Rui Wu,Ya-Ping Wang,Lin Shao,Wei Wang,Bi-Yu Tang,*

1 School of Chemistry and Chemical Engineering,Guangxi University,Nanning 530004,China

2 Institute of Biological Manufacturing Technology Co.Ltd,Guangxi Institute of Industrial Technology,Nanning 530022,China

Keywords:MgCaSi alloys Density functional theory Thermodynamic properties Structural stabilities Quasi-harmonic Debye-Grüneisen model

ABSTRACT The thermodynamic properties of MgCaSi and its mother phase Ca2Si are comparatively investigated from ab initio calculations and quasi-harmonic Debye-Grüneisen model.At 0 K,MgCaSi is more thermodynamically stable.Under high temperature,the advantage of higher thermodynamically stability of MgCaSi is reduced,originating from the less negative entropy contribution because the thermodynamic entropy of MgCaSi increases more slowly with temperature and the entropy values are slightly smaller.With increasing temperature,the anti-softening ability for MgCaSi is slightly smaller due to the slightly faster decrease trend of bulk modulus than that of Ca2Si,although the bulk modulus of MgCaSi is higher in the whole temperature range considered.The thermal expansion behaviors of both MgCaSi and Ca2Si exhibit similar increase trend,although thermal expansion coefficient of MgCaSi is slightly lower and the increases is slightly slower at lower temperature.The isochoric heat capacity CV and isobaric heat capacity CP of MgCaSi and Ca2Si rise nonlinearly with temperature,and both CV are close to the Dulong-Petit limit at high temperature due to the negligibly small electronic contribution.The Debye temperature of both phases decrease with increasing temperature,and the downtrend for MgCaSi is slightly faster.However,MgCaSi possess slightly higher Debye temperature,implying the stronger chemical bonds and higher thermal conductivity than the mother phase Ca2Si.The Grüneisen parameter of MgCaSi and Ca2Si increase slightly with temperature,the values of MgCaSi are slightly larger.The investigation of electronic structures shows that with substitution of partial Ca by Mg in Ca2Si,the stronger Mg—Si,Mg—Ca and Si—Si covalent bonds are formed,and plays a very significant role for the structural stability and mechanical properties.

1.Introduction

Magnesium alloys have been widely studied as potential aerospace and hydrogen storage materials due to their low cost and low density [1-6].In industry,some light metals such as silicon,aluminum,calcium and zinc are often added to magnesium alloys to improve their properties.Especially,calcium and silicon are important additives in magnesium alloys,which can significantly improve the strength and ductility of magnesium alloys [7-9].The addition of Ca can improve the strength and creep resistance of Mg alloys [10],and the addition of Si can improve the mechanical properties of Mg alloys [11],hence Mg-Ca-Si alloy systems have been extensively studied and developed [12-15].Except as a superior structural material,the Mg-Ca-Si alloys can be used as implant biomaterials in biomedical engineering due to their high mechanical strength and minimum risk of toxicity [15,16].Moreover,Mg-Ca-Si alloy can also serve as preferred thermoelectric materials due to their environmentally friendly and competitive price [16].Thus ternary Mg-Ca-Si alloys [17] have aroused significant research interest.

In Mg-Ca-Si alloys,several ternary compounds have been reported [9,11],including Ca7Mg7.5±δSi14,Ca2Mg3Si,Ca2MgSi3and MgCaSi.Especially ternary MgCaSi compound is a very typical phase and plays an important role in applications of Mg-Ca-Si alloys.Experimentally,the MgCaSi compound is observed as a precipitation phase and can increase the alloys tensile strength remarkably [18-21].Moreover,the MgCaSi compound exhibits excellent thermoelectric performance[22]and favorable hydrogen storage property [23].However,synthesis of single samples of MgCaSi phase is difficult and relevant research is very rare,because this ternary phase is often produced as by-products.Recently,a variety of synthetic techniques have been explored,for example,Whalenet al.[9] employs the metal flux synthesis method to fabricate the single crystals MgCaSi,Niwaet al.[22] synthesize MgCaSi phase by mechanical alloying (MA) and/or solid liquid reaction(SLR),Hosonoet al.[21]obtain MgCaSi at the Ca2Si/Mg2Si interface by heat treatment of Mg2Si/Si substrate in Ca vapor.Up to now,single-crystal MgCaSi have been produced by above experiments,and important physical and chemical properties such as hydrogen-storage behavior,resistivity [9],elastic properties [24]have been studied.To our best knowledge,the theoretical investigation on thermodynamic properties of MgCaSi is not reported,hence study of thermodynamic properties of this ternary phase should be important and necessary.

Density functional theory (DFT) is a powerful tool in studying the properties of condensed matter,which has become one of the most commonly used methods in the field of computational materials science.DFT calculations can predict various structures and properties of materials,and the results are consistent with the experimental measurements.Thus,in this paper,the fundamental thermodynamic properties including high temperature softening,thermal expansion,heat capacity,Debye temperature of MgCaSi and its mother phase Ca2Si are studied based on density functional theory calculation and quasi-harmonic Debye-Grüneisen model.The results are expected to be beneficial for future researches.

2.Computational Method

The present density functional theory (DFT) calculations were carried out applying ViennaAb initioSimulation Package (VASP)[25].The projector augmented wave (PAW) pseudo-potentials[26] was used with a 400 eV plane-wave cut energy in all calculations.And the valence electron configurations of MgCaSi are respectively 2p63s2for Mg,3s23p2for Si and 3p64s2for Ca.The electronic exchange-correlation energy is described by the generalized gradient approximation (GGA) [27] and parameterized by Perdew-Burke-Ernzerhof form [28].The Monkhorst-Pack scheme[29] of 8 × 12 × 8 grids were used to sample the Brillouin zone during structural optimization,and 10×14×10 grids for calculation of the electronic structure.The structural optimizations were performed by fully relaxation and the convergence criteria of self-consistent iterations are set as 10-6eV/atom,The linear tetrahedron method with Bl?chl corrections[31]was adopted for calculations of total electron energy and electronic density of states(DOS) [30].

During optimization of lattice structure,the energy dataversusvolume points obtained from first-principles calculations,were fitted by the third order Birch-Murnaghan equation of state (BM3-EOS) [31]:

whereE0,V0,B0correspond to the equilibrium energy,equilibrium volume,bulk modulus at 0 GPa,respectively.is the pressure derivative of bulk modulus atV0,and χ=(V/V0).To study the thermal properties,the Gibbs energy is obtained by Legendre transforming the volume-and temperaturedependent free energyF(V,T),which is computed within the quasi-harmonic approximation according to [32]:

whereE(V) denotes the static energy atT=0 K,Felstands for the electronic free energy and can be calculated by integrating the electronic density of states (EDOS) with Fermi-Dirac distribution [33],Fvibis vibrational free energies.

It should be noted that the vibrational free energy by phonon calculations is a dunting task,so it is treated by the quasiharmonic Debye-Grüneisen model,which considerably simplifies the phonon calculation while can be able to capture the essential features of lattice vibrations.The vibrational free energy is calculated as follows [31]:

wherekdenotes Boltzmann’s constant,nstand for the number of atoms per formula units,D y() is the Debye integral defined as:

where θDis the volume dependent Debye temperature.

In quasi-harmonic Debye model,the θDis described through the Grüneisen parameter γ [34,35].The Grüneisen parameter γ is expressed as follows:

Then the fundamental thermodynamic properties of materials,including bulk modulusB,volumetric thermal expansion coefficient ɑ,isochoric heat capacityCV,isobaric heat capacityCP,and entropySand others,are further computed as derivatives of Helmholtz free energyF(T,V),as implemented in GIBBS2 program [36].

3.Results and Discussion

3.1.Structure optimization

From the chemical stoichiometry of MgCaSi,this ternary compound appears to be related to the Zintl phase of Mg2Si and Ca2Si.In fact,the structure of this compound can be viewed as full substitution of Mg for Ca on one of the 4c sites of Ca2Si structure,which results in the formation of a three-dimensional Mg/Si network channels in which the Ca element resides.Meanwhile,both Mg and Si sites are bonded with highly distorted tetrahedral coordination [9].Finally ternary MgCaSi crystallize in orthogonal structure(space group No.62 Pnma),in which the Mg,Ca and Si atoms locate in three independent 4c sites in Wyckoff positions.The structural optimization of ternary MgCaSi and its mother phase Ca2Si are performed by full relaxation,and the optimized crystal structure are visually shown in Fig.1,and the optimized atomic internal positions of MgCaSi are tabulated in following Table 1.It can be seen that the calculated results are close to the experimental values,indicating that the calculating parameters which we selected are reasonable,and our calculated results are highly reliable.

Fig.1. Crystal structure of (a) Ca2Si and (b) MgCaSi.

Table 1 Atomic positions in the unit cell of MgCaSi

To investigate the ground state structure and properties of MgCaSi and Ca2Si,after the convergence test by full relaxation,the energiesE(Hartree,1 Hartree=27.2114 eV)as a function of volumesV(Bohr3,1 Bohr=5.29×10-11m) for two compounds are obtained,as displayed in Fig.2.Then the third-order Birch-Murnaghan equation of state(EOS)[31]are employed to fit the calculated energy versus volume data points.The optimized lattice parametera0,equilibrium volumeV0for two compounds at zero temperature are derived and listed in Table 2.

From the Table 2,it can be seen that the calculated results are in good aggreement with experimental data and other theoretical calculated value.Compared to the mother phase,the lattice constant of ternary MgCaSi is slightly reduced due to the atomic radius of Mg is smaller than that of Ca.This calculated result shows that the present calculations are accuracy and reliable.

Fig.2. The calculated E-V curve for (a) Ca2Si and (b) MgCaSi.

3.2.Phase stability

To further study the stability of studied phases,the formation energy is calculated according to following expression [41]:

whereEtotrefers to the total energydenoteds the energy of single atoms of theith pure elements solid states.Nirepresent the numbers ofith atoms in a single cell.The calculated formation enthalpy for MgCaSi and Ca2Si are -45.498 kJ·mol-1and -43.392 kJ·mol-1,respectively.The negative values indicate the thermodynamic stability of MgCaSi and Ca2Si,being in agreement with the available experimental and theoretical values [11].MgCaSi with a lower formation enthalpy than Ca2Si indicate the stronger thermodynamically stability,because the lower formation enthalpy means stronger stability.

3.3.Thermodynamic properties

The fundamental thermodynamic properties including entropyS,bulk modulusB,volumetric thermal expansion coefficient α,isochoric heat capacityCVand isobaric heat capacityCPfor MgCaSi and its mother phase Ca2Si are studied in tempreture range 0-1000 K based on their thermal stability range [42],and the main results are showed as follow.

3.3.1.The themodynamic entropy

Thermodynamic entropy is a fundamental property of materials,which describes the degree of chaos in system.The themodynamic entropy of a material can be computed asS=-?F/?T()P,and the obtained results for MgCaSi and mother phase Ca2Si are pictured in Fig.3(a).It can be seen that the systematic entropies for two materials increase rapidly at low temperatures,then slowlyat high temperatures.The value of Ca2Si is higher than MgCaSi,and the variation behaviors of the former is stronger whenT＜400 K,while the tendency became almost the same whenT＞400 K.It implies that two compounds are more stable under higher temperature because negative entropy contribution to Gibbs free energy.Moreover,the stability advantage of MgCaSi is less under high temperature.

Table 2 The calculated lattice parameters a (?),bulk modulus B0 (GPa) and its pressure derivative B0′ of Ca2Si and MgCaSi

Fig.3. Calculated entropy (a) S and (b) Svib with temperature for the Ca2Si and MgCaSi.

For nonmagnetic order compound materials,systematic entropy includes contributions from the vibrational entropySviband electronic entropySel.Since the density of states (DOS) at the Fermi level is the dominant factor in determiningSel,the calculated electronic entropy contributionSelbased on DOSs is negligible quantity.In this way,the total entropySvariation with temperature mainly originates from contributions of vibrationalSvib,namely,dS/dT=dSvib/dT,The obtained results of vibrational entropySvibfor MgCaSi and Ca2Si are pictured in Fig.3(b).It is clear that the tendency ofSvibis identical to the total entropyS,due to the negligible effect of electronic contribution.So the vibrational part of entropy might be main contribution to total entropy at high temperatures,and also the main contribution to the phase stability.

3.3.2.The bulk modulus sofenting

The temperature dependence of bulk modulusBcan be generally used to describe the high temperature softening effect of materials,so is an important thermodynamic quantity,which can be calculated as,whereB(T)stands for the isothermal bulk modulus,V0(T) is the equilibrium volume at zero pressure,F(V(P,T),T) is just the Helmholtz free energy.The derived results for MgCaSi and Ca2Si are plotted in Fig.4.One can see clearly that the bulk moduli for both MgCaSi and Ca2Si decrease with temperature increasing,exhibiting the general trends of softening behavior.At the lower temperatures,the bulk moduli decreases slowly,then declines rapidly at higher temperature.Especially,with temperature increasing the high temperature softening behaviors of MgCaSi seems to be stronger than Ca2Si.Moreover,the values of bulk modulusBfor MgCaSi are always higher than that of Ca2Si in the whole temperature range considered.The result also indicate that the strength of MgCaSi is always larger than Ca2Si,although bulk modulus of ternary MgCaSi drops slightly faster with temperature increasing.

3.3.3.The thermal expansion

The thermal expansion behavior of materials is common and important phenomena in engineering applications,which is often described by temperature dependences of volumes.For both MgCaSi and Ca2Si,the calculated temperature dependences of volumesVare plotted in Fig.5(a).It can be seen that the volumes for MgCaSi and Ca2Si exhibit the normal increasing trends with temperature.When temperatureTis lower,the volumes increase gently with increasing temperature,then increase more rapidly when temperatures is higher.Although volume expansion behaviors for MgCaSi and Ca2Si shows the essentially similar increasing trends,the volume increase of Ca2Si seems to be stronger,indicating that the volume of Ca2Si is more sensitive to high temperature.

The volume thermal expansion behavior is generally described in term s of the thermal expansion coefficient α,which can be computed as equation of[43],the derived thermal expansion coefficient α for MgCaSi and Ca2Si are plotted in Fig.5(b).Clearly,when the temperature is below～300 K,the thermal expansion coefficients α increase faster with increasing temperature,then this propensity of increase is relatively gentle at higher temperature.The variation behaviors of expansion coefficient α with temperature for MgCaSi and Ca2Si are essential analogical.Compared to Ca2Si,the expansion coefficient of MgCaSi is slightly smaller when temperature below 400 K,then become very close to one of Ca2Si when temperature above 400 K.

3.3.4.The heat capacity

Fig.4. Calculated of the bulk moduli B with temperature for the MgCaSi and Ca2Si.

Fig.5. (a) The volume V and (b) the thermal expansion coefficient α as a function of temperature dependence for MgCaSi and Ca2Si.

The heat capacity is also one of fundamental thermodynamic properties,and can reflect heat storage capacity of materials.The isochoric heat capacityCVfor both compounds is estimated with the expression ofCV=T?S/?T()V.The isobaric heat capacityCPare estimated with the expression ofCP=T?S/?T()P.The both isochoric heat capacityCVand isobaric heat capacityCPas a function of temperature are shown schematically in Fig.6 (a) and (b)respectively.It can be observed that asT＜400 K,theCVandCPvalues of MgCaSi and Ca2Si increase rapidly with temperature and then more gently increase when temperatures above 400 K.It is noted that comparing to Ca2Si,theCVandCPfor MgCaSi are smaller whenT＜600 K,then becomes very close to each other when the temperatures above 600 K.

It should be noted thatCVis proportional toT3at sufficiently low temperatures,naturally theCVvalues are much more sensitive to lower temperature,which is mainly originated from the exponential increase of the number of excited phonon modes from the longwave lattice vibration[44].For much higher temperature,the values ofCVapproaches approximately to a constant,and essentially obeys the well-known Dulong-Petit limit (CV=3nR≈300 J·mol-1·K-1,wherendenotes the number of atoms per unit cell)[33].

To isobaric heat capacityCP,it does not approach to a constant asCVdoes at very high temperature.This is because the relation between the isobaric heat capacityCPand isochoric heat capacityCVcan be expressed as:CP-CV=(3α)3BTV.Thus when the temperature up to 1000 K,the thermal expansion coefficient α is very low,so the values of (3α)3BTVbetween theCPandCVfor the studied materials are very small.

3.3.5.The Debye temperature and Grüneisen parameter

In present Debye-Grüneisen model,the Debye temperature θDis inherently related to volumeVthrough Grüneisen parameter γ,can be computed as expression in Refs.[34]and[35].The variation trends of θDfor MgCaSi and Ca2Si with temperatures are obtained and displayed in Fig.7(a).The results show that the θDdecreases slowly at low temperature,then the decrease trend become slightly faster with temperature increasing,although the effect of temperatureTon Debye temperatures θDis not very significant.Compared to Ca2Si,the decline tendency Debye temperatures θDfor MgCaSi is slightly faster at high temperatures,although the Debye temperatures θDof MgCaSi is larger in temperature range studied.Thus,the chemical bonding,hardness and thermal conductivity of MgCaSi are stronger than Ca2Si since a high Debye temperature implies the high thermal conductivity [45],stronger chemical bonds and larger hardness [46].

The variation trends of Grüneisen parameter γ for MgCaSi and Ca2Si at different temperatures are obtained and displayed in Fig.7(b).From the calculated result,the γ of two compounds demonstrates very similar trends that slowly increases with temperature.The γ values of MgCaSi are slightly higher than that of the Ca2Si in considered temperature range,and the variation range is also slightly larger than Ca2Si.Nevertheless,the present calculation results show that the influence of temperature on the Grüneisen parameter is not significant.

3.4.Electronic structure

Fig.6. Temperature dependence of the (a) isochoric heat capacity CV and (b) isobaric heat capacity CP for MgCaSi and Ca2Si.

In order to study the inherent mechanism for structural stability and thermodynamic properties from the chemical bonding,the electronic band structure and densities of states (DOSs) for both MgCaSi and the mother phase Ca2Si are calculated and plotted in Fig.8 for convenient comparison,in which the Fermi level is set as zero as marked by the vertical lines.It can be observed from Fig.8(a),there is obvious band gap between valence band conduction band,and the energy gap is 0.34 eV,indicating the semiconducting nature of Ca2Si.When the half of Ca atoms in Ca2Si is replaced by Mg,Fig.8(b) shows that the energy gap is zero and a small band go through the Fermi Level,indicating that MgCaSi is metallic material.More clearly,the DOSs of Ca2Si in Fig.8(c) also demonstrates the narrow band gap,and further indicating the semiconducting nature of Ca2Si.The calculated band gap is 0.34 eV,being very close to the result of 0.36 eV calculated from the local density approximation (LDA) method [47].In valence band region,especially near Fermi level,the contribution to total DOS mainly comes from the hybridization of Si 3p states and Ca 3p,3d and 4 s states.The results show that obvious covalent bonds are formed between Si—Ca atoms.

Fig.7. The temperature dependence of (a) Debye temperature θD and (b) Grüneisen parameter γ for MgCaSi and Ca2Si.

Fig.8. The band structures of(a) Ca2Si and (b) MgCaSi.The total and partial density of the states of (c) Ca2Si and (d)MgCaSi.In which the vertical dotted lines indicate the Fermi levels (state is the number of electronic states,dimensionless).

As shown in Fig.8(d),when the half of Ca atoms in Ca2Si is replaced by Mg,the band gap vanishes and there is a pronounced pseudo-gap at the Fermi level(Ef).The DOS value at the Fermi level is very close to zero,indicating that the MgCaSi exhibits semimetallic properties.Moreover,the Fermi level located at the bottom of the pseudo-gap implies the formation of covalent bonds between atoms,indicating that the MgCaSi phase is highly stable[48].In valence band region near the Fermi level,Ca,Mg,and Si contribute to the states almost equally,the hybridization of Mg 2p states with the Si 3p and Ca 3d states,as well as Mg 3 s with the Si 3p and Ca 3p,3d,4 s states are very stronger,implying stronger Mg—Si and Mg—Ca covalent bonding formed in the MgCaSi phase,so the stability of MgCaSi is higher.

To better uncover the bonding characteristics of the studied alloys,the charge density distributions are further investigated.Together with the atomic structures on (010) plane for Ca2Si and MgCaSi as shown in Fig.9(a) and (b),the charge density distributions contour plots on (010) plane for Ca2Si and MgCaSi are displayed in Fig.9(c) and (d),in which the charge density differences are plotted from 0 to 0.05 e·?-3with 0.005 e·?-3intervals.Obviously,the larger electron overlapping between Si and Ca atoms in Ca2Si is observed,implying the formation of stronger Ca—Si covalent bond.This covalent bond exhibits somewhat ionicity since the electronegativity of Si is larger than Ca.Noted that Si—Si and Ca—Ca bonds are relatively weak due to smaller electron accumulation between two neighboring M atoms.

With half of Ca atoms in Ca2Si is substituted by Mg in Ca2Si,as shown in Fig.9(d),there is larger charge overlap between Si atoms,suggesting that the stronger Si—Si covalent bond are formed,The Ca—Si bond is similar to one in Ca2Si.The intermediate election overlapping between Si—Mg indicates that the Si—Mg covalent bond is also formed,and this covalent bonding is similar to Ca—Si bond,exhibiting obvious ionicity.The above charge density features demonstrate that the stronger Si—Si,Mg—Si and Mg—Ca bonds are formed in ternary MgCaSi compound,and the results are in agreement with the conclusions obtained from the formation enthalpy and DOS analysis.Therefore,it further indicates that the stability of MgCaSi is higher.

4.Conclusions

Fundamental thermodynamic properties for MgCaSi and Ca2Si are comparatively studied in this work,using the density functional theory (DFT) in conjunction with quasi-harmonic Debye-Grüneisen model.At 0 K,the ternary compound MgCaSi is more thermodynamically stable due to a more negative formation enthalpy,the lattice constants is slightly reduced since the atomic radius of Mg is smaller than that of Ca.Thermodynamic entropy of MgCaSi and Ca2Si increases obviously with temperature,originating mainly from vibrational entropy contributions.Therefore,MgCaSi has slightly lower values,and the advantage of higher stability of MgCaSi is less under high temperatures,due to less negative contribution to Gibbs free energy.The bulk moduli of MgCaSi and Ca2Si decrease with increasing temperature,while high temperature anti-softening ability of MgCaSi is weaker because the decrease trends is slightly faster than Ca2Si,although the values of MgCaSi are higher in temperature range considered.The thermal expansion behavior of MgCaSi is slightly weaker in spite of similar trend.TheCVandCPof MgCaSi and Ca2Si exhibit the similar trend,and theCVfollows the Dulong-Petit limit at high temperatures because electronic contribution is negligibly small.Although the downtrend of Debye temperature of MgCaSi is slightly faster with temperature,MgCaSi possess slightly higher Debye temperature,implying the stronger chemical bonding and higher thermal conductivity than Ca2Si.The Grüneisen parameters of MgCaSi and Ca2-Si also increase slightly with temperature,and the values of MgCaSi are slightly larger.The electronic structures have been further investigated.When half of Ca in Ca2Si substituted by Mg,the stronger Mg—Si,Mg—Ca and Si—Si covalent bond are formed,which is responsible for the excellent stability and elastic properties of ternary compound MgCaSi.

Fig.9. The (010) plane section in crystal structures for (a) Ca2Siand (b) MgCaSi.The charge densitiy contour plots on the (010) plane for (c) Ca2Siand (d) MgCaSi.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

Authors gratefully acknowledge the support from Significant Project of Guangxi Scientific Foundation (2018GXNSFDA281010)and National Natural Science Foundation of China (51461002).