Structural and elastic properties of Mg3CuH0.6ternary hydride by ab initio study

Said Boucetta

Département de Physique,Laboratoire d’Elaboration de Nouveaux Matériaux et leurs Caractérisations(ENMC),Université SETIF-1,19000 Sétif,Algeria

1.Introduction

Magnesium(Mg)-based alloys have been attractive to researchers because of their remarkable physical,chemical and mechanical properties,such as high thermal conductivity,high specific strength,good ductility,high specific toughness,rigidity,good corrosion resistance,high recycling potential and also their importance as engineering materials.Magnesium alloys are used in a variety of industry,particularly for aerospace,defense and automotive applications[1,2].

In the lastyears,the researchers discovered that magnesium-based alloys possess unique properties in hydrogen capacity storage materials because of light weight and low cost[3].Intermetallic compound based magnesium,when reacting with hydrogen to form a hydride;it can have an influence on various physical properties of this compound[4].At high pressure,the atomic radius of metallic elements changes in the range of a few percent to more than 10%,depending on the nature of elements and the chemical potential of hydrogen can be enhanced[5].For that purpose,Mg-based alloys and their hydrides have been extensively investigated.

In the Mg-Cu-H system,anovelternary hydride(Mg3CuH0.6)has been synthesized recently,by a new method based on the direct reaction starting from mixtures of MgH2and Cu under high-pressure and moderate-temperature conditions[3].

To our knowledge,up to now,no experimental or other theoretical studies have been reported on this compound in literature.Thus,in the present work we have carried out a systematic theoretical investigation based on first-principles calculations,on the structural,thermal stability and elastic properties of Mg3CuH0.6compound,in order to provide a sounder basis and data for further experimental and theoretical studies.Then,the ductility and Debye temperature are further discussed.

The rest of this paper is organized as follows:the computational method is described in Section 2,the numerical results and discussions are given in Section 3 and finally,a conclusion is presented in Section 4.

2.Computational method

The first-principles quantum mechanics calculations are performed with the plane-wave pseudo-potential(PW-PP)total energy method implemented with the CASTEP(Cambridge Serial Total Energy Package)simulation program[6].This is based on the density functional theory(DFT)[7,8]which is,in principle,an exact theory of the ground state.We have used the local density approximation(LDA)developed by Ceperley and Adler and parameterized by Ceperley and Alder[9]and Perdew and Zunger[10],as well as the generalized gradient approximation(GGA)with the new functional of Perdew-Burke-Ernzerhof(PBE),known as PBE[11]for electronic exchange-correlation potential energy.The Coulomb potential energy caused by electron-ion interaction is described using the Vanderbilt-type ultrasoft scheme[12],in which the orbitals of Mg(2p63s2),Cu(3d104s1)and 1s for H,are treated as valence electrons.The cut-off energy for the plane-wave expansion was chosen at 500eV and the Brillouin zone sampling was carried out using the 4×4×4 set of Monkhorst-Pack mesh[13].

The structural parameter(a)of Mg3CuH0.6is determined using the Broyden-Fletcher-Goldfarb-Shenno(BFGS)minimization technique[14].This method usually provides the fast way of finding the lowest energy structure.

In the structural optimization process,the energy change,maximum force,maximum stress and maximum displacement are set as 5.0×10-6eV/atom,0.01ev/˚A,0.02GPa,and 5?10-4˚A,respectively.

The elastic constants were determined by applying a given homogeneous strain(deformation)with a finite value and calculating the resulting stress according to Hook’s law[15].The total energy is converged to 1.0×10-6eV/atom in the selfconsistent calculation from first-principles calculation.

3.Results and discussion

3.1.Structure and thermal stability

The new ternary hydride prepared in the Mg-Cu-H system by an original synthesis procedure has the crystallographic formula of Mg3CuH0.6.In this setting,the three nonequivalent Mg cations(Mg1,Mg2 and Mg3)are placed at 4a(0,0,0)for Mg1,4c(1/4,1/4,1/4)sites for Mg2 and 4d(3/4,3/4,3/4)for Mg3,the Cu cation is located at 4b(1/2,1/2,1/2)and the hydrogen,H,is situated at 24f(x,0,0)sites withx=0.301[3].Fig.1 shows a view of the cubic crystal structure of Mg3CuH0.6indexed in the F-43m(N°216)space group,with a lattice parametera=6.2876˚A of unit-cell according to the experimental data[3].

Fig.1.The cubic crystal structure of Mg3CuH0.6intermetallic hydride compound.

Table 1 Calculated and experimental values of the equilibrium lattice constant a(˚A)and enthalpy of formation ΔH(eV)in cubic structure of Mg3CuH0.6 compound.

The crystal structure was optimized at first.The obtained results of calculated lattice parameteraof Mg3CuH0.6intermetallic hydride compound using the(PW-PP)method within the LDA and GGA approximations are listed in Table 1,together with the available experimental data for comparison.From the present results in Table 1,it is clear that the calculated lattice constant agrees well with the experimental data ones in both approximations(LDA and GGA).To the best of our knowledge,there is no other calculation on this compound and there is no Mg3Cu compound in the Mg-Cu system to see the influence of hydrogen on the crystal cell after hydrogenation.

To estimate the thermal stability of above intermetallic hydride compound,the formation enthalpyΔH at zero temperature is calculated as follows[16]:

Where theEtotrepresents the total energy of unit cell.EMgandECuare electronic total energy per atom for pure Mg and Cu in ground state.EH2is the electronic total energy of single H2molecule.NMg,NCuandNHrefer to the number of composition elements in unit cell,respectively.The obtained result of calculation is listed in Table 1.The negative enthalpy of formation indicates that the ground state of this compound is thermodynamically stable.

Table 2 Calculated values of the elastic constants Cij(GPa),bulk modulus B(GPa),anisotropy factor A,shear modulus G(GPa),Young’s modulus E(GPa),Poisson’s ratio σ and B/G in cubic structure of Mg3CuH0.6compound.

3.2.Elastic properties

The elastic constants of a material describe its response to an applied stress or,conversely,the stress required to maintain a given deformation.The full elastic stiffness constants are evaluated to use the stress-strain method,in which the relationship between stress and strain is expressed by Hook’s law,σij=Cijklεkl,whereσijis the stress tensor,εklis the strain tensor andCijklis the elastic constants tensor[17].

The elastic constants can provide information on the stability,stiffness,brittleness,ductility,and anisotropy of a material.The elastic constants give also important information concerning the nature of the forces operating in solids.

The cubic crystal has three independent elastic constants,C11,C12andC44.The first two constants describe the crystal response to tension,whileC44describes the response to shear strain.

The Voigt-Reuss-Hill approximation gives the effective values of the bulk moduls.For the cubic system,the Voigt bounds of the bulk modulusBVand The Reuss boundsBRof the bulk modulus are equal(BR=BV)

The bulk modulusBbased on the Voigt-Reuss-Hill approximation is related to the elastic constants by:

In Table 2,the calculated elastic constants and the bulk modulus of Mg3CuH0.6at zero pressure are presented.For a cubic crystal,the obtained elastic constants meet the requirements of mechanical stability criteria:C11＞0,C44＞0,C11-C12＞0,C11+2C12＞0 and

C11＞B＞C12.From Table 2,one can see that the elastic constants of Mg3CuH0.6compound satisfy all of these conditions,suggesting that the structure of Mg3CuH0.6is mechanically stable.To the best of our knowledge,there are no experimental and other theoretical data in literature for the elastic constants(Cij)of Mg3CuH0.6for comparison,so we consider the present results as prediction study which still awaits an experimental confirmation.

The Zener anisotropy factorAis a measure of the degree of anisotropy in solid[18].It takes the value of 1 for an isotropic material.It provides a measure of the degree of elastic anisotropy,when theAvalues are smaller or greater than unity.The Zener anisotropy factorAOf Mg3CuH0.6compound is calculated by the following equation:

As shown in Table 2 that the calculated Zener anisotropy factor is larger than 1,which indicates that this compound is weakly anisotropic.The elastic constantsCijare estimated from first-principles calculations for monocrystal Mg3CuH0.6.However,the prepared materials are in general polycrystalline,and therefore it is important to evaluate the corresponding moduli for the polycrystalline phase.For this purpose we have applied the Voigt-Reuss-Hill approximation[19-21].For the cubic system,the Reuss and Voigt bounds on the shear modulus are given by:

Finally,the VRH mean value is obtained by:

We also calculated Young’s modulusEand Poisson’s ratioσwhich are frequently measured for polycrystalline materials when investigating their hardness.These quantities are related to the bulk modulus and the shear modulus by the following equations[22]:

The shear modulusG,Young’s modulusEand Poisson’s ratioσfor Mg3CuH0.6compound,calculated from the elastic constants are also listed in Table 2.The results demonstrate that the Mg3CuH0.6intermetallic hydride compound is stiffer in terms of bulk,Young’s and shear modulus.The ratioB/Ghas been proposed by Pugh[23],it is a simple relationship related to brittle or ductile behaviour of materials.A highB/Gratio is associated with ductility,whereas a low value corresponds to the brittleness.The critical value separating ductile and brittle material is 1.75.The calculated results are listed in Table 2.The results indicate that Mg3CuH0.6can be classified as ductile material at zero pressure.Moreover,the ductility or brittleness behaviour can be characterized usingPoisson’s ratio(σ)according to the criteriaσ＜1/3for brittle material andσ＞1/3for ductile material[24].

Table 3 The calculated density ρ,the longitudinal,transverse and average sound velocities vl,vtand vmcalculated from elastic moduli,and the Debye temperatures θDcalculated from the average sound velocity for Mg3CuH0.6compound.

Our calculated values of Poisson’s ratio confirm the ductile nature of Mg3CuH0.6compound.

3.3.Debye temperature

Debye temperature,which is connected directly with thermal vibration of atoms,it is an important fundamental parameter closely related to many physical properties such as specific heat and melting temperature.Debye temperature is determined using the calculated Young’s modulusE,bulk modulusBand shear modulusGfrom first-principles elastic constants calculations.At low temperature,the Debye temperature calculated from elastic constants is the same as that determined from specific heat measurements.Debye temperatureθDmay be estimated from the average sound velocityvmby the following equation[25]:

Wherehis Plank’s constant,kBBoltzmann’s constant,nis the atoms number per molecule andVais the atomic volume.The average sound velocityvmis given by[26]:

Wherevlandvtare the longitudinal and transverse sound velocity of an isotropic aggregate obtained using the shear modulusGand the bulk modulusBfrom Navier’s equation[18]:

The calculated Debye temperature and sound velocities as well as the density of Mg3CuH0.6intermetallic hydride compound in the LDA approximation are given in Table 3.Unfortunately,there are no experimental and other theoretical data to compare with our results,so we consider the present results as a prediction study.

4.Conclusion

In summary,the structural,thermal stability,elastic,and thermodynamic properties of Mg3CuH0.6intermetallic hydride compound have been investigated by means of the DFT within LDA functional.Our result for the optimized lattice parameter(a)is in good accordance with the available experimental data.The calculated enthalpy of formation indicates that this compound is stable.The elastic constantsCij,and related polycrystalline mechanical parameters such as bulk modulusB,shear modulusG,Young’s modulusEand Poisson coefficientσare calculated using Voigt-Reuss-Hill approximations.The Mg3CuH0.6hydride compound is mechanically stable according to the elastic stability criteria,while no experimental results of elastic moduli for comparison.The polycrystalline intermetallic Mg3CuH0.6turns out to be a low stiff material according to the calculated elastic constants.The calculated Zener factor indicates that Mg3CuH0.6compound is elastically anisotropic.The calculated values of the ratioB/Gand Poisson coefficientσshow a ductile manner for the Mg3CuH0.6compound.Finally,from the knowledge of the elastic constants and the average sound velocities,the Debye temperature has been predicted successfully.This is the first quantitative theoretical prediction of these properties which is valuable for further study as a reference.

Acknowledgement

This work is supported by the(ENMC)laboratory,University Ferhat Abbas Setif-1,Algeria.

[1]C.X.Shi,J.Magnes.Alloys 1(2013)1.

[2]A.A.Luo,J.Magnes.Alloys 1(2013)2-22.

[3]B.Torres,M.J.Martinez-Lope,J.A.Alonso,D.Serafini,M.T.Fernandez-Diaz,R.Martinez-Coronado,Int.J.Hydrogen Energy 38(2013)15264.

[4]S.M.Filipek,V.Paul-Boncour,R.S.Liu,I.Jacob,T.Tsutaoka,A.Budziak,A.Morawski,H.Sugiura,P.Zachariasz,K.Dybko,R.Diduszko,Appl.Surf.Sci.388(2016)723.

[5]H.Takamura,H.Kakuta,Y.Goto,H.Watanabe,A.Kamegawa,M.Okada,J.Alloys Compd.404-406(2005)372.

[6]M.D.Segall,P.J.D.Lindan,M.J.Probert,C.J.pickard,P.J.Hasnip,S.J.Clark,M.C.Payne,J.Phys.Condens.Matter.14(2002)2717.

[7]P.Hohenberg,W.Kohn,Phys.Rev.B 136(1964)864.

[8]W.Kohn,L.J.Sham,Phys.Rev.A 140(1965)113.

[9]D.M.Ceperley,B.J.Alder,Phys.Rev.Lett.45(1980)566.

[10]J.P.Perdew,A.Zunger,Phys.Rev.B 23(1981)5048.

[11]J.P.Perdew,K.Burke,M.Ernzerhof,Phys.Rev.Lett.77(1996)3865.[12]D.Vanderbilt,Phys.Rev.B 41(1990)7892.

[13]H.J.Monkhorst,J.D.Pack,Phys.Rev.B 13(1976)5188.

[14]T.H.Fischer,J.Almlof,J.Phys.Chem.96(1992)9768.

[15]J.F.Nye,Propriétés Physiques des Matériaux,(Dunod,Paris),1961.

[16]D.-H.Wu,H.-C.Wang,L.-T.Wei,R.-K.Pan,B.-Y.Tang,J.Magnes.Alloys 2(2014)165.

[17]N.W.Ashcroft,N.D.Mermin,Solid State Physics,(Holt Saunders,Philadelphia),1976

[18]C.Zener,Elasticity and Anelasticity of Metals,University of Chicago Press,Chicago,1948.

[19]W.Voigt,Lehrbuch de Kristallphysik,Terubner,Leipzig,1928.

[20]A.Reuss,Z.Angew,Math.Mech.9(1929)49.

[21]R.Hill,Proc.Phys.Soc.65(1952)349 London,Sect.A.

[22]E.Schreiber,O.L.Anderson,N.Soga,Elastic Constants and Their Measurements,McGraw-Hill,New York,1973.

[23]S.F.Pugh,Philos.Mag.45(1954)823.

[24]I.N.Frantsevich,F.F.Voronov,S.A.Bokuta,Elastic Constants and Elastic Moduli of Metals and Insulators Handbook,Naukova Dumka,Kiev,1983,pp.60-180.

[25]P.Wachter,M.Filzmoser,J.Rebisant,J.Phys.B 293(2001)199.

[26]O.L.Anderson,J.Phys.Chem.Solids 24(1963)909.