Elastic and thermodynamic properties of divalent transition metal carbides

Abstract:We have investigated the elastic and thermodynamic properties of transition metal carbides (TiC,ZrC,HfC,VC,NbC,and TaC)using the three body force potential model and density functional theory based ab initio approach.The com-puted second-order elastic constants (C 11,C 12,C 44)are in close agreement with the available experimental as well as theo-retical results.The other related elastic parameters (B 0,B 0′,C S ,C L ,G ,n ,E ,A ,b ,x ,l ,Cauchy pressure (C 12–C 44),B 0/G ,and E /r ratio)have also been computed successfully.The calculated Debye temperatures are in reasonably good agreement with the available data.PACS Nos:60:00B,79:00,62.20.de,62.20.dqRésumé:Utilisant l ’approche ab initio de fonctionnelle de densitéet un potentiel modélisant la force àtrois corps,nous étudions les propriétés élastiques et thermodynamiques de carbures de métaux de transition,TiC,ZrC,HfC,VC,NbC et TaC.Les valeurs calculées des constantes élastiques du second ordre,C 11,C 12et C 44,sont en bon accord avec les valeurs expérimentales disponibles et avec d ’autres résultats théoriques.Nous avons également calculéavec succès les autres pa-ramètres élastiques reliés,B 0,B 0′,C S ,C L ,G ,v ,E ,A ,b ,x ,l ,la pression de Cauchy (C 12–C 44),B 0/G et le rapport E /r .Les températures de Debye calculées sont en bon accord avec les données disponibles.[Traduit par la Rédaction]1.IntroductionRefractory compounds,particularly nitrides and carbides of transition metals,have attracted a great deal of attention from the scientific community because of their unique physical and chemical properties,such as high melting temperature,extreme hardness,electrical resistivity,magnetic susceptibility,superconducting,chemical inertness,etc.[1].These unique properties have made them most promising engineering mate-rials for a wide range of industrial applications like in elec-tronics,petroleum,aerospace,high power,high-temperature engineering,mechanical engineering,nuclear industry,and chemical industry [2–4].Many theoretical studies have been devoted to predicting their electronic structures [5–11]and revealed an unusual mixture of covalent,metallic,and ionic contributions to bonding,which must ultimately be the root cause of their unusual properties.Particularly the transition metal carbides (TMC),existing in their original rock-salt (B1)type phase,are widely used as coating metals in cutting tools,magnetic storage devices,generators,ma-glev trains,optoelectronic devices,and low temperature fuel cells [12].This has motivated researchers to investigate the bonding mechanism and cohesive properties of TMC series [13–15].Weber [16]has studied the lattice dynamics of ZrC,NbC,and HfC where the extreme physical properties are mainly at-tributed to strong covalent bonds between metal and nonme-tal atoms.Isaev et al.[17]have studied the phonon related lattice dynamics on body-centered cubic Vb-VIb group tran-sition metals,B1-type monocarbides,and mononitrides of IIIb-VIb transition metals,by means of first-principles density functional perturbation theory using ultra soft pseudopoten-tials,and generalized gradient approximation as the exchange-correlation functional.In another study,the electronic band structure,thermomechanical,and bonding properties of re-fractory metal carbides were studied by Zhukov et al.[18]by using the linear muffin-tin orbital (LMTO)method.Lu et al.[19]have calculated the thermophysical properties of cubic carbides and nitrides using the Debye –Gruneisen model combined with ab initio calculations.Recently,the elastic properties of selected transition metal nitrides and carbides in their B1type phase have been studied by Zhi-gang et al.[20]using ab initio density-function perturbation theory.Hollox [21]studied the factors affecting the micro-structure and mechanical behavior of carbides.Grossman et al.[5]and Li et al.[22]have reported the trend of bulk modulus and charge distribution to understand the hardness in transition metals and their carbides,using the full poten-tial linearized plane wave method.Srivastava et al.[23]and Singh et al.[24]have recently reported the structural prop-erties and the pressure-induced structural phase transitionin331C a n . J . P h y s .D o w n l o a d e d f r o m w w w .n r c r e s e a r c h p r e s s .c o m b y N o r t h w e s t e n P o l y t e c h n i c a l U n i v e r s i t y o n 04/01/17F o r p e r s o n a l u s e o n l y .TMCs by using ab initio methods and the rigid ion model,respectively.On the experimental side,efforts have been made to study the hardness and elastic properties of these materials using depth-sensing indentation [25]as well as by continuous indentation techniques [26].Motivated from this survey of the literature and the techno-logical importance of these materials,we thought it pertinent to explore the elastic and thermodynamic behavior of TMCs (TiC,ZrC,HfC,VC,NbC,and TaC).2.Theoretical methods2.1Three body force potential (TBFP)modelMost often,exposing a crystal to an applied pressure forces a particular phase of the solid to be unstable,neigh-boring atoms come closer to each other causing a decrease in the volume of the unit cell (change in the density),and in turn an increase in the coordination number of individual atoms.Therefore,the electron shells of the adjacent ions show overlapping,which causes lattice vibrations and is re-sponsible for the many body interactions (MBI),where three body interactions (TBI)are the most significant part.The TBI can be defined in terms of transfer of charge (charge transfer mechanism),interacting with other charge,and re-sponsible for MBI,where the governing part is TBI [27].These interactions become more significant under high com-pression.Besides enhancing the overlap energy,the trans-ferred charge due to the overlap in electron shells modifies the ionic charge,which in turn alters the coulomb energy.Thus,as the pressure increases,these NaCl (B1)type materi-als are expected to transform from their original NaCl (B1)type phase to CsCl (B2)type phase.The stability of the phase of the solid can be defined in terms of its Gibbs free energyG ¼U þPV ÀTSð1Þwhere S is the vibrational entropy at absolute temperature T ,V is the volume of the unit cell at pressure P ,and U is the internal energy,which at T =0K is equivalent to the cohe-sive energy due to the mutual interaction of the ions.Internal energies of the system can be expressed asU ðr Þ¼Àa m e 2z r ½z þ2nf ðr Þ ÀC r ÀDrþnb b þÀexpðr 1þr 2Àr Þrþn 0b 2b þþexpð2r 1Àkr Þr þn 0b 2b ÀÀexp ð2r 2Àkr Þr ð2Þwhere the first two terms are the potential energies due tolong-range coulomb and TBI [28];the third term is due to van der Waals (vdW)interaction governed by dipole –dipole (d –d)and dipole –quadrupole (d –q)interactions [29];the last three terms are the Hafemeister and Flygare type overlap re-pulsive interactions [30]effective up to the next nearest neighbour ions;k is the structure factor for the B1and B2type phases;C and D are the overall vdW constants,repre-sented in terms of vdW coefficients c ij and d ij weighted by the appropriate lattice sums (S ij and T ij )as reported by Tosi [29]for the B1and B2type phases,respectively.The values of these coefficients c ij and d ij have been evaluated by the re-lationship given in the Slater and Kirkwood variational ap-proach [28,31],and b ij (i =+;j =–)are the Pauling coefficients.b ij ¼1ÆZ i n i ÆZ jn j ð3Þwith Z as the ionic charge and n the number of electrons inthe outermost shell [32];Ze is the ionic charge with e as the electronic charge;b and r are the range and hardness para-meters;r is the nearest neighbor separation;r 1(r 2)the ionic radii of the ion i (j );and f (r )as the three body interaction parameter defined in terms of interionic separation,expressed by Cochran [33]asf ðr Þ¼f 0exp Àrrð4ÞCohesive energy as expressed in (2)contains three mate-rial-dependent model parameters b ,r ,and f (r ),calculated by making use of the equilibrium condition (i.e.,first derivative of lattice energy equal to zero)d Ud r r ¼r 0¼0ð5Þthe isothermal bulk modulus is defined asB 0¼C 11þ2C 123¼19kr d 2U ðr Þd r !r ¼r 0ð6Þand the Cauchy discrepancy C 12–C 44≠0from the knowl-edge of the lattice constant (2r 0).2.2Ab initio methodTo verify our results obtained through TBFP model calcu-lations,we have also carried out a first-principles calculation of the elastic and thermodynamical properties for TMCs in its original B1phase using CASTEP code [34].Within the framework of density functional theory [35],the norm con-serving nonrelativistic pseudopotential approach has been used and the effects of exchange-correlation interactions are handled by the generalized gradient approximation with Perdew –Burke –Ernzerhof [36]type parameterization.In these mate-rials,transition metal atoms occupy (0,0,0)while carbon atoms occupy the (0.5,0.5,0.5)position in the unit cell of their B1type phase.The electronic wave functions are ex-panded in a plane wave basis set with energy cut-off of 470eV for all the carbides.For k -point sampling,a grid of 8×8×8is used in the whole irreducible Brillouin zone as obtained by convergence test for these compounds.For the geometry optimization force tolerance of 0.01eV/Åand stress tolerance of 0.1GPa is used.2.3Computation of elastic properties2.3.1Computation of elastic properties using the TBFP modelTo understand the nature of interionic forces,we need to study the elastic constant and its pressure derivative at abso-lute temperature.The expressions for the second order elastic constants (SOECs),using the short-range potentials,are ex-pressed for both the NaCl (B1)structure as [37]332Can.J.Phys.Vol.90,2012C a n . J . P h y s .D o w n l o a d e d f r o m w w w .n r c r e s e a r c h p r e s s .c o m b y N o r t h w e s t e n P o l y t e c h n i c a l U n i v e r s i t y o n 04/01/17F o r p e r s o n a l u s e o n l y .C 11¼e 24a 4À5:112Z ½Z þ12f ðr Þ þA 1þA 2þB 22&þ9:3204zaf 0ðr Þ'ð7a ÞC 12¼e 24a 40:226Z ½Z þ12f ðr Þ ÀB 1þA 2À5B 24&þ9:3204zaf 0ðr Þ'ð7b ÞC 44¼e 24a 2:556Z ½Z þ12f ðr Þ ÀB 1þA 2þ3B 24&'ð7c Þand the CsCl (B2)structure as [37]C 11¼e 2À0:7009Z ½Z þ16f ðr Þ þA 1þ2B 1&þA 22þ3:1337zaf 0ðr Þ'ð8a ÞC 12¼e 24a À0:6897Z ½Z þ12f ðr Þ þA 1À4B 16&ÀB 22þ3:1337zaf 0ðr Þ'ð8b ÞC 44¼e 24a 4À0:3504Z ½Z þ12f ðr Þ þA 1þ2B 16þB 22&'ð8c Þwhere symbols A i and B i (i =1,2)are the short-range para-meter for the nearest and next-nearest neighbors,respectively,and are defined asA 1¼e24r d 2U SR ij d r "#r ¼r 0ð9a ÞA 2¼4r 4e 2 d 2U SR iid r 2þd 2U SRjj d r 2"#r ¼r 0ð9b ÞB 1¼4r 2=e 2ÀÁrd 2U SR ijd r 2"#r ¼r 0ð9c ÞB 2¼ð4r 2=e 2Þrd 2U SR ii d r 2þd 2U SR jj d r 2"#r ¼r 0ð9d Þin the terms of the short range (SR)potentials as given in U SRij ðr Þ¼b b þÀexpr 1þr 2Àr r ÀC r 6ÀD r8ð10ÞThese calculations are important to further check the cal-culation accuracy of the short range forces in materials.2.3.2Computation of elastic properties using the ab initio methodThe elastic constants C ij are defined by a Taylor expansion of the total energy E (V ,d )for the system with respect to a small strain d of the lattice primitive cell volume V .The en-ergy of a strained system is expressed as [38]E ðV;d Þ¼E ðV o ;d ÞþV oX i t i x i d i þ12X ij C ij x i d i x j d j "#ð11Þwhere E (V ,d )is the energy of the unstrained system withequilibrium volume V 0,t i is an element in the stress tensor,and x i and x j are the factors of Voigt index.For the cubic system,there are three SOECs C 11,C 12,C 44,and their combinations B S ,shear constant C S ,Bulk moduli B 0,and elastic stiffness constant C L that define the mechani-cal and dynamical behavior of these crystals.The combina-tion of these elastic constants provides valuable information about the nature of forces operating in solids and their inter-atomic behavior,bonding between adjacent atomic planes and the anisotropic character of the bonding and in turn the structural stability.To explain the strength and nature of the divalent TMC for different applications,the shear constant C S and elastic stiff-ness constant C L are expressed in terms of SOECs as C S ¼C 11ÀC 122andC L ¼C 11þC 12þ2C 442ð12Þthe bulk modulus B 0and the shear modulus G asB 0¼C 11þ2C 123andG ¼C 11ÀC 12þ3C 445ð13ÞYoung ’s modulus E ,Poisson ’s ratio n ,Zener anisotropyfactor A ,and Kleinman parameter x are thus written as E ¼9B 0G 3B 0þG andn ¼3B 0ÀE 6B 0ð14ÞA ¼2C 44C 11ÀC 12and x ¼C 11þ8C 127C 11þ2C 12ð15ÞUsing Poisson ’s ratio n ,Lamé’s coefficient l for TMCs is obtained as l ¼C 12ÀA 5¼C 12À2C 445ðC 11ÀC 12Þð16ÞTo calculate the pressure dependence of the bulk modulus we have used the following relation [39]:d B 0d P ¼1Àr 327VB 0d 3Ud rr ¼r o ð17ÞOr as given by Hildebrand [40]d B 0d P ¼1Àr 3d 3U d r =3r 2d 2U d r !r ¼r oð18ÞHere,P is the pressure,expressed as the negative deriva-tive of the cohesive energy;B 0is the equilibrium bulk modu-lus;and B 0′as its first pressure derivative.Srivastava and Diwan 333C a n . J . P h y s .D o w n l o a d e d f r o m w w w .n r c r e s e a r c h p r e s s .c o m b y N o r t h w e s t e n P o l y t e c h n i c a l U n i v e r s i t y o n 04/01/17F o r p e r s o n a l u s e o n l y .P ¼À@U @V B 0¼ÀV@P@VT¼r 29V d 2U d r 2ð19ÞThe compressibility is calculated using the following rela-tion:b ¼1B 0¼r 29V d 2U d r 2r ¼r oð20Þ2.4Computation of thermodynamic propertiesThe Debye temperature (q D )is a parameter intimately re-lated to the physical properties such as elastic constants,spe-cific heat,and melting temperature.At low temperatures the vibrational excitations arise solely because of the acoustic vi-brations,thus the Debye temperature calculated from elastic constants is the same as that determined from specific heat measurements.Another method to calculate the Debye tem-perature is to estimate it from the average elastic wave veloc-ity n m [2].q D ¼Z 3n N A r !1=3n m ð21Þwhere h is Planck ’s constant,k is Boltzman ’s constant,N A is Avogadro ’s number,n is the number of atoms per formula unit,M is the molecular mass per formula unit,r (=M /V )is the density,and the elastic wave velocity n m [41]is repre-sented asn m ¼2=n 3S ÀÁþ1=n 3LÀÁ3 !Àð1=3Þð22Þin terms of compressional wave velocity n L and shear wave velocity n S ,obtained from Navier ’s [42]equationsn L ¼B T þð4=3ÞG r !1=2ð23Þn S ¼G r1=2ð24Þwhere B 0is the adiabatic bulk modulus,and G is the shear modulus defined by G ¼G V þG R2ð25ÞHere G V is the Voigt shear modulus and G R is the Reuss shear modulus.G V ¼2C þ3C 445ð26ÞG R ¼15ð6=C Þþð9=C 44Þð27ÞC ¼C 11ÀC 122ð28Þ3.Results and discussion3.1Relative stability and structural propertiesThe vdW coefficients are calculated from the Slater and Kirkwood variational [28,31]approach and tabulated in Ta-ble 1along with the model parameters r ,b ,and f (r )calcu-lated using the equilibrium interionic separation r o ,SOEC C 44,and bulk modulus B 0and applying the equilibrium con-ditions given in (8)and (9).In comparison to a recent report by Singh et al.the model parameters calculated for TMCs show deviations due to the different assumptions taken into consideration.Singh et al.in their article have considered the same values of anion and cation radii (as interionic sepa-ration r )in both the phases as input,whereas,for the stable cubic type structure the ratio of anion radius to the cation ra-dius is between 0.41and 0.73for NaCl-type structures and greater than 0.73–1.00for CsCl type structures [29,54].Hence,the assumption that the cation and anion radii (as in-terionic separation r )have the same values in both phases is not at all justified.Because the lattice parameter a (as reverse interionic separation r )is inversely correlated with the bulk modulus B 0[55],we cannot take the same values of bulk modulus as input to compute the model parameters for both the phases.Therefore,the assumption that the bulk modulus B 0has the same value in both of the phases is not justified.In high pressure structural phase transition from B1phase to B2phase,the interionic potential reduces considerably and the coordination number increases.So the coordination num-ber plays an important role in the model calculation.In the present work we have used the ratio of nearest neighbors of both the structures to calculate the input parameter for B2type phase of TMCs.The stability of the TMCs in its B1as well as B2type phase have been analysed through energy as a function of volume relationship.3.1Elastic propertiesThe SOECs C 11,C 12,and C 44,bulk modulus B 0,pressure derivative of bulk modulus B 0′,shear constant C s ,elastic stiffness C L ,shear modulus G ,Poisson ’s ratio n ,Young ’s modulus E ,elastic anisotropy factor A ,Kleinman parameter x ,Lamé’s parameter l ,specific modulus of elasticity (E /r )ratio,Cauchy pressure (C 12–C 44),and compressibility b have been calculated using the expressions given in the pre-vious section from (7)–(20)for the TBFP model and their values are listed in Tables 2–4,along with our ab initio re-sults.For better justification of our findings the other avail-able experimental [1,16,43,44,46,47,49,53,61,63,65,66,68–70]and theoretical [17,19,20,23,24,50,51,56–60,62,64,67,71]results have also been given.The elastic constants C 11,C 12,and C 44of TMCs calculated using the TBFP approach are found to be in better agreement with experimental and other calculated data than the ab initio values,as reported in Table 2.This may be due to the differ-ent interactions used in these two approaches.All the C ij for TMC compounds are found to be positive in our calculations and satisfy the generalized Born criteria [72,73]proposed for mechanically stable cubic NaCl type crystals C S ¼C 11ÀC 122>0B T ¼C 11þ2C 123>0and C 44>0ð29Þ334Can.J.Phys.Vol.90,2012C a n . J . P h y s .D o w n l o a d e d f r o m w w w .n r c r e s e a r c h p r e s s .c o m b y N o r t h w e s t e n P o l y t e c h n i c a l U n i v e r s i t y o n 04/01/17F o r p e r s o n a l u s e o n l y .Such compounds are characterized by the positive values of the bulk modulus,shear modulus,and shear constant.The bulk modulus calculated from elastic constants for these ma-terials using the TBFP method is very close to other available data [1,16,17,19,20,24,43,44,47,49–51,57,58,60,63]as compared to our ab initio values.The values of shear modulus,Poisson ’s Ratio,Young ’s modulus,and compressi-bility for TMCs are also in good agreement with the data re-ported in the literature [16,19,24,44,46,47,49,58,63–66,68].To differentiate between brittleness and ductility with the use of Poisson ’s ratio,the concept of the critical value was proposed by Frantsevich [74]and is 0.33.For brittle materi-als,Poisson ’s ratio is less than 0.33;otherwise the material behaves in a ductile manner.The Poisson ’s ratios in the present work lie between 0.17and 0.24,by both the ab initioand TBFP approaches except for VC as given in Table 3,and are very close to the critical value,which shows the brittle-ness of these materials.The Cauchy pressure (C 12–C 44)can be used to describe the angular character of atomic bonding in metals and com-pounds as suggested by Pettifor [75].If the value of the Cauchy pressure is positive,the bonding of the material is metallic in character.In contrast,negative Cauchy pressure requires a directional character and low mobility in the bond-ing with a covalent-like contribution to atomic bonding.In our calculations we find that the Cauchy pressure is positive for these materials thus we conclude that these materials quite distinctly belong to the class of metallic bonding materials.Young ’s modulus or modulus of elasticity E is defined as the ratio of stress to strain,and used to provide a measure ofTable 1.vdW coefficients C and D ,interion spacing r o ,and SOECs C 44,bulk modulus B 0,hardness parameter r ,range parameter b ,and TBI parameter f (r )for TMCs.TMC C (10m J cm 6)D (10m J cm 6)r o (Å)C 44(100GPa)B 0(100GPa)r (Å)b (10–19J)f (r )(10–2)TiC 1819.233522.41 2.165*[43] 1.78*[16] 2.42*[44],2.4*[43]0.621,0.200[24]9.764,0.534[24]2.111ZrC1943.814261.09 2.349*[45] 1.573*[46],1.593*[46],1.60*[16] 2.30*[44],2.23*[47]0.598,0.202[24]9.964,2.978[24]2.206HfC 2095.565262.06 2.320*[48] 1.80*[16] 2.42*[49]0.586,0.211[24]9.82,0.731[24] 2.231VC 1957.213482.71 2.086 1.5*[16]3.51[50],3.48[51]0.5329.75–3.165NbC 3090.066027.36 2.235*[45] 1.60*[16]3.00*[49],3.02*[1]0.509,0.202[24]9.98,2.978[24]–1.065TaC4311.978260.592.235*[52]1.90*[16],1.53*[53]3.32*[63]0.56,0.2129.937,1.060 1.168Table 2.The calculated elastic moduli C 11,C 12,C 44,bulk modulus B 0,and their pressure derivative bulk modulus B 0′for TMCs.TMC C 11C 12C 44B 0B 0′TiC5.10,5.83,5.13[16],5.15[56],5.106[24],5.13*[16]1.05,1.22,1.06[16],1.06[56],0.874[24],1.06*[16]1.77,1.80,1.78[16],1.79[56],0.874[24],1.78*[16]2.40,2.76,2.55[16],2.42[57],2.401[58],2.42*[44],2.28[24],2.4*[2,43]2.40,3.24[23],3.44[23]ZrC4.68,4.83ab ,5.22[16],4.80[59],4.682*[46],4.72*[46],5.48[24],5.22[20],4.70*[16] 1.00,1.09ab ,1.10[16],1.02[59],0.997*[46],0.987*[46],0.87[24],1.10[20],1.00*[16] 1.59,1.42ab ,1.60[16],1.69[59],1.573*[46].1.593*[46],0.87[24],1.60[20],1.60*[16] 2.22,2.34ab ,1.87[16],2.22[17],2.17[19],2.30*[44],2.65[24],2.23*[47]2.72,3.44[23]HfC4.98,5.48ab ,5.74[16],5.74[20],4.54[24],5.00*[16]1.13,0.99ab ,1.07[16],1.07[20],0.865[24],1.14*[49]1.79,1.92ab ,1.80[16],1.80[20],0.865[24],1.80*[16]2.41,2.48ab ,2.62[16],2.43[16],2.63[20],2.36[19],2.18[24],2.42*[49]2.49,2.91[23],2.98[23]VC 4.98,5.40ab ,5.13[16],5.0*[16]2.89,1.45ab ,3.66[16],2.9*[16]1.49,1.53ab ,1.10[16],1.5*[16]3.58,2.77ab ,1.95[16],3.51[50],3.48[51],3.07[19]3.66,3.19[23],3.29[23]NbC6.17,4.60ab ,6.67[16],6.67[20],6.479[24],6.20*[16]1.99,1.59ab ,1.63[16],1.63[20],1.089[24],2.00*[16]1.59,1.03ab ,1.61[16],1.61[21],1.089[24],1.60*[16]3.38,2.59ab ,2.12[16],3.31[20],3.28[60],3.00*[49],2.88[24],3.02*[1]2.51,2.95[23]TaC5.47,3.70ab ,7.40[16],7.40[20],6.94[24],5.5*[61],5.95*[53]1.19,1.10ab ,1.65[16],1.2[62],1.27[24],1.65[20],1.50*[16]1.59,1.53ab ,1.76[16],1.6[62],1.76[20],1.90*[16],1.27[24],1.53*[53]2.62,1.97ab ,3.45[16],3.57[20],3.18[24],3.17[19],3.32*[63]2.59,3.04[23],3.14[23]Srivastava and Diwan 335C a n . J . P h y s .D o w n l o a d e d f r o m w w w .n r c r e s e a r c h p r e s s .c o m b y N o r t h w e s t e n P o l y t e c h n i c a l U n i v e r s i t y o n 04/01/17F o r p e r s o n a l u s e o n l y .the stiffness of the solid.The high values of the Young ’s modulus indicate that the TMCs are a stiffer material in cubic structure.The structural efficiency parameter (specific modu-lus of elasticity)E /r is important for comparing ultra light-weight metallic materials with excellent mechanical and func-tional properties for engineering and design purposes.This ratio quantifies the stiffness of a material per unit weight and is used to identify those materials and alloys having opti-mal mechanical properties for lightweight structural applica-tions.In our case,the calculated values of E /r vary from 2.23×10–2to 10.95×10–2GPa m 3/kg and follow the trend TiC >NbC >ZrC >VC >HfC >TaC,meaning TiC has highest and TaC lowest stiffness per unit weight.Zener anisotropy factor A is an indicator of the degree of anisotropy in the solid structure.For a completely isotropic material,A takes the value of 1,when A is smaller or greater than unity,then it is a measure of the degree of elastic aniso-tropy.Our calculated values of A by both the TBFP as well as ab initio methods are within the range 0.63–1.43,showing that the TMCs are elastically anisotropic in nature.Kleinman parameter x describes the relative positions of the cation and anion sublattices under volume conserving strain distortions for which positions are not fixed by symme-try.A low value of internal strain parameter implies that there is a large resistance against bond bending or bond-angle distortions while the reverse is true for a high value.The Laméconstant l has no physical interpretation,but it serves to simplify the stiffness matrix in Hooke ’s law and constitutes a parameterization of the elastic moduli for ho-mogeneous isotropic media,and is thus related to the other elastic moduli.The calculated values of x and l are given in Table 4.The bulk and shear moduli are used to compute the longi-tudinal (n l ),shear (n s ),and average (n m ),wave velocities andTable 3.Shear constant C s ,elastic stiffness C L ,shear modulus G ,Poisson ’s ratio n ,modulus of elasticity E for TMCs.TMC C sC LG (100GPa)nE (100GPa)TiC 2.02,2.30 4.85,5.33 1.87,1.99,1.88[64],1.886[58],1.37[24],1.87*[44],1.82[67],1.84±11*[63]0.17,0.17,0.2*[65],0.17–0.19*[66],0.146[24],0.187±0.01*[63] 4.74,5.40,4.57[58],4.5*[65],4.85[24],4.39*[66]ZrC 1.84,1.87ab 4.43,4.39ab 1.69,1.59ab ,1.51[16],1.697[58],1.436[24],1.65*[44],1.69.7[67],1.70*[47]0.18,0.18ab ,0.1*[65],0.19*[47],0.134[24],0.191–0.257*[66] 4.33,4.43ab ,5.5*[65],3.95*[44],5.25[24],4.07*[47]HfC 1.92,2.25ab 4.85,5.15ab 1.84,2.05ab ,2.01[16],1.85*[44],1.25[24],1.95*[49]0.190,0.15ab ,0.16[24],0.16*[44],0.166–0.18*[66]4.55,5.18ab ,4.264[24],4.61*[49],4.30*[44]VC 1.04,1.97ab 5.42,4.96ab 1.31,1.70ab ,1.92[16],1.71[19].2.09.1[67]0.37,0.21ab ,0.4*[65],0.19–0.32*[66] 2.86,4.78ab ,2.55–4.46*[66]NbC 2.09,1.51ab 5.67,4.13ab 1.79,1.20ab ,1.97[16],1.67[19],1.731[24],1.98*[49],171.7[67],1.97*[68]0.24,0.26ab ,0.1*[65],0.255[19],0.160[24],0.22*[49]5.20,3.78ab ,5.102[46],5.4*[65],6.165[24],5.14*[66]TaC2.14,1.30ab4.93,3.93ab1.81,1.44ab ,2.17[16],2.09[19],1.896[24],2.16*[49],2.02*[68]0.18,0.23ab ,0.154[24],0.172–0.24*[66]5.04,3.20ab ,5.67±68*[63],6.547[24],4.90*[68]Table pressibility b ,elastic anisotropy factor A ,compressibility (Mbar –1),Kleinman parameter x ,Laméconstant l ,Cauchy pres-sure (C 12–C 44),and specific modulus of elasticity E /r for TMCs.TMC b (100GPa)A x l (100GPa)C 12-C 44E /r (10–2GPa m 3/kg)TiC 0.42,0.360.87,0.78,0.419[24]0.36,0.360.88,1.43 3.33,4.029.67,10.95ZrC 0.45,0.43ab 0.86,0.76ab ,0.36[24]0.36,0.38ab 0.82,1.28ab 3.08,3.41ab 6.46,7.79ab HfC 0.41,0.40ab 0.93,0.85ab ,0.32[24]0.38,0.33ab 0.95,1.12ab 3.18,3.56ab 3.75,4.24ab VC 0.28,0.36ab 1.43,0.78ab 0.69,0.42ab 2.60,1.64ab 3.48,3.87ab 4.98,8.29ab NbC 0.30,0.38ab 0.76,0.69ab ,0.32[24]0.47,0.49ab 1.84,1.79ab 4.58,3.56ab 6.87,4.83ab TaC0.38,0.50ab0.74,1.18ab ,0.44[24]0.37,0.44ab1.05,1.01ab 3.88,2.17ab3.49,2.23abTable pressional wave velocity V l ,shear wave velocity V S ,elastic wave velocity V m ,Debye temperature q D for TMCs.TMC V l (103ms )V S (103ms )V m (103ms )q D (K)TiC 9.99,10.47 6.17,6.35 6.81,7.01936.24,965.48,943[58],947[70],940*[69]ZrC 8.17,8.85ab 5.02,5.28ab 5.54,5.84ab 702.22,703.52ab ,708[58],703[19],699*[70],700[69]HfC 6.33,6.53ab 3.90,4.09ab 4.30,4.50ab 551.17,570.48ab ,553*[69],545*[70]VC 9.61,9.33ab 4.75,5.42ab 5.33,6.00ab 761.79,857ab ,822[71],855[56],913[19],971[19]NbC 8.72,7.32ab 4.85,3.92ab 5.40,4.37ab 718.93,582.79ab ,740*[49],754*[70],742.76*[69]TaC6.24,5.20ab3.74,3.17ab4.13,3.49ab550.66,464.97ab ,573*[69],593±71*[63]336Can.J.Phys.Vol.90,2012C a n . J . P h y s .D o w n l o a d e d f r o m w w w .n r c r e s e a r c h p r e s s .c o m b y N o r t h w e s t e n P o l y t e c h n i c a l U n i v e r s i t y o n 04/01/17F o r p e r s o n a l u s e o n l y .。