechanical properties at room temperature.However,the poorthermalstability ofitsmain precipitation phase (β-Mg17Al12)limits its use in high temperature[5,6].

Rare earth elements are considered to be prime added elements to improve the performance of magnesium alloys at high temperature due to the solution hardening and precipitation strengthening effect,thus improving comprehensive mechanical properties of magnesium alloys[7–9].Y and Gd are commonly used due to their anomalous solid solution hardening effect in Mg[10].As added elements in Mg-Al alloy,Y and Gd have a priority to combine with Al to form Al-RE metal compound,whichAl2Y orAl2Gd phaseisthemain strengthening phase,resulting in a signifcant strengthening effect[11,12].Al also can form Al3RE compound in rapid solidifcation condition,which has a low density,high oxidation resistance,high melting point and good high temperature strength,hence,it can improve the mechanical properties of Mg-Al alloys[13].

The frst principles calculations are employed to systematically investigate the phase stability,electronic and elastic properties of binary Al2Y,Al3Y,Al2Gd and Al3Gd phases in this work.The thermodynamic properties were also discussed.The results of this work are compared with the available experimental and theoretical values,which can provide a theoretical guidance to design a new kind of magnesium alloy and beneft to understand the microscopic strengthen mechanism.

2.Computational details

Cambridge Sequential Total Energy Package(CASTEP) [14],a frst-principles plane wave pseudo-potential method based on density function theory(DFT),is used for geometryoptimization and energetic calculations.Generalized gradient approximation(GGA)of Perdew–Burke–Ernzerhof(PBE) [15,16]is used to describe the exchange-correlation energy function.The ultrasoft pseudo-potential[17]is used to describe the interaction between ion core and valence electron for all elements.

The cut-off energy of plane wave is set to 400 eV.The Monkhost–Pack scheme with a k-point separation for each phase is followed by system.In consideration of the stability deviation associated with atomic coordinates,all atomic positions within the cell of each phase were relaxed according to total energy and ionic force using the BFGS scheme.Calculation of total energy and related properties were then performed after geometry optimization with the SCF tolerance of 5×10?8eV.The maximum displacement is within 5×10?4?; and the maximum stress is within 0.02 GPa.

3.Results and discussion

3.1.Crystal structure and formation enthalpy

The crystal structures of Al2RE and Al3RE are shown in Fig.1.The lattice constants and structure parameters are listed in Tables 1 and 2.The calculated values are in good agreement with the experimental values.This good agreement indicates that the present calculation method is highly accurate and reliable.

The phase stability can be characterized by formation enthalpy.Formation enthalpy of Al-RE intermetallics is calculated by the following equation:

where ΔH is the formation enthalpy;Etotis the total energy of each Al-RE intermetallics;andare per atom energy of pure elements A and B,respectively;NAand NBare the number of A and B atoms in unit cell,respectively.The calculated formation enthalpy values of Al2Y,Al3Y,Al2Gd and Al3Gd are listed in Table 3.

Fig.1.Crystal structure of Al2RE(a)and Al3RE(b).

Table 1Structure parameters of Al2RE and Al3RE.

Table 2Equilibrium crystal parameters(a),unit cell volume(V0)and density(ρ)of Al2Y,Al3Y,Al2Gd and Al3Gd.

Table 3Formation enthalpy(ΔH)of Al2Y,Al3Y,Al2Gd and Al3Gd.

As can be seen from Table 3,the formation enthalpies of Al2Y,Al3Y,Al2Gd and Al3Gd are negative values,showing that these phases can stably exist.The lower the formation enthalpy is,the more stable the phase structure is.From Table 3,it can be concluded thatAl2Y has the stronger forming ability thanAl3Y, Al2Gd has the stronger forming ability than Al3Gd,and Al-Y phases are more stable than Al-Gd phases.

3.2.Electronic properties

The electronic properties were calculated to understand the mechanism about structure stability of these fourAl-RE phases and further reveal their bonding characteristics.The calculated total and partial density of states(DOS)of Al2Y,Al3Y,Al2Gd andAl3Gd are shown in Fig.2.The bonding peaks of these four phases,which mainly range from?10 to 15 eV,originate from the contribution of valence electron ofAl s,Al p,Y d orbits for Al-Y phases and Al s,Al p,Gd d,Gd f for Al-Gd phases.For Al2Y,Al p state andY p,d state mainly contribute to the bonds; forAl3Y,Al p state andY d state mainly contribute to the bonds; for Al2Gd,Al p state and Gd d,f state mainly contribute to the bonds;for Al3Gd,Al p state and Gd d,f state mainly contribute to the bonds.Gd f state has a signifcant effect on DOS curve. As can be seen in the total DOS comparison of these four phases,in the vicinity of the Fermi level,the occupied electron energy range of Al3Y is larger than Al2Y,Al3Gd is larger than Al2Gd;therefore,Al2RE is more stable than Al3RE,and Al-Y phases are more stable thanAl-Gd phases because of Gd with f valence electron.This result is in good agreement with the conclusion obtained from formation enthalpy.Pseudogap directly refects the covalent bonding strength of a phase,the wider the pseudogap is,the stronger the covalent bond is.There is no pseudogap in the DOS of Al-Y phases,but exists in the DOS ofAl-Gd phases.The performance of DOS curve at Fermilevel indicates that the covalent bond strength of Al-Gd phases is stronger than Al-Y phases.In addition,the width of pseudogap for Al2Gd is a little larger than Al3Gd.

Fig.2.Total and partial DOS of Al2Y(a),Al3Y(b),Al2Gd(c)and Al3Gd(d).

In order to fnd the ionic bond feature of the phases,the Mulliken populations of Al2Y,Al3Y,Al2Gd and Al3Gd were calculated.The calculated bond lengths of the four phases are positive,which indicates that the chemical bonds in the intermetallics can be able to build.The calculated electron occupation numbers ofAl2Y,Al3Y,Al2Gd andAl3Gd are shown in Table 4.For Al2Y,the valence electron confguration of Al atom is 3s23p1,while through bonding with Y,the electron confguration of Al atom changed to 3s1.23p2.16.The electron number localized inAl is 3.37,Al obtained 0.37e per atom;the valence electron confguration ofY atom is 4d15s2,while through bonding with Al,the electron confguration ofY atom changed to 4d2.075s0.27,the electron number localized in Y atom is 2.27, Y lost 0.74e per atom.The transfer number of electron is 1.48 in a cell.Similarly,for Al3Y,the electron number localized in Al is 2.89,Al lost 0.11e per atom;the electron number localized in Y atom is 3.33,Y obtained 0.33e per atom.The transfer number of electron is 0.66 in a cell.For Al2Gd,while throughbonding with Gd,the electron confguration ofAl atom changed to 3s0.893p1.64.The electron number localized in Al is 2.53,Al lost 0.47e per atom;the valence electron confguration of Gd atom is 4f75d16s2,while through bonding with Al,the electron confguration of Gd atom changed to 5p5.484f8.475d4.326s1.61,the electron number localized in Gd atom is 17.49,Gd obtained 1.88e per atom.The transfer number of electron is 2.35 in a cell.Similarly,for Al3Gd,The electron number localized in Al is 3.10,Al obtained 0.10e per atom;the electron number localized in Gd atom is 17.71,Gd lost 0.29e per atom.The transfer number of electron is 0.39 in a cell.Hence,judging from the electron’s transfer number we can draw a conclusion that the order of the iconic bond strength of the four intermetallic compounds from weak to strong is:Al3Gd,Al3Y,Al2Y and Al2Gd.

Table 4Mulliken charge of Al2Y,Al3Y,Al2Gd and Al3Gd.

Table 5Density of states at Fermi level Df,total number of valence electrons N,cell volume V and metallicity parameter fmof Al2Y,Al3Y,Al2Gd and Al3Gd.

The metallicities of Al2Y,Al3Y,Al2Gd and Al3Gd are calculated by the following equation[23]:

where Dfis the DOS value at Fermi level;T is the temperature; kBis the Boltzmann constant;nmand neare the densities of the thermal excited electrons and valence electron in the cell, respectively.neis calculated by ne=N/V,where N is the total number of valence electrons;V is the cell volume.Df,N,V and calculated fmare listed in Table 5,from which we can draw a conclusion that the order of the metallic bond strength of the four intermetallic compounds from weak to strong is:Al3Y Al3Gd,Al2Y andAl2Gd.Al-Gd phases are signifcantly stronger than Al-Y phases because of the existence of 4f electron in Gd atom.

3.3.Elastic properties

Elastic constants are often used to characterize the elasticity of materials;it is the most basic physical constants of materials.For cubic crystals,there are three independent elastic constants: C11,C12and C44.To stay stable,their elastic constants must satisfy the following stability conditions[23]:C11?C12＞0, C11＞0,C44＞0,C11+2C12＞0.The calculated elastic constants ofAl2Y,Al3Y,Al2Gd andAl3Gd are listed in Table 6.The elastic constants of Al2Y,Al3Y,Al2Gd and Al3Gd satisfy the stability conditions;therefore,they can stably exist.The calculations are in good agreement with the experimental value,indicating that the results are accurate and parameter settings are reliable.

Table 6Elastic constants(C11,C12and C44)of Al2Y,Al3Y,Al2Gd and Al3Gd.

Bulk modulus(B),Shear modulus(G),Young’s modulus (E),Poisson’s ratio(ν)and anisotropic coeffcient(A)ofAl-RE intermetallics can be deduced by the following equation[25]:

The calculated results are listed in Table 7.The bulk modulus B usually characterizes the resistance to deformation under an applied stress,the shear modulus G is a measure of deformation resistant capacity under shear stress[26].The larger the value,the stronger the capacity to resist deformation; Young’s modulus E denotes the stress and strain ratio,andPoisson’s ratio ν denotes the shear capacity of the material,the largerYoung’s modulus is,the tough the material is;the larger the Poisson ratio is,the better the plasticity is[24],it usually range from?1 to 0.5.As can be seen from Table 7,the B,G and E of Al2Y,Al3Y,Al2Gd and Al3Gd similar to each other,and all of them are large values,so they are all strong and hard phases. The small value of ν indicates the brittle nature of the four phases.With the addition of Gd andY elements,the precipitation of fne and dispersed Al-RE phases may remarkably improve the strength of Mg-Al-RE alloys.

Table 7The bulk modulus(B),shear modulus(G),Young’s modulus(E),elastic constants(Cij),G/B,Poisson’s ratio(ν)and anisotropic coeffcient(A)ofAl2Y,Al3Y,Al2Gd and Al3Gd.

Pugh[26]found that shear modulus and bulk modulus ratio (G/B)can refect the characteristics of the material during deformation,where shear modulus and bulk modulus of the material refect the resistance capacity to plastic deformation and resistance capacity to brittle fracture,respectively.The critical value is generally designated at 0.57.If G/B＞0.57,the material shows brittle fracture,while if G/B＜0.57,the material shows ductile fracture.The smaller the value of C11?C12is,the better the plasticity of material is.C12?C44characterizes the plasticity of materials[27,28],from which positive value indicates that the material is ductile;negative indicates that the material is brittle.As shown in Table 7,the G/B of Al2Y,Al3Y, Al2Gd and Al3Gd phases are larger than 0.5;C11?C12of these four phases are all large positive values;C12?C44of these four phases are all negative values;evidence shows that the four phases are brittle.This result is consistent with the former conclusions.Anisotropic coeffcients are close to 1;therefore, these four phases are all isotropic.The mechanical properties of Al2Y,Al3Y,Al2Gd and Al3Gd phases are similar,which shows that these four phases may have similar performance in Mg-Al-RE alloys.

3.4.Thermodynamic properties

Phonon calculations can be carried out to evaluate the enthalpy,entropy,free energy and heat capacity as a function of temperature for approximate quasi-harmonic crystal.The enthalpy E(T),entropy S(T),free energy F(T)and heat capacity Cv(T)are directly given by CASTEP based on the following equations[29]:

where Etotis the total energy,Ezpis the zero point energy,?is Planck’s constant,kBis Boltzmann’s constant,F(ω)is the phonon density of states and T is temperature.Moreover,the heat capacity curve can be divided into two parts.When at low temperature(T?ΘD),the change of heat capacity follows the law[23,30]:

Fig.3.The enthalpy E(T),entropy S(T),free energy F(T)ofAl2Y,Al3Y,Al2Gd and Al3Gd.

When at high temperature(T?ΘD),the change of heat capacity obeys the Dulong–Petit limit:

Fig.3 shows the calculated thermodynamic properties of Al2Y,Al3Y,Al2Gd and Al3Gd,from which we can obtain that enthalpy andT*entropy monotonously increase as the temperature increases,while free energy monotonously decreases with increasing temperature.There is no intersection in the curve, which indicates that the calculated values at 0 K are reliable and can be used to approximately instead the values at room temperature.The lower the free energy is,the stable the phase is. Therefore,Al2RE are more stable than Al3RE,Al-Y phases arestable than Al-Gd phases as mentioned above.As shown in Fig.4,Cvapproaches 0 as the temperature approaches 0 K, proportional to T3as the T rises at low temperature and converges to Dulong–Petit limit at high temperature,which obeys the change law.The heat capacity ofAl2RE is larger thanAl3RE; therefore,Al2RE phase is more stable thanAl3RE phase in high temperature,thus remarkably improving the elevated temperature strength of magnesium alloys.

Fig.4.The heat capacity Cv(T)of Al2Y,Al3Y,Al2Gd and Al3Gd.

4.Conclusions

(1)From the perspective of formation enthalpy,Al2Y has the stronger forming ability thanAl3Y,Al2Gd has the stronger forming ability than Al3Gd,and Al-Y phases are more stable than Al-Gd phases.

(2)The result through calculated DOS is in good agreement with the formation enthalpy.The bonding characteristics ofAl2Y,Al3Y,Al2Gd andAl3Gd are all covalent,ionic and metallic bonds.The ionic bond strength of Al-Gd phases is weaker than Al-Y phases;the covalent and metallic bond strengths of Al-Gd phases are stronger than Al-Y phases.

(3)The elastic properties of Al2Y,Al3Y,Al2Gd and Al3Gd show that these four phases are brittle,hard and isotropic. Their similar mechanical properties may lead to the similar performance in Mg-Al-RE alloys.

(4)The enthalpy,entropy,free energy of Al2Y,Al3Y,Al2Gd and Al3Gd are monotonously changed as temperature changes.From the perspective of free energy,Al2RE are more stable than Al3RE,Al-Y phases are stable than Al-Gd phases as mentioned above.The calculated Cvobeys the change law,which illustrates the computation is credible in this work.

Acknowledgements

This work is supported by the Key Technologies Research and Development Program of Liaoning Province (2013201018).

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Received 4 June 2015;revised 9 August 2015;accepted 25 August 2015 Available online 1 October 2015

*Corresponding author.School of Materials Science and Engineering, Shenyang University of Technology,Shenyang 110870,China.Tel.:+86 13591630166;fax:024-25496768.

E-mail address:chenhonglei250@163.com(H.L.Chen).

http://dx.doi.org/10.1016/j.jma.2015.08.003

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